/
Текст
INVERSE A N D ILL-POSED PROBLEMS SERIES
Ill-Posed and Non-Classical Problems of
Mathematical Physics and Analysis
Also available in the Inverse and Ill-Posed Problems Series:
Forward and Inverse Problems for Hyperbolic,
Elliptic and Mixed Type Equations
AG. Megrabov
Nonclassical LinearVolterra Equations of the
First Kind
AS. Apartsyn
Poorly Visible Media in X-ray Tomography
D.S. Anikonov, V.G. Nazarov, and I.V. Prokhorov
Dynamical Inverse Problems of Distributed
Systems
V.l. Maksimov
Theory of Linear Ill-Posed Problems and its
Applications
V.K. Ivanov.V.V.Vasin andV.P.Tanana
Ill-Posed Internal Boundary Value Problems for
the Biharmonic Equation
MAAtakhodzhaev
Investigation Methods for Inverse Problems
V.G. Romanov
Operator Theory. Nonclassical Problems
S.G. Pyatkov
Inverse Problems for Partial Differential
Equations
Yu.Ya. Belov
Method of Spectral Mappings in the Inverse
Problem Theory
V.Yurko
Theory of Linear Optimization
I.I. Eremin
Integral Geometry and Inverse Problems for
Kinetic Equations
AKh.Amirov
Computer Modelling in Tomography and
Ill-Posed Problems
M.M. Lavrent'ev, S.M.Zerkal and O.ETrofimov
An Introduction to Identification Problems
via Functional Analysis
A. Lorenzi
Coefficient Inverse Problems for Parabolic Type
Equations and Their Application
P.G. Danilaev
Inverse Problems for Kinetic and Other
Evolution Equations
Yu.E Anikonov
Inverse Problems ofWave Processes
AS. Blagoveshchenskii
Uniqueness Problems for Degenerating
Equations and Nonclassical Problems
S.P. Shishatskii, A. Asanov and ER. Atamanov
Uniqueness Questions in Reconstruction of
Multidimensional Tomography-Type
Projection Data
V.P. Golubyatnikov
Monte Carlo Method for Solving Inverse
Problems of Radiation Transfer
V.S.Antyufeev
Introduction to theTheory of Inverse Problems
A L Bukhgeim
Identification Problems ofWave Phenomena Theory and Numerics
S.I. Kabanikhin and A. Lorenzi
Inverse Problems of Electromagnetic
Geophysical Fields
P.S. Martyshko
Composite Type Equations and Inverse Problems
A.I. Kozhanov
Inverse Problems ofVibrational Spectroscopy
A.G.Yagola, I.V. Kochikov, G.M. Kuramshina andYuA
Pentin
Elements of theTheory of Inverse Problems
A.M. Denisov
Volterra Equations and Inverse Problems
A.L Bughgeim
Small Parameter Method in Multidimensional
Inverse Problems
A.S. Barashkov
Regularization, Uniqueness and Existence of
Volterra Equations of the First Kind
A. Asanov
Methods for Solution of Nonlinear Operator
Equations
V.P.Tanana
Inverse and Ill-Posed Sources Problems
Yu.E. Anikonov, B.A. Bubnov and G.N. Erokhin
Methods for Solving Operator Equations
V.P.Tanana
Nonclassical and Inverse Problems for
Pseudoparabolic Equations
A.Asanov and EUAtamanov
Formulas in Inverse and Ill-Posed Problems
Yu.E.Anikonov
Inverse Logarithmic Potential Problem
V.G. Cherednichenko
Multidimensional Inverse and Ill-Posed
Problems for Differential Equations
Yu.E Anikonov
Ill-Posed Problems with A Priori Information
V.V.Vasin andA.LAgeev
Integral Geometry ofTensor Fields
VA. Sharafutdinov
Inverse Problems for Maxwell's Equations
V.G. Romanov and S.I. Kabanikhin
INVERSE AND ILL-POSED PROBLEMS SERIES
Ill-Posed and Non-Classical
Problems of Mathematical
Physics and Analysis
PROCEEDINGS OF THE INTERNATIONAL CONFERENCE
SAMARKAND, UZBEKISTAN
Editor-in-Chief:
M.M. Lavrent'ev
Managing Editor:
S.Í. Kabanikhin
Editorial Board:
Akb.H. Begmatov, T.D. Djuraev,
S. Saitoh and N[. Yamamoto
///νsp///
UTRECHT · BOSTON
2003
VSP
Tel: + 3 1 3 0 6 9 2 5 7 9 0
P.O. B o x 3 4 6
Fax: +31 30 693 2081
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©VSP
www.vsppub.com
2003
First p u b l i s h e d in 2 0 0 3
ISBN 90-6764-380-7
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system,
or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or
otherwise, without the prior permission of the copyright owner.
Printed
in The Netherlands
by Ridderprint
bv,
Ridderkerk.
INVITED PAPERS
Problems of integral geometry on curves and surfaces in Euclidean
space
A. H. Begmatov and A. H. Begmatov
1
Existence of solutions of the first boundary-value problem for the
third order equations of mixed type in unbounded domain
T. D. Djuraev and A. R. Hashimov
19
On the monotone error rule for choosing
the regularization parameter in ill-posed problems
U. Hämarik and U. Tautenhahn
27
Well-posedness of one-dimensional inverse acoustic problem in L2
for small depth or small data
S. I. Kabanikhin, Κ. T. Iskakov, and M. Yamamoto
Fourier series in Banach spaces
Dj. Khadjiev and A. Çavu§
Ill-posed and inverse problems for hyperbolic equations
M. M. Lavrent'ev
Systems of linear integral equations of Volterra type with singular
and super-singular kernels
N. Rajabov
Inverses of a family of bounded linear operators,
generalized pythagorean theorems and reproducing kernels
S. Saitoh
Cauchy problem for the Helmholtz equation
Sh. Yarmukhamedov and I. Yarmukhamedov
57
71
81
103
125
143
CONTRIBUTED PAPERS
On problem for a third order equation with multiple characteristics
S. Abdinazarov and B. M. Kholboev
On uniqueness and stability of solution of the Cauchy problem
for pseudoparabolic equation
B. K. Amonov and S. S. Kobilov
Uniqueness theorem for an unbounded domain
Z. R. Ashurova and Yu. J. Zhuraev
Recovery of a function set by integrals along a curve family
in the plane
A. H. Begmatov and Z. H. Ochilov
Uniqueness of extension of solutions of differential equations
of the second order
A. Haidarov and D. Shodiev
173
179
185
191
199
This book is the Proceedings of the International Conference "Ill-Posed
and Non-Classical Problems of Mathematical Physics and Analysis" which
was held at the Samarkand State University, Samarkand, Uzbekistan from
11 to 15 September 2000.
The Conference was organized jointly by the Samarkand State University and Sobolev Institute of Mathematics, Novosibirsk, Russia. Advice and
general guidance were provided by the International Programme Comittee. More than 90 participants from Germany, Japan, Kazakhstan, Russia,
Tajikistan, Turkey, and Uzbekistan presented their lectures at the Conference.
The scientific program of the Conference covered the following topics:
• Theory of 111-Posed Problems
• Inverse Problems for Differential Equations
• Boundary Value Problems for Equations of Mixed Type
• Integral Geometry
• Mathematical Modelling and Numerical Methods in Natural Sciences
The Proceedings bring together fundamental research articles in the major areas of the numerated fields of analysis and mathematical physics. The
papers in this volume represent all plenary and some contributed lectures
presented at the conference. All the papers have undergone peer review.
We would like to thank the Conference paricipants for their interesting
reports and stimulating discussions. We express our sincere thanks to the
authors for submitting articles of such high quality and to the referees for
their thoughtful reviews. Finally, we would like to thank the staff at VSP
for their help in publishing the Proceedings.
M. M. Lavrent 'ev
S. I.
Kabanikhin
Akb. H.
T. D.
S.
M.
Begmatov
Djuraev
Saitoh
Yamamoto
El-Posed and Non-Classical Problems of
Mathematical Physics and Analysis, Samarkand, 2000, pp. 1-18
M.M. Lavrent'ev and S.I. Kabanikhin (Eds)
© VSP 2002
Problems of integral geometry on curves
and surfaces in Euclidean space
Akbar H. BEGMATOV* and Akram H. BEGMATOV*
1.
INTRODUCTION
Integral geometry studies the transformations assigning to functions on a
manifold X their integrals over submanifolds from certain set M [23]. This
important and intensively developing domain of modern mathematics is
closely connected with the theory of partial differential equations, mathematical physics, geometric analysis. This direction has many applications
in mathematical study of problems of seismic exploration, interpretation of
the data of geophysical and aerospace observations, in solving inverse problems of astrophysics and hydroacoustics (see [33] and the references given
there). Methods developed here are basic for solving the problems of medical
and industrial tomography [17, 18, 34].
The problem of recovery of a function from its integrals over all possible
hyperplanes in Euclidean space was considered by J. Radon [37]. In this
classical work the explicit inversion formulas for even and odd-dimensional
spaces were obtained. Also, the methods of solution of such problems were
developed and the various analogs of this integral transformation were considered [37]. Note that P. Funk [21], the year before, had investigated the
problem of recovery of even function in the sphere from its the integralsover
big circles.
'Novosibirsk, Sobolev Institute of Mathematics. E-mail: bai@math.nsc.ru
* Samarkand State University, Uzbekistan. E-mail: akrbegmat@mail.ru
The work was partially supported by State Commitee of Science and Technology of Uzbek
Republic (grants Nos. 15/99 and 2/01).
2
Α. Η. Begmatov
and Α. Η.
Begmatov
The connection of problems of integral geometry with differential equations was investigated by J. John [26, 27] (see, also, [28]). The transformation (called after by the ray transformation) which assigns to a function
in three three-dimensional space its integrals over all possible straight lines
was considered in [27].
Basing on the theory of Lie groups, I. M. Gel'fand with co-authors had
considered the problems of integral geometry on linear manifolds. This
had become the basic for investigation of integral geometry for wide classes
of homogeneous spaces. Interesting inversion formulas were obtained here.
The problems of reconstructing a function from its integrals over surfaces
of second order were also widely considered (see [33] and the bibliography
cited here).
The first important results for a multidimensional inverse kinematic seismic problem were obtained by M. M. Lavrent'ev and V. G. Romanov [32]
by the method of its reducing to the problem of determining the function
from its mean values along all possible circles with centers in a fixed line.
This work brought the attention to the cases when integration goes along
the manifolds of more complicated geometric structure. Namely, in [32]
the connection between multidimensional inverse problems for partial differential equations and the problems of integral geometry was revealed. The
manifolds which arise after reducing the inverse problems to the problems of
integral geometry are naturally connected with the initial differential equation. For the equation with variable coefficients these geometric objects may
be sufficiently complicated.
V. G. Romanov had investigated the problems of uniqueness and stability
in the case when the manifolds are invariant under the group of all motions
parallel to a (n — l)-dimensional hyperplane [38, 39]. The weight functions
are also assumed to be invariant under this group.
The important results on uniqueness and stability of solution were obtained by Yu. E. Anikonov and A. L. Bukhgeim for the following classes of
integral geometry problems: the solution is sought in the classes of functions analytic with respect to some variables; the manifolds along which
the integrations goes depend analytically on some variables (parameters)
(see [1, 16]).
Questions of uniqueness, obtaining stability estimates and inversion formulas for various classes of problems of integral geometry in Euclidean space
were investigated in [2-14],
In the second section of our paper uniqueness questions for a wide class
of problems of integral geometry in the plane are considered. Unlike the
works mentioned above we do not impose the conditions of invariance or
analyticity. The uniqueness theorem is established for smooth convex curves.
Problems of integral
geometry
3
The third section is devoted to weakly ill-posed problems of integral
geometry on curves and surfaces with singularities in the vertices. The
existence and uniqueness theorems and the explicit inversion formulas are
obtained as well as the stability estimates in Sobolev spaces. For the problem
of integral geometry on the family of cone surfaces in the even-dimensional
space very simple representation of solution was obtained. On the basis
of these results the stability estimates in Sobolev spaces were obtained;
therefore, weak ill-posedness of the problem was established. The existence
theorem was obtained also.
Note that in our case (the cone surfaces) the odd case and the even
case differ essentially. Such situation occurs frequently in the problems of
integral geometry.
Finally, in the fourth section we consider the problem of recovery of the
function given the integrals of this function on a family of lines in threedimensional space which are generatrices of cones. Such integral transformations are called ray transformations [25] and they have wide applications
in computer tomography [34].
The inversion formulas of the ray transformation connected with the
cone scheme of scanning the computer tomography may be found in the
works of H.K. Tuy [40], D.V. Finch [20], P. Grangeat [24] (see, also, the
surveys of F. Natterer [35], V. P. Palamodov [36] and the references cited
there). A.A. Kirillov [29], A.S. Blagoveshchenskii [15], I.M. Gel'fand and
A.B. Goncharov [22] axe investigated various settings of the problems of
recovery of a function from its integrals taken along the lines intersecting a
certain set in the space. The explicit formulas of effective determining the
desired function were obtained.
The problem that we consider is connected with auxiliary problems of
analytic extension. Unlike the problems considered in [11-13] this problem is
strongly ill-posed. The uniqueness theorem for solution of the problem in the
class of continuous finite functions is established. The estimate of conditional
stability of solution of the logarithmic type problem was obtained.
2.
THE PROBLEM OF INTEGRAL GEOMETRY
OF THE VOLTERRA T Y P E OF GENERAL FORM
IN THE P L A N E
Let £ = (&>&),!/ = (2/1,2/2), χ = (χι,χί), R+ = {x = (xi,x2) • X2 > 0}. We
consider the family of the curves {Γ(ζ)} in the strip L = {x e R+ : 0 < X2 <
h, h < 00}. The curve from this set is defined by the equation £2 = Ψ(Χ, £i)·
We assume also that for any χ € L and any angle a G [0, αχ] U [02, π], where
A. H. Begmatov and A. H. Begmatov
4
O < αχ < π/2 < c*2 < π there exists a unique curve from {Γ(χ)} passing
through the point χ at the angle a with positive direction of the axis OX\.
We consider the problem of recovery of a function of two variables u(x) if
we know the integrals of this function with a given weight function p(:r, ξι) on
the family {Γ(χ)}. This is the problem of solution of the operator equation
in the function u(x):
ί
ρ(χ,ξΜΟάξι
J Γ(χ)
= ί(χ).
(1)
We can represent equation (1) as follows:
¡•xi+hi
/
ρ(χ,ξιΗξ)άξ1=!(χ),
where h\ — h\(x), h2 =
(2)
h2(x).
T h e o r e m 1. Let the right-hand side f(x) of the equation (2) be known
for all χ from the strip L; hk G C 2 , k — 1,2, ρ e C3, φ G C 7 and the
following conditions hold
ρ(χ,ξι)
φ(χ, Çi-hi)
> ¿i > 0;
= φ(χ, ξι + h2),
φ(χ,χι)
φ{χ, xi-hi)
-οο < 63 < S
dtì
< ¿2 <
= x2;
= φ(χ, χι + h2) = 0;
-2,
where δι, δ2,63(63 < 62) are constants.
Then the solution of equation (2) is unique in the class
Cq(L).
Proof. Pass in (2) to integration with respect to £2
rx 2
/ Φ,6)[«(®i
Jo
- <P(X>6)) + φ ι + <p(x,6))]
=
fix)·
Fix a certain point xq € L. Taking into account the conditions imposed
on the curves of the family {Γ(χ)}, we can select the subset Γ C {Γ(χ)} of
the curves being tangent from inside to the parabola P(x°) = {x G L : x2 =
χ2 — (xi — ^î) 2 }. Let t = (ti,t2) be the point of touching the curves Γ(χ)
and P(x°). Then
tx=ßi{x0,x),
t2=ß2{x°,x)
(3)
ί2 =
(4)
where βι, ß2 are smooth functions and
t2 = x°2-{x°1-t1)2,
φ(χ°,ίι).
Problems
of integral
geometry
Formulas (3), (4) and the inverse function theorem yield
xi = ipi{x°,ti)
X2 =
(P2{x°,h),
where φι,ψ2 are smooth functions.
As a result we obtain £2 = Φ{χι,χ2,ζι)
= Ψ{ψi(£0,íi)j<¿>2(z°,íi),£i)
=
0
<¿>(z ,fi,£i), where φ is a smooth function. Thus, we pass from the
parametrization of the curve of the family Γ(χ) using the coordinates of
its vertex £1,2:2 to the parametrization by means of the three parameters
Analogously, 0(2,£1) = g (φ ι,<Ρ2,ξι) = 0 i ( ® V i » f i ) , where gì is a
smooth function.
We have
rx 2
/ 9(x,b)[u(x
JO
φ2(χ°Μ)
ι-φ(χίξ2),ξ2)+φΐ+φ(χ,ξ2),ξ2)]άξ2
folfofrM^l
-Φΐι & )
+^2,6)1^2,
where φι, φ2, φι and Ψ2 are smooth functions.
Thus, the initial problem is reduced to the following problem of integral
geometry
l><p2(x°,ti)
/
J0
- Φι,ξί) +92{χΛ2)η{φι
+ V
s
,
=
f(x°,ti).
(5)
on the family of curves { Γ ( ι ° , ί ι ) } . It is easy to verify that the right-hand
side of equation (5) satisfies certain differential equation of the first order
with partial derivatives; therefore, the problem of its solution is not overdetermined.
We apply now to both sides of (5) the integral operator
Ih(x°,t1)=
rf+yfi°
K{x°,ti)h{x°,ti)dti.
Jxi-V4
As a result we obtain
Ifi(x°,ti)=
/ 1
JA-yR
-,
2
w(x°,ti)
K(x°,ti)fi(x°,ti)dti
pP 2(*°>ίι)
/
[9ι(χ,ξ2)η(φι-Λι,&)
+ 92(x,ξ2)η(φι
+ / i i , 6 ) ] <%2dti
6
A. H. Begmatov and A. H. Begmatov
+ w(x°,ti)g2(x,h)u(<Pi
= ff
J Jp(x°)
+ äi,£ 2 )]
&dti
d
Ψ(χ°,ξ)η(ξι,ξ2)άξ1άξ2.
Here P(x°) is a part of the strip L bounded by OXi-axis and the
parabola V(x°) with the vertex in the point x°:
ν(χ0) = {(ξΐ,ξ2)·.χ°2-ξ2
=
(χ'ί-ω2}·
We choose the kernel Κ of the integral operator I so that the weight function
W(a: 0 ,£) = Wq(x0,()/^x°2
- &- £i) 2 , where W 0 (z 0 ,£) is smooth
function.
Then, applying the integro-differential operators, we can reduce the
problem of solution of the latter equation (therefore, the initial problem
of integral geometry) to the problem of integral geometry in the strip in the
set of parabolas with the perturbation:
f
η(ξι,ξ2)άξι+
JV(x0)
ff
G(x°,()u((1,t2)d(ld(2
J Jp(X°)
= F(x°).
(6)
The uniqueness of the solution of such problem of integral geometry with
perturbation was investigated by M. M. Lavrent'ev [31]. The results of this
paper imply that if the conditions of Theorem 1 hold, then the solution of
equation (6) is unique in the class Cjj (L). Hence Theorem 1 follows.
•
3.
W E A K L Y ILL-POSED P R O B L E M S :
UNIQUENESS, T H E STABILITY ESTIMATES
AND T H E INVERSION FORMULAS
Denote by Ω the strip
Ω = {(χ, y) : χ Ε R 1 , ye (0,1), I < οο}.
We consider the operator equations is a function u(x,y) G CQ (Ω):
ry
/ [ιι(χ + ϊι,η)+u(x-Ιι,η)]άη
Jo
= fo(x,y)
(7)
Problems of integral geometry
7
and
ry
I
\u{x
+
77) +
u(x — h, η)] άη
Jo
Π
χ+h
_—h Κ(χ,ν,ξ,η)ν{ξ,η)<1ξ<1η = f f a y ) , (8)
where h — (y —η)2, (x, y) 6 Ω. The function Κ(· ) G Cp (Ω χ Ω) and vanishes
on the curves V(x,y) = {{ξ,η) • {y — η)2 = \χ — 0 < η < y, 0 < y < 1}
together with its derivatives up to the fifth order inclusively.
Suppose λ e R 1 , ρ G C (p = ρι + Ψ2)· Introduce the functions
roo
cos {λζ2) άζ,
J(\,p)=
1
Jo
/,α+»00
do
jxxMKy)
and denote
d¡fo(x,y) =M*,V)
=
d3f
dyzy),
d2xr0(x,y)
= (e-
where E is the unit operator.
Theorem 2. The solution of equation (7) in the class ο(Ω) is unique
and it has the following representation
Π
00
χ
-00Μ
- i» y - ^(¿xdy/oXi, η) d£ άη,
and satisfies the inequality
ΙΜΙΖ,2(Ω) < Co II/ο II ivi (Ω)·
Here Co is some constant; W2 (Ω) is the Sobolev space.
In the proof of uniqueness of the solution of equation (7) we shall use
the integral
fOO
J{X,p)=
e-^cos(ÀC2)dC·
Jo
Α. Η. Begmatov and Α. Η. Begmatov
8
Making the change -\/T^ÍC = τ, ρ =
where
ΛΟΟ
we obtain J(X,p)
= <Ji{p)/
Ji(p)=
e~pT cos (τ 2 ) dr,
peC(p
= x + iy).
Jo
It occurs that the integral J\ (p) satisfies the estimates obtained below
in Lemmas 1-3 (the proof of the lemmas can be found in [13]).
Lemma 1. For Ji(x)
Ji(x)
><
the following estimate
holds
'l/(4v^F),
[0,y/2fr),
if χ e
1/(4πχ),
if χ 6 [\/2/π, oo).
Lemma 2. For each χ > 0, Ji(x)
satisfies the
inequalities
Moreover, the integral J\{p) is separated from zero in the right halfplane.
Lemma 3. Let ρ G C+ = {ρ = χ + iy : χ > 0, y G R } .
integral
Ji(p)=
satisfies the
inequality
ç oo
Jo
Then the
e~pT cos τ 2 dr
\JÁP)\ > o.
Proof of Theorem 2. At first we suppose that I = oo. Applying
the Fourier tranformation with respect to χ and the Laplace transformation
with respect to y to both sides of (7), we obtain
v(\,p)j(\,p)
= i>(\ip).
(9)
Here v(X,p) and φ{\,ρ) are the Fourier—Laplace images of the desired
function and the right-hand side of equation (7) respectively, and
J(X,p)=
poo
Jo
e_pi» cos (Χζ2) άζ.
We try to obtain now the estimates from below of the function J(X,p).
This will allow us to obtain the estimates for v(X,p) and, finally, the estimates for the desired function u(x, y). Taking into account Lemmas 1-3, we
obtain
I/\J\<MV\M
+ \P\)·
(io)
Problems of integral geometry
9
Apply to equation (9) the inverse Laplace transformation with respect
to ρ and the Fourier transformation with respect to λ. Taking into account
the inversion theorem and the convolution theorem and using the properties
of these transformations, we obtain
Π
00
-00
Μ* ~ξ,ν-
η)(ά*8ξ/ο)(ξ,η) άξάη,
(11)
where Τι{χ — ξ, y — η) and {d2dyfo)(^, η) axe defined above.
The inversion formula (11) has the local character with respect to y.
Taking into account the condition suppu C Ω, we have that representation (11) holds also for an arbitrary I < oo. Then it follows from (9)—(11)
that the solution of the initial problem of integral geometry (1) is unique in
the class Cq (Ω), and the following estimate holds:
oo na+ioo
/
/
\v(X,p)\2dpdX
•oo J a—ioo
oo ra+ioo
/ -oo Ja—too
/
\Ml + \X\ +
\p\)\2mX,p)\2dpdX.
This inequality and (11) yield the stability estimate for solution of equation (7).
•
Theorem 3. The solution of equation (8) is unique in the class Cq (Ω)
and satisfìes the following estimate
ΙΜΙμω) <
c\\f\\wi{n),
where C is a constant.
Proof. Consider the second term on the left-hand side of equation (8):
n
x+h
2
h. K(x, y, ξ, η) u{Ç, η) άξ άη = fi {χ, y), h = {y-η) .
For y < yo, yo being sufficiently small, and taking into account the conditions
that the weight function K{-) satisfies, we obtain the estimate
ΙΙ/ι|Ινν2ΐ(Ω) <e|MUa(n).
(12)
where ε -»• 0 as yo 0.
The estimates from Theorem 2 and the triangle inequality for the norms
yield
ΙΜΙΜΩ) ^ σ ο(||/||^ΐ(Ω) + ll/illwi(n))·
(13)
Α. Η. Begmatov and Α. Η. Begmatov
10
Inequalities (12) and (13) yield the inequality
ΙΜΙμω) <
(14)
c\\f\\wHü),
where
C = C0/(l - C0e).
Hence the existence of solution of equation (8) follows for y sufficiently
small. Since (8) is a Volterra equation [33], the uniqueness and stability
estimate (14) will hold not only for small y, but in the whole strip Ω. The
theorem is proved.
•
Introduce the notations χ — (xi,x2, •.. ,χ η _ι), ξ = (ζι,ζ2, • · · ,ζη-ι)ι
η > 2; y e R 1 , η e R 1 , y > 0, η > 0, Ω = {(χ,y) : χ e R n _ 1 , y Ε (0,h),
h< oo}, Ω = {(χ, y) : χ e R n _ 1 , y G [0,h]}.
In the layer Ω consider the set of cones {K(x, y)} with the vertices in
the points (x, y)
K(x,y)
-ξπι)2 = {y-η)2,
= {(ξ,7?) :
0 < τ? < y, y < h, h< oo
oo}.
m= 1
We denote by Q(x,y) the part of n-dimensional space bounded by the surface K(x, y) and the hyperplane y — 0. Consider the following problem
Find the function u(-) dependent on n variables if for all (x, y) e Ω the
integrals of this function are known along the surfaces K(x,y):
if
u(t,V)dk
J J K{x,y)
= f(x,y).
(15)
Theorem 4. Suppose n is even and the function /(·) is known in the
layer Ω. Then the solution of (15) in the class Cq (Ω) is unique, the following
representation
U
(16)
dy
>
J0
and the inequality
\\u(x,y)\\w%
ο
·\η) - Cill/foyJIIw?
hold. Here Cι is a constant,
1
Co = η - 1 / 2 ( η - 1 ) / 2 ( / ) '
2
π
Γ η 2
η
(Ω)·
11
Problems of integral geometry
The theorem can be proved analogously Theorem 2.
The problem of existence of solution of equation (15) is formulated as
follows.
T h e o r e m 5. Let the right-hand side of equation (15) satisfy the following
conditions:
1) f(x,y)
is fínite with respect to x;
2) f(x,y)
has all continuous partial derivatives up to n-th
Qk
order;
Qk
3)-^f(x,y)
= -^f(x,y)
= 0 (0
<k<n).
oyK
y=0
dyK
y=h
Then the solution of equation (15) exists in the class of continuous
tions fínite with respect to χ and is defined by formula
4.
I N V E R S I O N OF T H E X - R A Y
WITH INCOMPLETE
func-
(16).
TRANSFORMATION
DATA
Let a; = (χι,χ2,χ3)
G R3, ξ = ( 6 , 6 , 6 ) e R3, λ = (λι,λ2,λ3) G R3,
χ = ( χ ι , χ ί ) G R 2 , ξ = ( 6 , 6 ) G R 2 , λ = ( λ ι , λ 2 ) G R 2 , Q = R 3 χ Θ,
V = R 2 χ θ , Θ = { α : α Ε [0,27γ]}, μ = (cos α, sin α ) , Ω = {(χ, y) : |χ| <
1, \y\ < I, 0 < I < οο}; Ω = {{χ, y) : \χ\ < 1, |y| < /}.
Denote by Κ (a;) the family of two-sheeted cones with vertices in the
points (x):
Κ(χ)
= {ξΕ
R 3 : | z 3 - 6 | = |¿-<£!}·
We consider the operator equation in a function u(x)
L
R1
tt(a;i + s cos a, X2 + s sin α,
+ s)ds — f(x, a ) .
(17)
We assume that the function / ( · ) is known for all ( χ , α ) e Q.
Thus, we consider the problem of integral geometry for the family of lines
which are generatrices of two-sheeted cones K(x).
The desired function u(-)
is the function of three variables and the right-hand side of equation (17)
depends on four parameters X\,X2,xz, and a.
Nevertheless, the problem
of solution of equation (17) is not overdetermined since its right-hand side
satisfies the equation with partial derivatives of the first order
df
df
.
df
n
- — cos a + - — sin a + - — = 0.
OX\
OX2
ox3
Moreover, the following theorem on uniqueness of solution of equation (17) holds.
12
A. H. Begmatov and A. H. Begmatov
Theorem 6. Solution of equation (17) is unique in the class of continuous fìnite functions with the support in Ω.
Proof. Applying to both sides of equation (17) the Fourier transformation with respect to the variables xi,x2, we obtain
/ e~is^w(
J R1
Αχ, λ 2 , s + ®3) de = ψ(Χ, χ3, α).
We denote here by w(X, x¿) and ψ(\, x3, a) the Fourier transformations with
respect to xi,x2 of the functions u(x) and f{x,a) respectively, and (·, ·) is
the scalar product.
Changing the variables s + £ 3 = t and introducing the notation λι cos a+
λ2 sin α = — Ä3 we obtain
v{\) = eiX3X3iP(\,a),
(18)
where v(X) is the Fourier transformation of the function ιυ(λι, Λ2, X3) with
respect to X3.
In the space of the Fourier variables (λι, À2, λ3) consider the two-sheeted
cone
K = {(λι,λ 2 ,λ 3 ) € R.3 : |λ| = |λ3|>
with the vertex in the origin.
Denote
it = {X G R 3 : |λ| < |λ3|}.
Taking into account that (18) holds for all a G [0;2π] and ψ(Χ, X3, a) is
the Fourier transformation with respect to the variables (xi,x2) of the given
function f(x, a), we see that the values of υ (λ) are uniquely determined from
equation (18) for all λ € R 3 \ K. Thus, the problem of finding u{x) from
equation (17) is reduced to the problem of extension of its Fourier image v(X)
from R 3 \ K, to the interior of the cone /C.
Since u is finite, w(A) can be extended analytically with respect to the
first argument into the complex plane. Therefore, the initial problem of
integral geometry is reduced to the following problem of analytical extension:
to find the function v(v\ + iv2) in the interval {u G C :
< Λ3,
= 0}
which is analytical in the whole complex plane if its values in the rays
L\ — {(^1,^2) :
> Ag, V2 = 0} and L2 = {(^1,^2) :
< -λ^, v2 = 0} are
known.
The solution of this problem, as is known, is unique. Hence Theorem 6
follows.
•
Problems of integral geometry
13
We shall formulate now one auxiliary statement which will be useful for
deduction of stability estimate for the problem in question.
In the complex plane C we consider the strip
S = {ζ = zi + iz2 : zi G R 1 , \z2 | < απ, a > 0}
and the rays
ri = {z : —oo < z\ < —a, Z2 = 0},
Γ2 = {ζ : a < χ < oo, y — 0}.
Suppose that G = S \ {ri U Γ2}. In other words, the domain G is the
strip S with the cuts along the rays ri and Γ2Introduce the notations E = ri U r2, G is the closure of the domain G,
dG is the boundary of G, ω = ω(ζ, E, G) is the harmonic measure of the
set E with respect to the domain G.
Lemma. For ω = ω(ζ, E, G) the following estimate holds
2/3 < ω(ζ, E, G) < 1.
In the space of the functions f(x, α) we introduce the norm
Wf{xi,x2,y,a\\1
= Wh(xi — X3 cos α, X2 — xs sin a, a)\\c(v) •
Theorem 7. Suppose the function u(x,y) belongs to the class CQ (Ω)
and the inequalities
IK)llcg(n) < 1.
ll/(-)lli<e,
hold, where ε > 0 is sufficiently small.
Then the following estimate of conditional stability holds for the solution
of the problem of integral geometry (17)
||u(.)||c<ai|ln(l/e)|-1>
where α ι is some constant.
Proof. The stability estimate can be found analogously Theorem 6 by
reducing the initial integral geometry problem to the problem of analytical
extension of the function v{u\ + iv-ì) in the strip Π =
1^2) :
G R1,
\v2\ < απ, 0 < α < oo} from the rays pi = {(1/1,^2) • > a,
= 0} and
P2 = {(^1,^2) : "ι < —a, i>2 = 0} into the interval {(^1,1^2) : —a < ι>ι < a,
U2 = 0}.
14
Α. Η. Begmatov
and Α. Η.
Begmatov
Introduce the domain Λ = Π \ {ρι U Ρ2}· The harmonic measure of the
set {pi U P2} with respect to the domain Λ is ω = ω(ν,ρι Up2,A).
From the lemma follows the estimate for the harmonic measure ω:
2/3 < ω(ν,ρι Up2,A) < 1.
(19)
The function v(v) satisfies the conditions of the theorem on two constants; therefore, taking into account inequality (19), we obtain
ν e π,
(20)
where ει and M\ are the estimates from above of the modulus of the function v(v,\2,\3)
in the rays p\ and p2 and in the lines {u : v\ 6 R 1 ,
V2 — —απ}, {¡/ : ¡/[ e R 1 ,
= απ} respectively.
In the rays p\ and P2 we have
|«(λ)| < α 2 ε,
(21)
where 02 is a constant.
It is not hard to prove the following estimate for |ι>(· )| in the boundary
of the strip Π:
κ · ) | <α3ε α π /(απ),
(22)
where 03 is a constant.
Estimates (20)-(22) yield the estimate
Μ < a 4 a- 1 / 3 e a i r / 3 e 2 / 3 ,
(23)
where 04 = (0^03/π) 1 / 3 .
We obtain now the estimates of |υ(λ)| inside the ball GR = {λ : |λ| < R,
0 < R < 00} and outside it. Applying the inverse Fourier transformation,
we obtain
\η(χ)\<α5ΗΆ^Η/3ε2/*+α6/Η,
where 05, ae are constants , Ή =
R/V2.
We obtain now the estimate from above for the sum from the right-hand
side of this inequality. To this end, we choose Η so that
a6ft8/V*/3e2/3
=
a7/n.
Estimating the order of principle part of Ή, we have
|u(œ)| < αι| In ( 1 / ε ) | - 1 .
Theorem 7 is proved.
•
Problems
of integral
geometry
15
R e m a r k 1. All the results obtained above may be easily propagated
for the case of generatrices of the cones of more general form
2
Σ am(X™ - trn)2{y ~ η)2,
m=1
where am
eR1.
R e m a r k 2. We may consider also such generalization of problem of
solving equation (17) when the generatruces of cones are smooth closed
convex plane curves.
REFERENCES
1. Yu. E. Anikonov, Some Methods for the Study of
Multidimensional
Inverse Problems for Differential Equations. Novosibirsk, Nauka, 1978
(in Russian).
2. Akbar H. Begmatov, One class of problems of integral geometry in the
plane. Dokl. Ros. Akad. Nauk (1993) 331, No. 3, 261-262 (in Russian).
3. Akbar H. Begmatov, On certain classes of polysingular integral equations. Sib. Mat. Zh. (1994) 35, No. 3, 515-519.
4. Akbar H. Begmatov, Reductin of the integral geometry problem in
the three-dimensinal space to the polysingular integral equation with
perturbation. Dokl. Ros. Akad. Nauk (1998) 360, No. 5, 583-585 (in
Russian).
5. Akbar H. Begmatov, On some classes of polysingular integral equations. Sib. Zh. Ind. Mat. (1999) 2, No. 2, 8-14 (in Russian).
6. Akbar H. Begmatov, The problem of integral geometry with perturbation in the three-dimensional space. Sib. Mat. Zh. (2000) 41, No. 1,
3-14 (in Russian).
7. Akbar H. Begmatov, One problem of integral geometry with perturbation in the three-dimensional space. Dokl. Ros. Akad. Nauk (2000)
371, No. 2, 155-158.
8. Akbar H. Begmatov, Integral geometry problems of Volterra type. In:
Integral Methods in Science and Engineering. B. Bertram, C. Constanda and A. Struthers (Eds.). Chapman&Hall/CRC Research Notes
16
Α. Η. Begmatov and Α. Η. Begmatov
in Mathematics Series, 418. Chapman&Hall/CRC, Boka Raton, Fl,
2000, 46-50.
9. Akbar H. Begmatov, Problem of integral geometry on paraboloids with
perturbation. In: Proceedings of the Second ISAAC Congress. Begehr
et al. (Eds). Kluwer, Dordrecht, Vol. 1, 97-103.
10. Akbar Η. Begmatov, On recovery of surfaces from contours of their
shadows. J. Inv. Ill-Posed Problems (2002) 10, No. 3, 213-220.
11. Akram H. Begmatov, Two classes of weakly ill-posed problems of integral geometry in the plane. Sib. Mat. Zh. (1995) 36, No. 2, 243-247
(in Russian).
12. Akram H. Begmatov, Problems of integral geometry for the cone set
in n-dimensional space. Sib. Mat. Zh. (1996) 37, No. 3, 500-505 (in
Russian).
13. Akram H. Begmatov, Volterra problem of integral geometry in the
plane for the curves with singularities. Sib. Mat. Zh. (1997) 38, No. 4,
723-737 (in Russian).
14. Akbar H. Begmatov and Akram H. Begmatov, Inversion of the X-ray
transform and the Radon transform with incomplete data, In: Integral
methods in science and engineering. B. Bertram, C. Constanda and
A. Struthers (Eds). Chapman& Hall/CRC Research Notes in Mathematics Series, 418. Chapman&Hall/CRC, Boka Raton, Fl, 2000,
51-56.
15. A. S. Blagoveshchenskii, On recovery of a function given its integrals
over ruled manifolds. Mat. Zametki (1986) 39, No. 6, 841-849.
16. A. L. Bukhgeim, Volterra Equations and Inverse Problems. Novosibirsk, Nauka, 1983 (in Russian).
17. A. M. Cormack, Representation of a function by its line integrals, with
some radiological applications. J. Appi. Phys. (1963) 34, 2722-2727.
18. A. M. Cormack, Representation of a function by its line integrals, with
some radiological applications II. J. Appi. Phys. (1964) 35, 2908-2912.
19. R. Courant, Partial Differential Equations. Interscience, New York,
1962.
Problems of integral geometry
17
20. D. V. Finch, Cone beam reconstruction with sources on a curve. SIAM
J. Appi. Math. (1985) 45, 665-673.
21. P. Funk, Uber eine geometrische Anwendung der Abelschen Integralgleichung. Math. Ann. (1916) 77, 129-135.
22. I. M. Gel'fand and A. B. Gonchaxov, Recovery of a compactlysupported function from its integrals over the straight lines intersecting a given set of points in space. Dokl. Akad. Nauk SSSR (1986) 290,
1037-1040.
23. I. M. Gel'fand, S. G. Gindikin, and M. I. Graev, Some Inverse Problems
of Integral Geometry. Moscow, Dobrosvet, 2000 (in Russian).
24. P. Grangeat, Mathematical framework of cone beam 3D reconstruction via the first derivative of the Radon transform. In: Mathematical
Methods in Tomography. Herman et al. (Eds). Springer, 1991.
25. S. Helgason, The Radon Transform. Birkhauser, Boston, 1980.
26. F. John, Bestimmung einer Funktion aus ihren Integralen über gewisse
Mannigfaltigkeiten. Math. Ann. (1934) 109, 488-520.
27. F. John, The ultrahyperbolic differential equation with 4 independent
variables. Duke Math. J. (1938) 4, 300-322.
28. F. John, Plane Waves and Spherical Means Applied to Partial Differential Equations. Interscience, New York, 1955.
29. A. A. Kirillov, On a problem by I. M. Gel'fand. Dokl. Akad. Nauk SSSR
(1961) 137, No. 2, 276-277 (in Russian).
30. M. M. Lavrent'ev, Integral geometry and inverse problems. In: IllPosed Problems of Mathematical Physics and Analysis. Novosibirsk,
Nauka, 1984, 81-86 (in Russian).
31. M. M. Lavrent'ev, Integral geometry problems with perturbation on
the plane. Sib. Mat. Zh. (1996) 37, No. 4, 851-857 (in Russian).
32. M. M. Lavrent'ev and V. G. Romanov, Three linearized inverse problems for hyperbolic equations. Dokl. Akad. Nauk SSSR (1966) 171,
No. 6, 1279-1281 (in Russian).
18
A. H. Begmatov and A. H. Begmatov
33. M. M. Lavrent'ev, V. G. Romanov, and S. P. Shishatskii, Ill-Posed
Problems of Mathematical Physics and Analysis. Nauka, Moscow,
1980; English transi.: AMS, Providence, Rhode Island, 1986.
34. F. Natterer, The Mathematics
Stuttgart, 1986.
Computerized
Tomography. Teubner,
35. F. Natterer, Recent developments in X-ray tomography. In: Tomography, Impedance Imaging, and Integral Geometry. Quinto et al. (Eds).
AMS, 1994.
36. V. P. Palamodov, Some mathematical aspect of 3D X-ray tomography.
In: Tomography, Impedance Imaging, and Integral Geometry. Quinto
et al. (Eds). AMS, 1994.
37. J. Radon, Uber die Bestimmung von Funktionen durch ihre Integralwerte längs gewisser Mannigfaltigkeiten. Math.-Nat. Kl. (1917) 69,
262-277.
38. V. G. Romanov, Some Inverse Problems for Equations of Hyperbolic
Type. Novosibirsk, Nauka, 1972; English, transi.: Integral Geometry
and Inverse Problems for Hyperbolic equations. Springer-Verlag, 1974.
39. V. G. Romanov, Inverse Problems of Mathematical Physics. Moscow,
Nauka, 1984 (in Russian).
40. H. K. Tuy, An inversion formula for cone-beam reconstruction. SIAM
J. Appi. Math. (1983) 43, 546-552.
Ill-Posed and Non-Classical Problems of
Mathematical Physics and Analysis, Samarkand,
M . M . Lavrent'ev and S.I. Kabanikhin (Eds)
© VSP 2002
2000, pp. 19-26
Existence of solutions of the first boundary-value
problem for the third order equations of mixed type
in unbounded domain
T. D. DJURAEV and A. R. HASHIMOV
Abstract — In this paper the solutions of the first boundary-value problem for
the third order equation of mixed type are considered. The existence theorem is
established in the classes of functions growing in infinity.
Intensive development of theory of generalized functions had allowed to
establish the theorems of uniqueness and existence in the classes of increasing
functions for linear equations of elliptic and parabolic type for domains with
noncompact boundary depending on geometric domain characteristics [1,2].
For equations of odd order which fill a highly important place in applications
the analogous research up to nowadays were absent.
In our paper we consider an unbounded domain Ω C R" = {χ \ x\ > 0}
and the existence of generalized solution of the problem
lAu + Bu =
=0,
lLJ
Au = at:i(x)uXiXj
+ al(x)uXi
+ a(x)u,
lu = IQU + a(x)u,
k
σ0 = {x Ε Γ I a (x)uk{x)
f(x),
(1)
/0«|σι=0,
(2)
Bu = blj(x)uXiXj
+ bl{x)uXi + a(x)u,
IQU — ak(x)uXk,
Γ = 9Ω,
k
= 0},
σι = {χ € Γ | a (x)uk{x)
k
σ 2 = {ζ G Γ I a {x)vk(x)
> 0},
< 0},
u{x) = (ui,... vn) is unit vector of internal normal to Γ in the point x. Here
and further in formulas the summing goes along the repeating indices.
20
T. D. Djuraev and A. R.
Hashimov
Note that the existence of generalized solution of problem (1), (2) in a
bounded domain is researched in [3].
We assume that in a certain neighborhood of each its point the hyperplane Γ is represented in the form Xj = χ(χι,...,
Xj-i, • • • Xj+i, · · ·, Xn) f° r
2
some j, where χ belongs to the class C .
Suggest that all the coefficients in (1), the functions a(x), ak(x) (k =
1 , 2 , . . . , n), and their derivatives which met further are bounded and measurable in each bounded subdomain of Ω.
We assume further that
¿i = a?\
α0|ξ|2<α^<Ωι|£|2,
¿[α*(ζ)]2^0,
k=l
Qij = ci - c\Xi/2 + ¿¿iX./2
& = <P\
¿i - ota - ci < 0,
+ (cSa) X i /2 < - c 0 < 0,
for χ G Ω υ Γ , ξ G M"; ÜQ, οχ, <¿o, di-, cío, cu, CQ are positive constants. Here
& = c?' - {<¿ak>)Xk + ota? +
c*J = bij + aaij
- akXkaij,
{aka^)xJ2,
c\ = b< + ota* -
akka\
ci = b + aa — cnXlca.
We assume that { Ω τ } is the set of bounded subdomains of domain Ω
depending on the parameter τ 6 Π = { τ | 0 < τ < τ ° } , r ° < o o . We shall
assume that Ω τ C Ωτ/ if r < τ'. Denote ST = <9ΩΤ \ «9ΩΤ/. We assume
that ST is connected (n — l)-dimensional surface with the same smoothness
that dÜ and dST C 3Ω.
Set Γ τ = Γ Π 0Ω Τ , σ 0)Τ = {χ G Γ τ | akvk = 0}, σλ,τ = {χ G Γ τ | akuk >
0}, σ-ι)Τ — {χ € Γ τ I akVk < 0}. For h > 0, we define
= { ζ G σι )Τ |
p(x, dai r) > h}, σ£τ = σ1]Τ \ al¡htT.
Suppose that E ( ü r ) is the set of the functions ϋ from the class (7 2 (Ω Τ )
such that ϋ = 0 in Γ τ and for some h > 0 we have IQÛ = 0 in σοιΤ1)σ2,τΙ>σι)Τ·
Denote by Η(Ω Τ ) the closure of Ε(Ω Τ ) in the norm
IMIif(nT) = [ [ (^^xi^xj
lJn
T
+u2)dx+
[
akukaZJuXiuXj
J σι,τ
ds]
1
/
.
21
Existence of solutions
Consider the bilinear form
ai(tt,i?) = / [akaljuXiûXjXk
JnT
- c\juXidXj
+ (akatj)XjuXiûXk
-akaiUXitiXk
+ ( 4 . - a* a - c\)uûXi + (Cl - c\Xi + c\jXiX.)utf] dx.
(3)
Definition We define the generalized solution of the problem (1), (2) in
the domain Ω each function u(x) such that for each bounded subdomain Ω τ
of the domain Ω, u(x) 6 Η(ΩΤ) and
ai(ti,0)=/
Jfir
fêdx
(4)
for any function ϋ(χ) e E(ÜT), ê = 0 in ST.
I. First we suppose that ak — const, k = Ι,η, αϊ > 0. In [3] it was
shown that the generalized solution satisfies the second condition in (2) on
the average.
For simplicity of presentation we assume first that ST = Ω Π {χ | x\ =
r + 7} for each τ G [0, το], 7 = const.
Introduce the notation
Q(u) = dLjuXiuXj - qlju2,
g=
P(r) = sup B(x),
aid^^ia1)2,
(5)
Sr
B(x) = max {2 _1 (α 1 α 1 + cf - (alaij)Xj,
0}.
(6)
Set
0 < λ(τ) < inf { Í Q{v) ds
J St
where Ν is a set of functions υ continuously
hood of Sτ for χ G Ω and equal to zero in ST
We suppose that Φ (τ) is positive for r e
!
(7)
* ST
differentiable in the neighborΠ Γ.
Π and
Φ Μ ^ λ - ^ Μ + ΡΜλ"1^).
(8)
Depending on the type of the domain the considerations of the work will
differ. We shall consider the two classes of domains Ω for which one of the
below conditions will hold
Α) ΘΦ(τ)/δτ < ε for all τ € Π, ε = const, 0 < ε < 1;
Β) ΘΦ(τ)/δτ > ε for all r 6 Π
T. D. Djuraev and A. R. Hashimov
22
and satisfying the condition r(0) = 0. Here the function Φ is such that the
right-hand sides of the below formulas (9), (10) will be absolutely continuous.
For problem (1), (2) the following theorems hold [4].
Theorem 1 [the analog of the Sen-Venan principle]. Suppose that u(x)
is the generalized solution of problem (1), (2) in the domain Ω of the class ),
where
( a V ) X i - (aW)XiSj
+ 3c[ j x . x . - 2aia -2c\>0
and f(x) = 0 in Ωτο. (9)
Then for each RQ and R such that 0 < Ro < R the following
holds
[
exp {-(R
Q{u) dx <
- Ro)} f
Q(u) dx.
estimate
(10)
Theorem 2 [the analog of the Sen-Venan principle]. Suppose that u(x)
is the generalized solution of problem (1), (2) in the domain Ω of the class B),
and condition (9) holds. Then for each RQ and R such that 0 < RQ < R the
following estimate holds
f
JNT(RO)
Q{u) dx < exp { f ( Ä ) g ^ }
L JT(RO)
[
ΦΙ«)J
Q{u) dx.
(11)
JNT(R)
Now, we establish the existence of solution of problem (1), (2).
Lemma 1. We suppose that there exists an infìnite sequence of bounded
subdomains
of the domain Ω from the class ) such that Ωτ(^) C
Ωτ(λγ+ι) for Ν = 1,2,... ,Ω — (J"=1 Ωτ^. Suppose that for each fìxed Ν for
each function ω which is a generalized solution of equation (1) in Ωτ(
ΠΓ
the following estimate holds
[
+ 1 } ' ! 1 exp" 1 [
Q(")dx<T{Ni
Q(u)dx.
(12)
Suppose also the the function f(x) is defìned in Ω and satisfies the estimate
Í
f2 dx < ΜχΛ(Ωτ(Λτ)) exp [(1 — δ)Ν],
η = 1,2,...,
(13)
•'ΩΤ(Ν)
where δ = const, 0 < δ < 1, the constant M\ is independent of Ν,
Λ(Ωτ(λ0)=
inf
¡ Í
Q<û)dx( [
tf2^-1).
(14)
23
Existence of solutions
Then there exists a unique generalized solution u(x) of problem (1), (2)
in Ω which satisfìes the estimate
f
JnΩ
g(w)ffa<M2exp[(l-¿)iV],
Ν =1,2,...,
(15)
Τ(ΛΓ)
where M2 is the constant independent
of Ν.
Proof. Note that Λ(Ωτ(ΛΓ)) > 0. Denote by uljn'(x) the sequence of
functions from
such that u™ converges to m for m
00 in the
norm Η(Ωι)·, u¡ is the generalized solution of problem (1), (2) in the domain Ω; [3]. We fix an arbitrary subdomain
from the sequence
Ω τ (ΐ) C Ωτ(2) C . . . and consider the sequence of subdomains
k
oo.
Taking into account (4), for Ω χ = QT(N+k)i u = uN+ki $ =
integrating by parts and passing to the limit for m ->• 00 we obtain
/
Q(uN+k) dx<JftriN+k)
f • uN+k
dx.
JttriN+k)
Hence, taking into account the Cauchy-Bunyakovskii inequality, we have
Í
Q(uN+k)
JftriN+k)
d x < ( f
f2dx)
Jttr(N+k)
/
Q{uN+k)dx
J^TiN+k)
' ( Í
u2N+k dx)
^r(N+k)
< Λ_1(Ωτ(^)) /
1
,
f2dx.
Conditions (13) and this inequality yield
[
ώ < Μι β χ ρ { ( 1 - ί ) ( Λ Γ + *:)}.
(16)
Set
It is easy to see that the function ω = u^ + k+1 — ujv+fc in the domain ^N+k
is the generalized solution of equation (1) for f(x) = 0 satisfying conditions (2). Therefore, applying subsequently estimate (12) to the domains Ωτ(λγ) j
• · · » ^ T (N+k) a n d taking into account (16) we find
\\uN+k+i
- uN+khT(N)
< M s ^ / j i y v ^ ,
where the constant M3 is independent of Ν and k.
(17)
24
T. D. Djuraev and A. R.
Hashimov
Inequality (17) for each natural ρ > 0 and q > 0 yields
< M4(T(^)1||1)P/2e-^2,
-
(18)
where the constant M4 is independent of N, p, and q. Therefore ||u^+fc+i —
ujv+fc||nT(Ar)
0 f° r e a c h 9 > 0 and ρ -> oo.
Since Η(Ωτ(Ν)) is a Hilbert space, then
\\uN+p
-
«ΛΓ+Ρ+9||Η(ΩΤ(Ν)) <
M5\\Un+p
- ΪΧΛΓ+Ρ+9||Ωτ(λγ))
where the constant M5 is independent of p. Hence it follows that the sequence {tip} converges in the norm H ( i i T ^ ) for ρ
oo to the function
u(x) e Η(Ωτ(Ν)). As Ν > 0 is taken arbitrarily, then u(x) is defined in Ω
and u(x) 6 H(Ú') for each bounded subdomain Ω' of the domain Ω.
Taking into account the imbedding theorems, the trace of the function u(x) in dO.' Π Γ exists and the first condition from (2) holds in this
set. As we have noted the second condition from(2) holds following [3].
It is easy to see that «;v+pm satisfies integral identity (4) when Ω' = Ω τ (^)
and ϋ(χ) E Ε(ΩΤ^).
Passing to the limit for pm —> 00 we see that u(x)
satisfies integral identity (4) for Ω τ = Ω τ (#) and ΰ G Ε ( ί ί ν ) . Since iV > 0 is
arbitrary, then, for each subdomain Ωλγ e Ω and the function ϋ 6 Ε (Ω Ν),
integral identity (4) holds. Therefore, u(x) is the generalized solution of
problem (1), (2) in the domain Ω.
Setting in relation (18) ρ = 1, tending q to 00 , and taking into account
estimate (16) we find
I M I n r ( J V ) < A;exp{(l -δ)Ν}
+ ||ι**+ι||η*
< kexp {(1 - S)N/2} + h exp {(1 - δ)(Ν + 1)/2}.
Hence relation (15) follows.
We show now that the solution u(x) of problem (1), (2) satisfying conditions (15) is unique. As it follows from inequality (12), for each two
generalized solutions u and ϋ of problem (1), (2) in the domain Ω for which
estimate (15) holds the following estimate holds for each I > 0
i
Q(u -ΰ)άχ<
2 j 'M 3 exp ( - j ) Í
·!Ωτ(,1)
Q(u - ΰ) dx
J^T
< 2jM3 exp{-j} [ Í
JnT(l+j)
(l+j)
Q{ 1?) dx1 < 2jM6 exp ( - 6 j ) ,
Q{u) dx + Í
Jcïi+i
•3
25
Existence of solutions
where the constant Mq is independent of j. Passing to the limit when j —oo
in this estimate, we obtain ||u — #||ωτ(ι) = 0 for all I > 0. Since Λ(Ωτ(ί)) > 0
then u — ϋ = 0 in Ωτ(/); therefore, u = ϋ in Ω.
•
As it follows from this lemma, the existence of the generalized solution
of problem (1), (2) in Ω is based on the assumption that relation (12) holds.
We show now how to construct the sequence of domains {Ω τ (^)} so that
relations (12) holds. Suppose {Ω τ } is a set of bounded subdomains of the
domain Ω and let all the conditions of Theorem 1 hold. Then, for each Rq
and R such that 0 < Rq < R the following estimate holds
/
JNT(RO)
^^
d x
- ^tIt+7
T{RQ) +1
e_(Ä
~'Ro) f
JUT(R)
Q(u)dx.
(19)
The lemma and estimate (19) yield the following theorem of existence
and uniqueness for generalized solution (1), (2).
T h e o r e m 3. Suppose for each domain Ωτ(#) we have Α(ΩΤ(Λ)) > 0; the
function f(x) is defìned in Ω and satisfìes the relations
Λ _1 (Ω τ( *) j
f2 dx < M2 exp(l — S)k,
k = 1,2,...,
where δ = const, 0 < δ < 1, the constant M2 is independent ofk, r(k) is the
function defìned in (19). Then there exists a unique generalized solution u(x)
of problem (1), (2) and the inequalities hold
j
Q(u) dx < Μγ exp {(1 — ö)k},
^T(k)
k = 1,2,....
Proof. Set Ω , - ^ = Ωτ^·) in Lemma 1. Inequality (12) for the functions
Ui — u¿_ 1 follows from Theorem 1. Thus, Theorem 3 follows from Lemma 1.
•
For the domains of the class B) the following theorem holds.
T h e o r e m 4. Suppose for each domain ΩΤ(Λ) we have Λ(Ωτ(·^) > 0
and f(x) defìned in Ω satisfìes the relations
A_1(íÍT(fc))
L
f2dx M8exp (1 δ)
-
{ " Io{k) ^y}'
k=K
-•·'
where δ = const, 0 < δ < 1, ε = const, 0 < ε < 1; the constant Ms is
independent of k; r(k) is the function defined in (11). Then there exists
26
T. D. Djuraev and A. R. Hashimov
a unique generalized solution u(x) of problem (1), (2) which satisfies the
inequalities
f
Q(u)dx<M9exp{(l-S)
JQtW
1
Γ™
Jo
Φ (β) J
k = 1,2,....
Theorem 4 is proved analogously Theorem 3.
II. Suppose ak(x)fk(x) < 0 in Γ. Then problem (1), (2) becomes the
Dirichlet problem; the operator IQ, in this case, has the variable coefficients
and the solution u(x) of problem (1), (2) belongs to the space
[3].
In this case we can obtain the analog of the Sen-Venan principle and the
existence theorems. In the classes of functions growing in infinity similarly
as in point I. The proof of these statement is just the same as in point I.
Remark. The analogous results hold when ST = Ω Π {|x| = τ + 7}.
REFERENCES
1. O.A. Oleinik and G.À. Iosif'ean. Uspekhi Mat. Nauk (1976) 31,
No. 113, 142-166 (in Russian).
2. O. A. Oleinik and G.À. Iosif'ean. Mat. Sbornik (1980) 4, 588-610 (in
Russian).
3. A. I. Kojanov, Boundary- Value Problems of the Odd Order for Mathematics Physics. Nauka, Novosibirsk, 1990 (in Russian).
4. A. R. Hashimov. Uzb. Mat. Zh. (2001) 5-6, 64-71.
Ill-Posed and Non-Classical Problems of
Mathematical Physics and Analysis, Samarkand,
M . M . Lavrent'ev and S.I. Kabanikhin (Eds)
© VSP 2002
2000, pp. 27-55
On the monotone error rule for choosing
the regularization parameter in ill-posed problems
U. HÄMARIK* and U. TAUTENHAHNt
Abstract — We consider linear ill-posed problems Ax = y in Hilbert spaces with
minimum-norm solution x* and suppose that instead of y noisy data ys are given
satisfying ||y — 2/5|| < δ with known noise level (5. For the stable numerical solution
regularization methods are considered including continuous regularization methods
such as ordinary Tikhonov regularization x\ — (A*A + r~1I)~1A*ys and iterative
regularization methods. For the proper choice of the regularization parameter r
(which is the stopping index in iterative methods) we study the monotone error rule
(ME rule): Choose r = ΓΜΕ as the largest r-value for which it can be guaranteed
that the error
— a;t|| is monotonically decreasing for r e (0,ΓΜΕ]· We compare
this rule with other a posteriori rules and give conditions for which convergence and
order optimal convergence rate results can be guaranteed. For the computation of
ΓΜΕ in Tikhonov methods some nonlinear equation has to be solved. Newton's
iteration for this equation appears to be globally and monotonically convergent.
Numerical experiments are provided that verify some of the theoretical results.
1.
INTRODUCTION
In this paper we consider linear ill-posed problems
Ax = y
(1.1)
'University of Taxtu, Department of Mathematics, Liivi 2, 50409 Tartu, Estonia. Email: Uno.Hamarik@math.ut.ee
^University of Applied Sciences Zittau/Görlitz, Department of Mathematics, P.O. Box
1455, 02755 Zittau, Germany. E-mail: u.tautenhahn@hs-zigr.de
The work was supported by Deutsche Forschungsgemeinschaft (grant 436 EST) and by
the Estonian Science Foundation (grant 4345)
28
U. Hämarik and U. Tautenhahn
where A E C(X,Y)
is a bounded operator with non-closed range R(A)
and X, Y are infinite dimensional real Hilbert spaces with inner products
(·, ·) and norms || · ||, respectively. We are interested in the minimum-norm
solution x^ of problem (1.1) and assume that instead of exact data y there
are given noisy data ys Ε Y with ||y — ys\\ < δ and known noise level <5.
Ill-posed problems (1.1) arise in a wide variety of problems in applied
sciences. For their stable numerical solution regularization methods are
necessary, see [4, 19, 26, 39]. Regularization methods can be divided into
continuous regularization methods that include ordinary Tikhonov regularization x\ ·.— Rrys — {A*A + r~1I)~1A*ys,
and iterative regularization
methods where the stopping index plays the role of the regularization parameter. The element xf — Rrys is called regularized approximation for the
minimum-norm solution χt of problem (1.1) provided
1. for any r > 0, Rr : Y —• X is continuous and
2. for arbitrary y Ε Y with Qy E R(A),
lim \\Rry —
r—yoo
= 0, where Q is
the projection operator onto R{A)
(see [36]). Traditional regularization methods possess the property that in
the case of exact data the error
— a;t|| as a function of r is monotonically
decreasing for r —> oo. This property is no longer true for the error
In the case of noisy data the situation is as follows:
(i) If r becomes too large, then the error \\xf —
increases due to the
fact of ill-posedness of the operator equation (1.1).
(ii) If r becomes too small, then the error \\xf — || increases due to the
fact that the error
— a;t|| is monotonically increasing for decreasing
r-values.
The monotone decrease of the error
—
for growing r-values can only
be guaranteed for small r. Typically
a;t|| diverges for r —> oo. Therefore
a rule for the proper choice of the regularization parameter r is necessary.
In the monotone error rule for choosing a proper regularization parameter the idea consists in searching for a largest computable regularization
parameter r = r^E for which we can guarantee that the error \\xf — rtr11 is
monotonically decreasing for r Ε (0, ΓΜΕ\·
For continuous regularization
methods this means that
^-Hxf-xtf <0
for all
re(0,rMÊ],
(1.2)
Οτι the ME rule for choosing the regularization
parameter
29
for iteration methods this means that
<
\\ΧΌΓ-Ι
— z^ll for all
r = 1,2,...,
ΓΜΕ
·
(1.3)
Similar rules for the choice of the regularization parameter which are based
on monotonicity properties of the error were proposed and studied for some
iterative methods in [1, 2, 13, 15, 16], for the method of ordinary Tikhonov
regularization in [34] and for the method of iterated Tikhonov regularization
and some other continuous regularization methods in [14, 15, 35].
In this paper we study the ME rule for continuous regularization methods including Tikhonov methods and asymptotical regularization, and for
iterative regularization methods including gradient type methods (Landweber's method, steepest descent method, minimal error method) and implicit
iteration methods. We compare the ME rule with other a posteriori rules
including Morozov's discrepancy principle and study questions concerning
convergence xfME —> x^ for δ ->· 0 and concerning order optimal error bounds
under certain source conditions. For the computation of ΓΜΕ in Tikhonov
methods some nonlinear equation ¿MEÍ^) = Í has to be solved. The function
¿ME appears to be strictly monotonically decreasing and strictly convex.
These properties guarantee global and monotone convergence for Newton's
iteration. In the final section numerical examples are provided which verify
some of the theoretical results.
2.
CONTINUOUS REGULARIZATION METHODS
2.1.
Continuous regularization methods and the ME rule
In continuous regularization methods which include the method of ordinary
Tikhonov regularization xf = (A*A + r~1I)~1A*ys
we use in this section for
the regularization parameter the traditional notation a — 1/r instead of r.
We consider continuous regularization methods of the general form
xsa=ga(A*A)A*y5.
(2.1)
Here # α (λ) : [0, α] -» R with a — ||A[|2 is a family of piecewise continuous
functions depending on a positive regularization parameter a > 0 and the
operator function ga is defined according to
where A*A = /0° λ dE\ is the spectral decomposition of the operator A*A.
For the functions ga(X) we assume as in [4, 26, 38, 39] that there exist
30
U. Hämarik and U.
Tautenhahn
constants 7, jp and po such that for a > 0
sup |</β(λ)| < Ί α ~ ι
(2.2)
0<λ<α
and
sup Λρ|1 — λ0 α (λ)| < τ p a p
for
0 < ρ < p0 .
(2.3)
0<λ<α
The largest constant po in assumption (2.3) is called qualification of the
regularization method (2.1) (see [39]). Three well known a posteriori rules
for choosing the regularization parameter a in continuous regularization
methods (2.1) are:
1. Morozov's discrepancy principle [4, 27, 38, 39]. In this principle (D
principle) the parameter a = a η is chosen as the solution of the equation
dD(a) := ||yÄ - AxsJ = CÔ with C > 1 .
2. Rule of Raus [30]. In this rule, which we call R rule, the regularization
parameter α = (XR is chosen as the solution of the equation
dR(a)
:= ||(/ -
δ
- Αχδα)|| = CS
9α(ΑΑ*)ΑΑψ^(υ
with
C > 1.
Here po is the (largest) constant from assumption (2.3).
3. Rule of Engl and Gfrerer [3]. In this rule, which we call EG rule, the
regularization parameter A = O¿EG is chosen as the solution of the
equation
dEG(a):=j-1/2a^Axi-ys,-^ga(AA*)y6y/2
= CÔ
with
C>
1.
Here 7 is the (smallest) constant from assumption (2.2).
For ordinary and iterated Tikhonov methods the R - and EG rules coincide. The resulting rule was also proposed in [8] and we call this rule
Raus—Gfrerer rule (RG rule).
Now let us turn over to the ME rule. The general idea of this rule (see
Chapter 1) consists in searching for a largest computable r = ΓΜΕ for which
we can guarantee that the error
— a;t|| is monotonically decreasing for
r G (0,γμε]. The reformulation of this idea in terms of α = 1/r means:
Search for the smallest computable regularization parameter a — OÎME for
which we can guarantee that
•J-
—
> 0
for all
A G \OÌMEÌ00) ·
On the ME rule for choosing the regularization
parameter
31
In order to guarantee this property we use the identity ga{A*A)A* —
A*ga(AA*) and estimate the derivative of the squared error \\x&a —
with respect to a as follows:
\-L
H3* - ^ l
2
= (x°
-¿,A*^9«(AA*)ys)
= (Αχδα-νδ
>
+
(νδ-ν)±9α(ΑΑ*)νδ)
' (Αχδα-νδ,±9α(ΑΑ*)νή
'
¿9a(AA*)ys
ì&gaiAA-tf
This estimate leads us to following ME rule for continuous regularization
method (2.1):
M E rule. For regularization methods with monotonically increasing
functions (IME{A) in (2.4), choose A = AUE as the solution of the equation
dME(a)
2.2.
:=
\Axsa-y',^ga{AA*)/)=&.
(2.4)
The ME rule for ordinary
and iterated Tikhonov regularization
In these methods we start with χδα 0 = 0 and compute the regularized solution χδα := xSa rn recursively by solving the m operator equations
(A*A + aI)x*atk
= A'ys
+ axeatk_lt
k = l,2,...,m.
For m = 1 this method is the method of ordinary Tikhonov
(2.5)
regularization.
X/a)~m]/X
In these methods we have <7α(λ) = [1 — (1 +
with some fixed
positive integer πι > 1. Let r Q j m denote the discrepancy of the regularized
solution χδα = χδα ηι, i.e.
ra,m =
y ~ Axa
m
.
Using the identities
l - W * )
= (
^
r
and
we obtain
ra,m = [I-
AA*ga(AA*)]ys
= [a(AA*
+ α/)"1]
V
32
U. Hämarik and U. Tautenhahn
and
^
— ga(AA*)ys
do;
=
=· r Q , m + i .
ο.Δ
Prom these representations we conclude that the functions άϋο{α) and
<ÌME{α) for the R G - and ME rules have the form
duda)
= {ra,m,ra,m+l)ll2
and
dME{a)
=
,
(2.6)
The function dME{&) given in (2.6) possesses following properties, see [34]
for m = 1 and [14, 35] for m > 1:
Theorem 2.1. Let Ρ denote the orthoprojection
R(A)± and let A*ys φ 0. Then:
(i) dME{&) is strictly monotonically
ΊΜΕ{0) = ||¿VII
increasing and
and
lim dME(a)
Q—>00
of Y onto Ν (A*) =
obeys
= ||ytf||.
The equation DME{&)
— δ has a unique solution
\\Py*\\ < δ < \\y*\\.
(ii) For all Α G (α/ιÍE, oo) there holds
(iii) There
provided
OÍME
^
holds
dRG(ct) < dME(a)
<
dD(a).
If C = 1 in the D principle and in the RG rule, then an < a m E <
OÍRG-
Prom parts (ii) and (iii) of the theorem there follows
II4mb-^II<II4äg-^II·
(2.7)
Hence, the ME rule provides always a smaller error than the RG rule. Exploiting the monotonicity property (ii) we obtain for the parameter choice
α = OÍME order optimal error bounds (see [35]):
Theorem 2.2.
ρ G (0,2m]
Assume χΐ = (A*A)p/2w
with ||ω|| < E.
Then
\\xíME - ¿W < V p / i p + 1 ) + 2 - 1 / ( p + 1 ) 7 * ( 7 p / 2 ) 1 / p } £ 1 / ( p + 1 ) ¿ p / ( p + 1 )
with 7 , = Vm and
7p/2
= [p/(2m)]f/ 2 [l - p/{2m)]m-P/2
< 1.
for
(2-8)
Οτι the ME rule for choosing the regularization parameter
2.3.
33
The ME rule for asymptotical regularization
In this regularization method the regularized solution is given by χδα = x Ä (r)
where xs(r) is the solution of the initial value problem
^-xs{t)+A*Axs{t)
di
= A*y5
for 0 < i < r ,
^(0) = 0
with r = 1/a. In this method we have <?α(λ) = (1 — β - λ / α ) / λ . Using the
identities
- ^ α ( λ ) = e - V " = 1 ( 1 - λ 5Ω (λ))
da
er
ar
we obtain that for the method of asymptotical regularization there holds
dR{a) = cIeg{q0 = dME{ot) = dD(a).
Prom the identity do(a) = dME{a) and the monotonicity property of our
ME rule we conclude that for the method of asymptotical regularization the
best constant C in the D principle is C = 1:
Theorem 2.3. Let A*yδ φ 0, let xsa be the regularized solution obtained
by the method of asymptotical regularization and let an be the regularization parameter of the D principle with C = 1. Then
\\*iD -® f ll < 1 1 4 - ^ 1 1
for all
a>aD.
From the identity dji(a) = dßG(a) = ^ M E ( Ö ) = do (a) we conclude
that all results known for the D principle (see, e.g., [4, 38, 39]) are also true
for the R rule, the EG rule and the ME rule, respectively. Exploiting the
monotonicity property of our ME rule we obtain order optimal error bounds
for ||a;£ —
with a chosen by the ME rule, or equivalently, the R rule, the
EG rule or the D principle, respectively.
Theorem 2.4. Assume x* = (A*A)p/2w with |M| < E. Let χδα the
regularized solution obtained by the method of asymptotical regularization
and let α be chosen by the ME rule. Then for all ρ e (0, oo) the order optimal
error estimate (2.8) holds true with 7» « 0.6382 and yp/2 = (p/(2e)) p .
Proof. Let us introduce the two operators
Ka = I - A*Aga(A*A),
Ka = I -
AA*ga(AA*).
Then we obtain from (2.1)
χδα-χ1
= -Kaxi
+ ga(A*A)A*(ys
- Axt),
(2.9)
34
U. Hämarik and U. Tautenhahn
Αχδα -ys
= -ΚαΑχϊ
+ Ka(Axi
- ys).
(2.10)
The method of asymptotical regularizaron is characterized by ga(Λ) = (1 —
e~~x!a)/\. For this function there holds for arbitrary a > 0 the estimate
sup \/λ | 5α (λ)| < 7 * / \ / ^
0<λ<α
(2.11)
with 7 , « 0.6382 (see [38, 39]). Prom (2.9) and (2.11) we obtain
114-^11 < 1 1 - ^ 1 1 + 7 , 5 / ^ .
(2.12)
Now we exploit the assumption χt = (A*A) p l 2 w with \\w\\ < E and obtain
from (2.3), which holds with ρ ο = oo and ηρ = (p/e) 2p , the estimate
\\Ka¿\\ < Ί Ρ / 2 α ρ / 2 Ε .
(2.13)
For arbitrary D = D* > 0 and 0 < s < t there holds the moment inequality
||Z>Äu|| < ρ ^ Ι Ι ^ Ι Ι υ Ι Ι 1 " ' / *
(2.14)
(see, e.g., [22]). We apply this inequality with D = (A*A)1!2, v — Kaw,
s = p, t — ρ + 1 and obtain due to the relations Dp+1Kaw
= KaAxt and
\\Ka\\ < 1 that
\\Καχ1\\ = \\DpKaw\\
< \\DP+1
Kaw\\p/(p+1ï\\Kaw\\l/(p+li
< WKaAx^WP^+^E1/^ .
(2.15)
For A = Α-ME — OLD we obtain due to the relations \\Ax^
— ys\\ — J,
IIΑχϊ - y5II < δ, \\Κα\\ < 1 and (2.10) that
\\ΚαΜΕΑχϊ\\ < \\AxiME-ys\\
Consequently, (2.15) attains for
A
=
CÍME
+ \\KaME(A¿
- ys)\\ < 2δ.
the form
II^omb^II < (25)P/(P +1 )£; 1 /(P+1).
(2.16)
Let a = a* be chosen such that the right hand sides of the estimates (2.13)
and (2.16) coincide, i.e., let α» = (jp/2)-2/pE-2Hp+1î{2δ)2^ρ+ι1
If
ΑΜE <
then the desired estimate (2.8) follows from the monotonicity
property ||£αΜβ —
< ||χα. —
(see Theorem 2.3) and from the estimates (2.12), (2.13) with a = a*. If ο,με >
then the desired estimate (2.8) follows from (2.12) with A = < X M E and from the estimates (2.16)
and j*ö/y/oiME < 7*à/\/^**· Hence, the proof is complete.
•
35
On the ME rule for choosing the regularization parameter
2.4.
Newton's iteration for the ME rule in Tikhonov methods
Let us consider the methods of ordinary and iterated Tikhonov regularization (2.5). For choosing the regularization parameter a according to the D
principle, the RG rule and the ME rule, respectively, the following nonlinear
equations
do{oi) — Cô,
άϋο{οή — C6
and
cImeÌ®) = S
have to be solved numerically. For the iterative solution of these nonlinear
equations the change of the variable a by r = 1/a is reasonable since the
functions
dD(r) := dD{l/r),
dRG(r) := dRG(l/r),
:=
¿ΜΕ(Τ)
dME{l/r)
are monotonically decreasing and convex for all r > 0. These two properties
guarantee global monotone convergence, e.g., for Newton's iteration. For
the function dME{r) we prove the properties of monotonicity and convexity
in Theorem 2.5. In order to prove these properties for the functions cfo(r)
and dRo{r) we use the representations
dD(r) - \\R?y*\\,
dRG(r) = | | ^ + 1 / Υ I I
with Rr = (I + rAA*)~l, exploit the two identities
^ | | E r y II2 = -2k\\A*Rkr+^2ys\\2
,
II2 = —(2k + l ) | | ; L 4 * i ^ + y II2
dr
(2.17)
(2.18)
that hold true for arbitrary r > 0 and k > 0 (compare [35]) and obtain
d'D(r)< 0, 4 g ( 0 < 0 , 4 ( 0 > 0 and ¿'^(r) > 0.
(2.19)
For the proof of (2.19) see also [12] and [4]. In our next theorem we prove
that the function
dME(r)
= i i i c + i / y i i 2 / R m + v il
has analogous properties.
Theorem 2.5. The function dME(r)
analytic, strictly monotonically
decreasing and strictly convex. For all r > 0 there hold the estimates
d'ME(r) < -m\\A*RT+1yS\\2/\\RT+1yS\\,
(2-20)
36
U. Hämarik and U. Tautenhahn
·
d"ME{r) > m(m +ΐ)\^+1/2υψ\\ΑΑ^^+2νδ\\2/\^+1ν{\\3
(2.21)
P r o o f . Prom [35] we have that (2.20) holds true and that d'ME is given
by
d'uEir) = (m + l ) | | Ä r 1 V l l 2 | | ^ Ä r 3 / 2 » , l l a / l | Ä ? , + V l l 3
-(2m +
| | 2 | | A * J ^ + V | | 2 / | | J ^ + y | | 3 • (2·22)
In order to prove (2.21) we differentiate (2.22) as follows. We use (2.17),
(2.18) with k = m + 1/2 and k = πι + 1, respectively, apply the identity
= (3/2)||Ä™+y ||(-2m - 2)\\A*Rm+^2y5\\2
,
use the notations
h := i i i c + i / y i i 2 i i A * i c + 3 / y il4
t2 :=
\\A*R™+lys\\2\\A*R™+3/2ys\\2\\R™+lys\\2
t3 := | | i ^
+
y ||4||^*iC+3/V
f
and obtain
^ e ( r ) | | Ä r V l l 5 / ( m + l)
= 3(m + 1)¿1 - 2(2m + 1 )f 2 + 2(2m + l)i 3 - (2m + 3)t A
= 3(m + 1)(ίχ - Í2 + <3 - U) + {m-
1 )(ί 3 - t2) + mí 4 .
(2.23)
We use the abbreviation D := (AA*) 1 / 2 , apply the identity
\\z\\2 = \\RlJ2zf
with ζ = R™+1ys and ζ = DR™+ly6,
+ r\\DRlrl2z\\2
(2.24)
respectively, and obtain
i 3 - ta = | | J C + V l l 2 ( l l - ^ 3 / V l l 2 P ¿ * i ^ + 3 / V l l 2
-||A*iC+3/V||4) >0.
(2.25)
Repeated use of (2.24) yields
llDiRT+2-k/2y5[{2 =
£ (k)¿ai+j
j=o
with
a, = IIDiR? + 2 y 5 \\ 2 .
On the ME rule for choosing the regularization
37
parameter
We apply this formula for k = 1,2,3 and i = 0 , . . . , 3 — k and obtain
t\ - h + ¿3 - ti = (oo + 3rai + 3r2a2 + r3a3)(ai
- ( a o + 2ra\ + r2a2)(ai
+
ro2)2
+ 2ro2 + r 2 03)(ai +
ra2)
+(ao + 2ra\ + r a ) ( a 2 + ra 3 )
2
2 2
-02(00 + 1ra\ + r2a2)(a0
+ 3rai + 3r202 +
r3a$)
= (r 4 a 2 + 3r 3 ai + 3r 2 a 0 )(o3ai - 02)
+ r ( a j / 2 - O3 / 2 a 0 ) 2 + 2ra 0 oi[(a 3 ai) 1 / 2 - a 2 ] .
(2.26)
Prom the Cauchy—Schwarz inequality we have
αζαχ-αΐ
= \\DzR^+2ys\\2\\DR^+2y5\\2 - (D3R?+2ys,
DR?+2ys
)2
>0,
and due to (2.26) we obtain t\ — t2 + i 3 — Í4 > 0. From this inequality,
(2.25) and (2.23) we conclude that d"ME{r)\\Rf+lysf
> m(m + 1 )i 4 which
is equivalent to (2.21).
•
Let us rewrite the derivative (2.22) into some equivalent form which is
suitable for numerical computations. We apply the identity ra>k = ys —
Ax5a k = K*y& = R*ys with Ka = a(AA* + al)~l and obtain from (2.22)
d
'ME( r ) - [ ( m + l ) ( ^ a , m , r a , m + i ) ( A * r a > m + i , > l V Q ) m + 2 )
- (2m + 1)IIA*r e > m + 1 ||2||rQ,m+11|2] / ||r a , m+ i ||3
with a = 1 jr. This representation shows that the evaluation of the derivative d'ME(r) requires the computation of r a > m , ra¡m+i
and r a ¡ m + 2 · Note that
an efficient evaluation of the derivatives d'D (r) and d'RG (r) requires only the
computation of r Q ¡ m and r a ¡ m + 1 (see [4]). As an alternative to Newton's
method applied to the equation dMß(^) — δ = 0 one could use the secant
method in which the expensive evaluation of d'ME(r) is not necessary.
3.
ITERATIVE REGULARIZATION METHODS
3.1.
Iteration methods and the M E rule
For approximately solving linear ill-posed problems (1.1) with noisy data
ys Ç. Y we consider iteration methods of the general form
4
= 4-1 + ^*^-1,
η = 1,2,.
(3.1)
38
U. Hämarik and U. Tautenhahn
with zn 6 X and initial guess xf¡ = 0. The elements zn characterize the special iteration method. For example, zn = ß(ys — Axsn) with β G (0,2/||Α|| 2 ]
leads to the well-known Landweber iteration.
Iteration methods for approximately solving ill-posed problems are especially attractive for large scale problems (cf. [19]). Such problems arise
e.g. in the field of parameter identification in differential equations. Exploiting ideas from control theory, in such identification problems the elements
A*zn can effectively be computed by solving one direct problem and one
associated adjoint problem.
The iterates χδη of the iteration process (3.1) generally diverge. Nevertheless these iterates allow a stable approximation of χt provided the iteration
is stopped after an appropriate number of iteration steps. Two well-known
a posteriori rules of choosing the stopping index τι — n(S) are:
1. Morozov's discrepancy principle [27, 38, 39]. By this principle (D principle) the stopping index in (3.1) is chosen as the first index η = ηρ
satisfying
dD(n) :=\\rn\\<CS
with
rn = ys - Αχδη
and
C > 1.
(3.2)
2. Rule of Engl and Gfrerer [3]. This rule may be applied to iteration
methods (3.1) where zn has the special form zn = hn(ßn, AA*)rn with
rn — y5 — Axsn. Here hn is some operator function which depends on
ßn G M and ßn is allowed to depend on the noisy data ys. In this rule
(which we call EG rule) the stopping index in (3.1) is chosen as the
first index η = Η EG satisfying
dEG{n) :=
——
< CS
with
C > 1.
(3-3)
Here κ η is a constant with
Kn = SUp{h n (ß n , Λ) | 0 < λ < ||A|| 2 }.
(3.4)
In the monotone error rule (ME rule) for choosing the stopping index we
focus our attention on the monotonicity property (1.3). Our aim consists
in searching for a largest computable iteration number TIME for which (1.3)
can be guaranteed. Exploiting (3.1) and using the notation rn = ys — Αχδη
39
On the ME rule for choosing the regularization parameter
we obtain
114 - ^ l l 2 - 1 1 4 - 1 - * f ll 2 = 114-1 +
= 2 ( 4 _ ! - x \ A*Zn^)
- * f ll 2 - I l 4 - i - ^ l l 2
+ P^n-lll2 = ( 4 - 1 + 4
-
á
= (2(y -y)-(rn_i+rn),zn_i)
fn—1 ι rnizn-l
)/(2|k-il|)].
(3.5)
This estimate leads us to the following ME rule for iteration methods (3.1):
M E rule. Choose
as the first index
TIME
Η
satisfying
dME{n) ·.= (Γη + Γη+ι,2 η )/(2||2 η ||) <δ.
(3.6)
By this a posteriori choice of the stopping index in iteration methods (3.1) the monotonicity property (1.3) can be guaranteed:
P r o p o s i t i o n 3.1. Let ||ζ π || φ 0 for η = 0 , 1 , 2 , . . . and let time be
chosen by the ME rule (3.6). Then the monotonicity property (1.3) holds
true.
P r o o f . From the ME rule for iteration methods we have that ÚmeÍ^ —
1) > δ for η = 1 , 2 , . . . ,ΠΜΕ• Consequently, from (3.5) there follows for all
η = 1 , 2 , . . . , TIME that
I l 4 - z f l l 2 < \\χδη_1-χΗ\2
which completes the proof.
3.2.
< | | 4 - ι - ^ Ι Ι 2 (3-7)
+ 2\\ζη-1\\{δ-άΜε(η-1)}
•
T h e M E r u l e in g r a d i e n t t y p e m e t h o d s
Let us consider gradient type methods of the form (3.1) with zn = βnrn
where rn = ys — Αχδη is the discrepancy and βη > 0 is some properly chosen
stepsize. For such methods the iteration (3.1) attains the form
4 = 4 - 1 + ßn-iA*(ys
- Αχ*η_λ),
η - 1,2,... .
(3.8)
Since in the EG rule hn(ßn, λ) = βη, for the constant κη of (3.4) we have
^n — βη· Consequently, the functions dEG{n) and c/mjs(«) of the EG- and
ME rules, respectively, attain the form
j / \
\rn + ^Vi+1) rn)
dEG{n) =
j=
V2
ι
and
ι
/ \
\rn + rn+li rn)
dME{n) =
tt¡—¡¡
.
2||r n ||
For gradient type methods (3.8) following properties are valid:
/o n\
(3.9)
40
U. Hämarik and U. Tautenhahn
Theorem 3.2. Let Ρ denote the orthoprojection of Y onto Ν (A*) =
.R(A) -1, let A*ys φ 0 and let βη be chosen such that
0<ßn<\\A*rn\\2/\\AA*rn\\2.
(3.10)
Then for the iterates of (3.8) following properties are valid:
(i) The function dß(n) = ||r n || is strictly monotonically decreasing and
obeys
Ikn+ill2 < ( r n , r n + i ) < I M 2 .
(ii) The functions dME(n) and d^G(n) are strictly monotonically decreasing and obey
do(n + 1) < dME{n) <
< cfo(n).
(Hi) Let ßn > c > 0 with some positive constant c, then
lim dMEÍ n ) = l i m άκοίη) = li m d o M = Il Pi/II.
n—>00
n->oo
η-too
(3.11)
(iv) If\\Pys\\ < CS, then the stopping indices np and ueg are well defined.
For C = 1 also ume is well defined and there holds
no - 1 < time < riEG < «£> ·
(v) If\\Axsn-ys\\>0,
then
114-^11 <114-1-^11·
Proof. From (3.8) we conclude that r n +1 — (I — ßnAA*)rn.
quently,
Conse-
= \\rn\\2 - ßn\\A*rn\\2 ,
(a)
(rn,rn+i)
(b)
||r n + 1 || 2 = ||r n || 2 - 2ßn\\A*rn\\2 + ß2\\AA*rn\\2 .
From (a), ßn > 0 and A*y5 φ 0 we obtain the right inequality of (i). Combining (a) and (b) we have
Ikn+ill2 = ( r „ , r n + 1 ) -ßn\\A*rn\\2
+
ß2JAA*rn\\2.
From this equation and assumption (3.10) we obtain that the left inequality
of (i) holds true. In order to prove the left inequality of (ii) we use the
On the ME rule for choosing the regularization parameter
41
inequality 2ab < a2 + b2 with α φ b and obtain with the help of the left
inequality of (i) that
2||r„||||rn+1|| < ||rn||2 + ||r n+ i|| 2 < \\rn\\2 +
{rn,rn+1),
which is equivalent to d,£>(n+1) < <¿M£(")· The middle and right inequalities
of (ii) are equivalent to ( r n , r n + i ) < ||r„|| 2 and follow from part (i). For
proving (3.11) we proceed according to (3.5) and obtain for arbitrary w E X
and arbitrary iteration methods (3.8) with βη > 0 that
114 - w\\2 - 114.! - w\\2 = (2{y s - Aw) - (r n _i + r„), A , _ i r n _ i )
< 2 / ö n _ 1 ||r n _ 1 ||{||y i - Aw\\ - dME(n
- 1)} .
(3.12)
For iteration methods (3.8) with stepsizes ß n satisfying (3.10) there holds
INI2
- ||rn+1||2 = ßn {2\\A*rn\\2 - ßn\\AA*rn\\2)
> ßn\\A*rn\\2 .
Passing to the limit on both sides yields due to ßn > c > 0 that
l i m ^ o o ||A*r„|| = 0. From ys = Axsn + rn and the Cauchy—Schwarzinequality we have
||rn||2 - IIA«; -ySW2 = 2{rn, Aw - Αχδη) - \\Αχδη - Aw\\2 < 2\\A*rn\\\\x{ - ω||.
(3.13)
Now we will use a contradicition argument and assume that linin-j.oo άη(η) >
llPy^ll· Under this condition there exists some element w G X with the
property that
lim ||r„|| > \\Aw — j/ 5 ||.
(3.14)
n—too
From (3.14) and (i) we have ||r n || > ||Aw — ys\\ for all η G Ν. This property
provides together with (3.12) and (ii) that
IIxi - HI 2 - 114-1 - HI 2 < 2Ai-i||r„-i|| {||rn|| - dME{n - 1)} < 0 .
Hence, | | 4 — ^11 is bounded by ||ω|| for all η G Ν. Passing to the limit on
both sides of (3.13) provides due to | | 4 — H I — ll^ll a n d hmn^oo m * r n | | = 0
that limn-xx, ||r n || < ||Aw — ys\\. This contradicts (3.14) and shows that our
assumption limn-^oo άο(η) > ||Py á || cannot hold true. Hence, due to (ii)
we obtain (3.11). Assertion (iv) follows from (3.11), the definition of the
stopping rules and (ii). In order to prove (v) we use the first part of (3.7),
property (ii) as well as the assumption d,£>{n) > δ and obtain
114 - χψ
- 114.! - xt||2 < 2||z n _ 1 ||{¿ - dME{n
- 1)}
<2||zn_i||{Ä-dß(n)} < 0 ,
which finishes the proof.
•
(3.15)
42
U. Hämarik and U. Tautenhahn
R e m a r k 3.1. The limit relation (3.11) can not only be guaranteed for
strictly positive stepsizes ßn > c > 0, but also for stepsizes tending to zero
not too fast and satisfying
1
0 < /3i < Tj-rfis·
PII2
and
71
lim V ß { = oo.
For the proof of this result we use the identity rn+1 — {I — ßnAA*)rn
gives
η
rn+1 =rn(AA*)ys
with rn(\) = J ] ( l - β λ ) .
i=0
Hence, for η —> oo we have for all Λ Ε (0, ||Λ||2]
|r n (A)|->0
η
lnJJ|l-ßA|->-oo
i=0
which
η
¿=0
- βλ|
-oo.
Using the estimate In |1 —
< —ξ for ξ G [0,1] with ξ = ßiX we obtain the result that for 0 < ßi < 1/||Λ||2 there holds |rn(A)| - » 0 provided ΣΊ—ο ßi —> oo. Finally we conclude that from |rn (λ) I
0 for all
2
Ä
λ G (0, ||A|| ] there follows lim^oo ||rn|| = ||P3/ || and the proof is complete.
R e m a r k 3.2. Part (v) of Theorem 3.2 shows that the iteration (3.8)
should not be stopped as long as \\Ax^ — ys\\ > δ holds. Let us modify the D
principle as follows: Choose η = no as the first index η satisfying
\\Axi+l-yS\\<CS
with
C>
1.
Let πχ and «2 the stopping indices of this modified D principle with C — C\
and C = 02·, respectively. If 1 < C\ < C2, then, since dp (a) is monotonically decreasing there follows n\ > «2, and due to part (v) of Theorem 3.2
we obtain
—
< ||x£ —
Hence, the best possible choice for the
constant C in this modified D principle is C = 1.
Before we will study some special methods that fit into the framework of
Theorem 3.2 let us derive some useful inequality that is helpful for checking
condition (3.10).
Proposition 3.3. Let D € C(Y, Y), D = D* > 0 and η > 0. Then for
all ν Ε Y
Il-D^IHI-Dwll < ||£)7?+1υ||||υ||.
(3.16)
On the ME rule for choosing the regularization
parameter
43
Proof. We apply the moment inequality (2.14), first with s — η and
t — η + 1, second with s = 1 and t = η + 1 and obtain the inequalities
\\Όην\\ < Ι Ι ^ + ^ ρ / ^ Ι Η Ι 1 / ^ ) and ||D«|| < { { Ό ^ υ ^ ^ Μ ^ ^
We multiply both inequalities and obtain (3.16).
.
•
Now we are ready to study some special gradient type methods (3.8)
that fit into the framework of Theorem 3.2. In the methods M3 - M5 below
we assume that ys is not an eigenelement of the operator AA* since in the
opposite case there follows ri = 0 and the stepsizes ßn, n = 1 , 2 , . . . are not
defined.
Method Ml: Landweber's method with ßn = β e (0,1/||Λ||2]. This
method may actually be applied with stepsizes ßn = β E (0,2/||Λ.||2), see
[1, 4, 5, 13, 17, 23, 25, 28, 30, 38, 39]. However, due to the inequality
||·Α·Α*7"η|| < ||A||||A*rn|| we realize that condition (3.10) and hence the results
of Theorem 3.2 hold true for β G (0,1/P|| 2 ].
Method M2: Nonstationary Landweber's method. This method is characterized by (3.8) with variable ßn G [c, 1/||Λ||2] and c > 0, see [32], As
in method Ml we conclude that condition (3.10) and hence the results of
Theorem 3.2 hold true for ßn G [c, 1/||Λ||2].
Method Μ3: Steepest descent method (see [1, 2, 7, 9,10,11,16, 20, 22, 31,
32]). This method is characterized by (3.8) with ßn = ||AVn||2/||>L4*rn||2.
Since this stepsize satisfies (3.10), the results of Theorem 3.2 hold true for
this method.
Method M4: α-processes with stepsizes ßn = \\Da+1rn\\2/\\Da+2rn\\2,
D = (AA*)1/2 and a > 0. These methods may actually be applied with
a > —1, see [10, 16, 22, 31, 32]. However, assumption (3.10) holds true
only for a > 0. In order to check (3.10) we apply inequality (3.16) with
D = (AA*)1/2, ν = (AA^^rn
and η — a and obtain the inequality
\\ϋ«+^η\\\\Ό2Γη\\ < ||-Οα+2Γη||||£>Γη|| which gives ßn < ||£rn||/||£>2rn||.
Since furthermore ßn > ||Da+1rn||2/(j|£)||2||£>a+1rn||2) = 1/P|| 2 we obtain
l/\\A\\2<ßn<\\A*rn\\2/\\AA*rn\\2
(3.17)
and hence (3.10). Since the application of (3.16) requires η > 0 we realize
that Theorem 3.2 holds true for ct-processes with a > 0.
Method M5: Method (3.8) with ßn = max{||Z)a+2rn||2/(||.D||4x
IIDa+1rn\\2),ß}
where D = (AA*)1!2, β 6 (0,1/||A||2] and α > - 1 (see
44
U. Hämarik and U. Tautenhahn
[16, 32]). For this method there holds
From this estimate it follows as in method Ml that condition (3.10) and
therefore the results of Theorem 3.2 hold true.
Note that method M3 is a special case of α-processes with a — 0. In
this method the stepsize βη minimizes the functional g{ßn)
||^n+i||2 =
\\{I-ßnAA*)rnf.
The results of Theorem 3.2 can be used to establish convergence- and
convergence rate results for the above discussed a posteriori stopping rules
in iteration methods M1-M5.
Theorem 3.4. Assume A*ys φ 0 and ||Py á || < δ. Letxsn the regularized
approximation obtained by one of the methods Ml - M5 and let NO, UEG
and TIME the stopping indices according to the discrepancy principle with
C > 1, the EG rule (3.3) with C > 1 and the ME rule (3.6), respectively.
Then for all η G {nn^MEi^EG}
there holds:
(i) 1 1 4 - ^ 1 1 —> 0 for <5 —> 0.
(ii) I f x t e R{{A*A)P/2),
then
114-^11
= 0(áp/(í,+1))
for all p> 0.
Proof. First, let us discuss the case C > 1. In this case the proof for
η = n o is known from the literature, see [38] for method Ml, [31, 32] for
methods M2-M5 and [9, 10] for method M3. Let ΠΜΕ,Ο the stopping index
of the ME rule (3.6) with δ replaced by Cö. From the validity of Theorem 3.4
for η = n o and part (ii) of Theorem 3.2 we obtain that assertions (i), (ii)
are valid for TIEG and TIME,C
as well. Second, let us consider the case
C = 1. From the monotonicity of <1ME{N) we obtain that ΠΜΕ,Ο < i m e , ι ·
Consequently, due to Proposition 3.1 there follows
Hence, the assertions (i) and (ii) of the theorem are true for Η =
TIME,I·
From this result and part (iv) of Theorem 3.2 we conclude that assertions
(i) and (ii) of the theorem are also valid for Η = n p and Η = TIEGΠ
Some weaker results compared with the results of Theorems 3.2 and 3.4
can be proved for method M4 with α G [—1,0). This class of methods
On the ME rule for choosing the regularization
parameter
45
contains for a = — 1 the minimal error method in which the stepsize ßn is
given by ßn = ||rn||2/||A*rn||2, see [1, 2, 6, 9, 10, 11, 16, 22, 31, 32], In the
minimal error method the stepsize ßn minimizes the norm
— A~lys\\
l
5
provided A~ y exists. For this method there holds (rn,rn+1)
= 0 which
shows that the left inequality of part (i) of Theorem 3.2 generally does not
hold.
T h e o r e m 3.5. Let A*y5 φ 0. Then in method
following properties are valid:
(i) For all η Ε Ν there
dD(n)/2
M4 with a e [ - 1 , 0 )
holds
< dEG(n)/V2
< dME{n)
<
<
dD{n).
(ii) Denote by UME,C the stopping index of the ME rule with δ replaced by
CS. Then, IÏME,C j s weJi defined for C > 1. In addition, UEG,C '·— KEG
is well defìned for C > y/2 and η^β '•— nD is well defìned for C > 2.
For arbitrary C > 1 there holds
ND,2C
(iii) For η — np^c,
<
NEG,V2C
η — nßG V2c
-
NME,C
<
<
an<^ n = nME,C w¡th
Wxi-x^W - > 0
for
·
(3.18)
C > 1 there
holds
¿->0.
P r o o f . By elementary computations it can be shown that the first two
inequalities of assertion (i) are equivalent to (rn,rn+1) > 0. This inequality,
however, is equivalent to
ßn < i|r„ii7llAvn||2
and follows from (3.16) with η = α + 1 and ν = rn.
The final two
inequalities of assertion (i) are both equivalent to ( r n , r n + i ) < ||rn||2.
This inequality, however, holds for arbitrary stepsize ßn > 0 since due to
rn+1 = [I - ßnAA*]rn and A*y& φ 0 we have
( r „ , r „ + i ) = ||r„||2 -^η||Ανη||2 < ||rn||2.
Now let us prove indirectly that for C > 1 there exists a finite stopping index
UME,C· For this aim we assume that D^EIP) > CS for all η G Ν. Then, due
to (3.5), ||x£ — art J! <
— 1|, and the limit lim^oo \\x„ — zt|| exists.
U. Hämarik and U. Tautenhahn
46
We summize the inequalities (3.5) with zn
due to Zq = 0 that
\\χψSince
the
lim
T—>00I
η
right
.
hand
= ßnrn
for all
η>
1 and obtain
2
\\xi-xi\\
>2Y
ß \\rn\\{dME(n)-ö}.
—
•
• / n
n=0
side
is
finite
there
follows
that
Due to ßn > 1 / P | | 2 , ||r n || > dME(n)
(compare (i)) and the assumption dME(n) > CS there follows
lining ßn\\rn\\{dME{n)-δ} = 0.
A.||r„H > lknll/ΐμΐΙ 2 >
dME{n)/\\A\\2 > Οδ/\\Α\γ .
Consequently, lim^oo
— = 0. This contradicts our assumption
dME{n) > C6 for all η G Ν and proves that there has to exist some finite
stopping index ume,c for C > 1. Hence, due to the first two inequalities
of (i) there follows that ueg,c is well defined for C > y/2 and η ρ β is well
defined for C > 2. Now the proof of (3.18) follows from the definition of
the corresponding stopping rules and (i). In order to prove (iii) we conclude
n
from riDflC < nEG ^c — ME,c a Q d Proposition 3.1 that
1
1
\\xi
11 n
| | xIIi n G,V2C
- a ^ l l 11< —liar'
nME,C- x | II| <—
o,2C "
E
Since assertion (iii) holds true for no,2C with C > 1 (see [10]), we
that (iii) holds also true for nEG
and ume,c with C > 1.
•
t
conclude
Remark 3.3. The estimate (3.5) which led us to the ME rule can
also be exploited for finding stepsizes βη > 0 in iteration methods (3.8)
which guarantee that χδη+χ is a better approximation for x^ than x&n. Here
the stepsize βη may not only depend on ys, but also on the noise level δ.
Exploiting (3.7), (3.6) and r n + i = (I — ßnAA*)rn we obtain for βη > 0
iFn+l
x til
2
-
-
χψ < 2βη\\τη\\{δ -
|2 .
IMI +
ß n ^ ^ }
·
(3.19)
The right hand side of (3.19) is negative for ||r n || > δ and 0 < βη <
(2||r n ||/||>l*r n || 2 )(||r n || — δ) which shows that for such stepsizes the element
xsn+i is a better approximation for x¡ than xsn. Minimizing the right hand
side of (3.19) with respect to β η yields
βη = {\\τη\\/\\Α*τη\\2){\\τη\\-δ).
Substituting into (3.19) shows (see also [1], p. 69) that for this stepsize the
improvement of the squared error can be estimated by
llxjUl - ^ n 2 - 1 1 4 - ^ l l 2 < - ( I k n l l 7 p * r n | | 2 ) ( | | r n | | _ & f .
On the ME rule for choosing the regularization
47
parameter
Some gradient type methods that do not fit into the class of methods (3.8) are conjugate gradient methods. The conjugate gradient method
for the normal equations A* Ax = A*ys (see [1, 2, 9, 10, 11]) is known as
powerful method for the approximate solution of ill-posed problems. The
advantage of this method over methods of type (3.8) consists in the fact
that the stopping index is generally much smaller. In the standard variant
of this method (see [1], p. 51) the regularized approximations x5n have the
form
^n+l
= x
n "I" ßnPn ι
ßn = {A*rn,pn)/\\Apn\\2
Pn = A rn + 7n-lPn-l j
,
r
n = if ~ Axn ,
7„_1 = ||A*r n || 2 /pV n _i|| 2 ,
Po
= A*r0 .
It can easily be realized that this method fits into the framework of methods (3.1). Hence, the ME rule (3.6) for choosing an appropriate stopping
index can be applied. In [1] it is shown that for this method the ME function
(3.6) has the form
,
iME{n) =
I M ' + lh+ill 2 TT2
¿
1
P
^
/ A IMI
/ à P ^ '
This representation for dME{n) has been used in [1] to prove that the regularized approximation x&nME converges to x^ for δ —> 0. Unfortunately, we
don't know results concerning convergence rates although such results are
known for the D principle (see [4]).
3.3.
The M E rule in implicit iteration methods
Let us consider implicit iteration methods of the form (3.1) with zn =
hn(ßn,
AA*)rn
= (ßnI + AA*)~lrn
where rn = ys — Αχδη is the discrep-
ancy and βη > 0 is some suitably chosen real number. For such methods
the iteration (3.1) attains the form
+ A*(AA* + ßn-il)-l{ys
xi -
= (A*A + ßn-iI)-Hßn-ixi-i
-
Axi^)
+¿V),
η - 1,2,... . (3.20)
For methods (3.20) the constant κη in (3.4) is given by κη = β'1. From
(3.20) we obtain that rn = βη-ι (AA* +ßn-1I)-1 rn-χ. Consequently, the
element zn = (ßnI+AA*)~1rn
has the form zn = ß~lrn+i and the functions
dEG{n) and
of the EG- and ME rules of Subsection 3.1 attain the
form
dEG{n)
=
^
and dME{n)
=
,
(3.21)
48
U. Hämarik and U. Tautenhahn
respectively. For implicit iteration methods with arbitrary positive parameters βη > 0 following properties are valid:
T h e o r e m 3.6. Let A* yδ φ 0 and βη > 0 arbitrary. Then for the iterates
of (3.20) following properties are valid:
(i) The function d£>(n) = ||r„|| is strictly monotonically decreasing and
obeys
Ikn+ill2 < (rn,rn+1) < ||r n || 2 .
(ii) The functions d,ME(n) and d,EG{n) are strictly monotonically decreasing and obey
(a)
dD(n + 1) < άΕσ(η) < dME(n) < dD(n)
(b)
lim
η—>oo
) = lim ¿ e g M = lim dpin)
η—too
η—too
(iii) Let lim^^oo do (η) < CS and C > 1. Then the stopping indices tid
and time are well defìned. If C = 1, then also time is well defìned and
there holds
nu — 1 < UEG < KME < nD •
(iv) If\\Ax5n-ys\\
> δ, then
114 -® f ll < 1 1 4 - 1 - ^ 1 1 ·
Proof. From (3.20) we conclude that rn = (β~ιΑΑ* + I)rn+\.
quently,
(a)
( r n , r n + 1 ) = ||r n + i|| 2 + ß?\\A*rn+1\\2
(b)
||r n || 2 = \\rn+1\\2 + 2ß-^A*rn+1\\2
Conse-
,
+
ßn2\\AA*rn+1\f.
From (a), (b) and A*y5 φ 0 we obtain assertion (i). From (3.21), the CauchySchwarz inequality, the triangle inequality and inequality (i) we obtain
dME(n)
:=
(r +
"
+ 1
n+l)
2 | [; n + ;¡'
< ì ( | r . | | + ||r„ + i||) < ||r B || =
dD(n).
Hence, the right inequality of (ii) holds true. The two other inequalities
of (ii) are equivalent to ||r n + i|| 2 < (rn,rn+i) and follow from part (i). Now
part (b) of (ii) is a consequence of part (a) of (ii) and assertion (iii) of the
On the ME rule for choosing the regularization parameter
49
theorem is a consequence of assertion (ii). The proof of (iv) can be done in
analogy to the proof of part (iv) of Theorem 3.2.
•
As in explicit iteration methods, property (iv) of the theorem shows
that also in implicit iteration methods (3.20) with arbitrary ß n > 0 the
iteration should not be stopped as long as ||Axsn — y&|| > δ holds. Here
the same discussion as in Remark 3.2 can be made. Further note that the
limit lim^oo dj}(n) always exists since du(n) is monotonically decreasing
and bounded by zero. However, under the additional condition
η 1
lim V - = oo,
(3.22)
which is necessary for convergence xn —> x^ for η -» oo in the case of exact
data, it can be shown that Πη^-κ*, d£>(n) = H-Pj/ll where Ρ denotes the
orthoprojection of y onto N{A*) = R(A)1 (cf., e.g., [18]). Condition (3.22)
excludes that the parameter sequence {β η } is growing faster than the sequence {n a } with a > 1.
Our next aim in this section is to establish convergence and convergence
rate results for implicit iteration methods (3.20) provided the stopping index
is chosen from the D principle, the EG rule and the ME rule, respectively.
For the D principle there is known that the proof of convergence rates can
be done under the additional assumption
1
Pn
n_1
j=0
1
Pi
for some
c>0
(3.23)
(see [18]). These results and the results of Theorem 3.6 allow the proof of
convergence and convergence rate results for the EG- and ME rules.
Theorem 3.7. Assume A*yδ φ 0, ||Py 5 || < δ and (3.22). Let χδη be the
regularized approximation obtained by (3.20) and let no, η eg and time be
the stopping indices according to the discrepancy principle with C > 1, the
EG rule (3.3) with C > 1 and the ME rule (3.6), respectively. Then for all
η £
there holds:
(i) \\xsn - ®t||
o for δ
0.
(ii) If xt G R((A*A)P/2) and (3.23) hold, then
Wxi-x^W = 0(<J p/(p+1) )
for all p> 0.
50
U. Hämarik and U. Tautenhahn
P r o o f . The proof is along the lines of the proof of Theorem 3.4 and uses
that for C > 1 the results of the theorem for η = η ρ are known from [18].
•
Now let us study some special implicit iteration methods (3.20) that fit
into the framework of Theorem 3.7.
Method M6: Stationary implicit iteration method (see [13, 24, 28, 30,
38, 39]). This method is characterized by (3.20) with fixed βη := β > 0.
Obviously, condition (3.22) holds and condition (3.23) is satisfied with c = 1.
Hence the results of Theorem 3.7 hold true for this method.
Method M7: Nonstationary implicit iteration method with ßn = ßqn,
β > 0 and q 6 (0,1). For this method condition (3.22) can easily be checked
and condition (3.23) holds true with c = 1/q (see [18]). Hence, Theorem 3.7
applies for this method.
Method M8: Nonstationary implicit iteration method with ßn G [ci,c2¡,
0 < ci < C2- For this method (see [32]) property (3.22) is valid due to
β η < C2- From
n-l
we obtain (3.23) with c = ci¡c\. Consequently, Theorem 3.7 applies for this
method.
Method M9: Nonstationary implicit iteration method with
ßn = m^{\\Da+2rn\\2/\\Da+1rn\\2,ß},
β>0,
α >-1
and D = (AA*) 1 / 2 . A modification of this method with a = — 1 and without
lower bound β has been studied in [29, 33, 37]. Method M9 (see [16, 32, 37])
is a special case of method M8 with
ß<ßn<max{\\D\\2,ß}
.
(3.24)
The inequalities (3.24) follow from the definition of β η . Hence, the results
of Theorem 3.7 hold true for this method.
4.
NUMERICAL EXPERIMENTS
In this section we summarize some of our numerical results for integral
equations of the first kind
< s< 1
(4.1)
On the ME rule for choosing the regularization parameter
Table 1. Errors in the method of ordinary Tikhonov regularization
δ
eD
efiG
IO-1
IO-2
IO"3
IO"4
IO -5
.05878
.02332
.00896
.00338
.00119
.05982
.01905
.00589
.00183
.00055
35
35
eME eD/öV2 eRo/δ ' eME/ö '
.05355
.01710
.00527
.00163
.00049
.1859
.2332
.2833
.3380
.3763
.2132
.2710
.3325
.4094
.4900
.2381
.3019
.3716
.4597
.5500
in a L 2 -space setting with X = Y = L 2 (0,1). We discretized problem (4.1)
by the collocation method with 100 piecewise constant spline basis functions on a uniform mesh. Instead of y randomly perturbed data ys with
\\y — ysII < δ have been used. Our test problem is taken from [26] and corresponds to equation (4.1) with
p 2
For this test problem we have art e R((A*A) ! ) for all ρ Ε (0,3/2).
In a first experiment the discretized problem was regularized by the
method of ordinary Tikhonov regularization (2.5) with πι = 1. The regularization parameter a has been chosen according to the ME rule, the RG rule
and the D principle, respectively. For C we have used the constant C = 1.
In Table 1 the errors
:t||
and
βΜΕ =
<
.5
—X
&ME
are given. The results in Table 1 verify the theoretical results of the estip 2
mate (2.8) of Theorem 2.2 which tells us that for x* e R{(A*A) / ) with
ρ < 2 there holds e ^ E = 0(S P ^ P+1 ^). Table 1 also illustrates the well known
ρ+1
1 2
results that for 1 < ρ < 2 there holds eRG = 0(δΡ^ ^) and eD = Ο {δ 1 ).
In addition we observed that always cme < eRG holds true.
In a second experiment we solved the discretized problem by one explicit
and one implicit iteration method, namely by Landweber's method with
β = 1 / | μ | | 2 and by the stationary implicit iteration method with β = 1.
Both iterations were stopped with index n* +10, where n* is the first η with
ll^n+i ~ 2fll >
— a;t||. In all experiments the final inequality appeared
to be true for all η = η* + 1 , . . . , η* + 10. In Table 2 comparisons of the
indices time, n * and the corresponding errors
eME ' I\XnME - ^11
and
e
* = I\χη. ~
51
52
U. Hämarik and U. Tautenhahn
Table. 2. Indices
TIME,
n
* and errors
CME-,
Landweber's method
δ
IO- 1
2
IOIO- 3
IO -4
IO- 5
e» in iteration methods
Implicit iteration method
n\fE
n*
ZME
e»
TIME
η*
βΜΕ
e.
15
16
.0630
.0621
32
41
.1131
.1060
38
160
947
4794
67
332
1545
8643
.0453
.0137
.0033
.0009
.0400
.0111
.0030
.0008
103
495
2422
12951
166
750
3741
19667
.0406
.0097
.0031
.0008
.0356
.0087
.0029
.0008
are given. Note that in these and in all other numerical experiments we
observed that TIME < N* and TIME = ηd or TIME = ηd — I , which is in
agreement with part (iv) of Theorem 3.2 and part (iii) of Theorem 3.6. We
made some further experiments in which instead of random perturbations
some special perturbed data y δ have been used. In these experiments we
observed essentially smaller quotients (n* — TIME)/N*
compared with those
which follow from Table 2. For some further numerical experiments for
comparing the ME rule with the RG rule and the D principle in the method
of ordinary Tikhonov regularization see [14, 16, 21, 35].
Acknowledgement
The authors acknowledge assistance from Reimo Palm, University of Tartu,
for providing a MATLAB-code for the numerical computations. Parts of
this work have been done during a visit of the first author at the University
of Applied Sciences Zittau/Görlitz (June-July, 2000).
REFERENCES
1. 0 . M. Alifanov, E. A. Artjukhin, and S. V. Rumjancev, Extremal Methods for Solving Ill-Posed Problems and its Applications to the Inverse
Problems of the Heat Exchange. Nauka, Moscow, 1988 (in Russian).
2. Ο. M. Alifanov and S. V. Rumjancev, On the stability of iterative
methods for the solution of linear ill-posed problems. Soviet Math.
Dokl. (1979) 20, 1133-1136.
3. H. W. Engl and H. Gfrerer, A posteriori parameter choice for general
regularization methods for solving linear ill-posed problems. Appi. Numer. Math. (1988) 4, 395-417.
On the ME rule for choosing the regularization
parameter
53
4. H. W. Engl, M. Hanke and A. Neubauer, Regularization of Inverse
Problems. Kluwer, Dordrecht, 1996.
5. V. M. Fridman, The method of successive approximations for the Fredholm integral equation of the first kind. Uspekh. Mat. Nauk (1956) 11,
233-234 (in Russian).
6. V. M. Fridman, New methods of solving a linear operator equation.
Dokl. Akad. Nauk SSSR (1959) 128, 482-484 (in Russian).
7. V. M. Fridman, On the convergence of methods of steepest descent
type. Uspekh. Mat. Nauk (1962) 17, 201-208 (in Russian).
8. H. Gfrerer, An a posteriori parameter choice for ordinary and iterated Tikhonov regularization of ill-posed problems leading to optimal
convergence rates. Math. Comp. (1987) 49, 507-522.
9. S. F. Gilyazov, Methods for Solving Linear Ill-Posed Problems. Moscow
Univ., Moscow, 1987 (in Russian).
10. S. F. Gilyazov, Approximate
Solution of ΠΙ-Posed Problems. Moscow
Univ., Moscow, 1995 (in Russian).
11. S.F. Gilyazov and N. L. Goldman, Regularization
of Ill-Posed
Prob-
lems by Iteration Methods. Kluwer, Dordrecht, 2000.
12. V. I. Gordonova and V. A. Morozov, Numerical parameter selection
algorithms in the regularization methods. USSR Comp. Math.
Math.
Phys. (1974) 13, 1-9.
13. U. Hämarik, Monotonicity of error and choice of the stopping index
in iterative regularization methods. In: Differential and Integral Equations: Theory and Numerical Analysis. A. Pedas (Ed). Estonian Math.
Society, Tartu, 1999, 15-30.
14. U. Hämarik and T. Raus, On the a posteriori parameter choice in
regularization methods. Proc. Estonian Acad. Sci. Phys. Math. (1999)
48,133-145.
15. U. Hämarik and U. Tautenhahn, The monotone error rule for parameter choice in regularization methods. In: ENUMATH 99-Proc. of the
3rd European
Conf. on Numer.
Math, and Advanced Appi. P. Neit-
taanmäki, T. Tiihonen, and P. Tarvainen (Eds). World Scientific, Singapore, 2000, 518-525.
54
U. Hämarik and U. Tautenhahn
16. U. Hämarik and U. Tautenhahn, On the monotone error rule for parameter choice in iterative and continuous regularization methods. BIT
(2001) 41, 1029-1038.
17. M. Hanke, Accelerated Landweber iterations for the solution of illposed equations. Numer. Math. (1991) 60, 341-373.
18. M. Hanke and C. W. Groetsch, Nonstationary iterated Tikhonov regularization. J. Optimization Theory Appi. (1998) 98, 37-53.
19. M. Hanke and P. C. Hansen, Regularization methods for large-scale
problems, Surv. Math. Ind. (1993) 4, 253-315.
20. W. J. Kammerer and Μ. Ζ. Ν ashed, Steepest descent for singular linear
operators with nonclosed range. Appi. Anal. (1971) 1, 143-159.
21. E. M. Kiss, Ein Beitrag zur Regularisierung und Diskretisierung des
inversen Problems der Identifikation des Memorykernes in der Viskoelastizität. Dissertation. Techn. Univ. Bergakademie Freiberg, Shaker,
Aachen, 2000.
22. M. A. Krasnoselskij, G. M. Vainikko, P. P. Zabreiko, et al, Approximate Solution of Operator Equations. Wolters-Noordhoff, Groningen,
1972.
23. L. Landweber, An iteration formula for Fredholm integral equations
of the first kind. Amer. J. Math. (1951) 73, 615-624.
24. L. J. Lardy, A series representation for the generalized inverse of a
closed linear operator. Atti della Accademia Nazionale dei Lincei, Rendiconti della Classe di Science Fisiche Matematiche, e Naturali. Serie Vili (1975) 58, 152-157.
25. M. M. Lavrent'ev, Some Improperly Posed Problems of Mathematical
Physics. Springer, New York, 1967.
26. A. K. L. Louis, Inverse und schlecht gestellte Probleme.
Stuttgart, 1989.
Teubner,
27. V. A. Morozov, On the solution of functional equations by the method
of regularization. Soviet Math. Dokl. (1966) 7, 414-417.
28. V. A. Morozov, On the regularizing families of operators.
Metody Program (1967) 8, 63-95 (in Russian).
Vychisl.
On the ME rule for choosing the regularization parameter
55
29. S.V. Pereverzev and E. Schock, Brakhage's implicit iteration method
and information complexity of equations with operators having closed
range. J. Complexity (1999) 15, 881-890.
30. T. Raus, On the discrepancy principle for solution of ill-posed problems
with non-selfadjoint operators. Acta et Comment. Univ. Tartuensis
(1985) 715, 12-20 (in Russian).
31. L. Sarv, A certain family of nonlinear iterative methods for solving
ill-posed problems. Izv. Akad. Nauk Est. SSR. Fiz. Mat. (1982) 31,
261-269 (in Russian).
32. L. Sarv, Iteration Methods for Linear Ill-Posed Problems. Dissertation.
Sverdlovsk, 1984 (in Russian).
33. V. I. Starostenko and S. M. Oganesjan, Regularizing iterative processes. In: Methods for Solution of Ill-Posed Problems and its Applications. A.N. Tikhonov (Ed). Novosibirsk, 1982, 119-128 (in Russian).
34. U. Tautenhahn, On a new parameter choice rule for ill-posed inverse
problems. In: Hellenic-European Research on Computer Mathematics and its Applications. E. A. Lipitakis (Ed). Lea Publishers, Athens,
1999, 118-125.
35. U. Tautenhahn and U. Hämarik, The use of monotonicity for choosing
the regularization parameter in ill-posed problems. Inverse Problems
(1999) 15, 1487-1505.
36. A. N. Tikhonov and V. Y. Arsenin, Solution of Ill-Posed
Wiley, New York, 1977.
Problems.
37. V. N. Trushnikov, A non-linear regularizing algorithm and some of its
applications. USSR Comp. Math. Math. Phys. (1979) 19, 22-31.
38. G. M. Vainikko, The discrepancy principle for a class of regularization
methods. USSR Comp. Math. Math. Phys. (1982) 22, 1-19.
39. G. M. Vainikko and A. Y. Veretennikov, Iteration Procedures in IllPosed Problems. Nauka, Moscow, 1986 (in Russian).
Ill-Posed and Non-Classical Problems of
Mathematical Physics and Analysis, Samarkand,
M.M. Lavrent'ev and S.I. Kabanikhin (Eds)
© VSP 2002
2000, pp. 57-69
Well-posedness of one-dimensional inverse acoustic
problem in Li for small depth or small data
S.I. KABANIKHIN*, K . T . ISKAKOV t , and M. YAMAMOTO*
Abstract — We investigate a system of nonlinear Volterra integral equations. In
previous publications [1, 2] we showed that this system can be derived from an
inverse acoustic problem and from an inverse electromagnetic problem respectively.
Using the method of weighted estimates and the Banach fixed point theorem we
prove that the inverse problem is well-posed if either the domain is sufficiently small
or the data is sufficiently small. Moreover we find explicit constants which allow us
to estimate the depth of well-posedness provided the data are arbitrary but fixed
or to estimate the norm of the data which guarantees the existence while the depth
is arbitrary but fixed. In [3] Z^-approach has been applied to general nonlinear
Volterra operator equation, but without specifications of the constants. In [4] the
similar technique allowed to justify the method of steepest descent. Multidimensional analog of our problem was considered in [5].
1.
INTRODUCTION
We consider a system of nonlinear Volterra integral equations
Φ(ζ,ί)
= G{x,t)
+ Β{Φ),
{x,t)eA{l).
(1.1)
'Sobolev Institute of Mathematics, Acad. Koptyuga prosp., 4, Novosibirsk, 630090,
Russia. E-mail: kabajiikh@math.nsc.ru
^Buketov Karaganda State University, Universitetskaya str., 28, Karaganda, 470074,
Kazakhstan. E-mail: ka2izat@mail.ru
* Graduate School of Mathematics, University of Tokyo, Komaba, Tokyo, Japan.
E-mail: myama@ms.u-tokyo.ac.jp
S. I. Kabanikhin, Κ. T. Iskakov, and M. Yamamoto
58
Here
Φ(χ,ί) = (Φι(χ,ί),Φ2(ζ),Φ3(ζ))Τ,
G{x,t) = {G1{x,t),G2,G3(x))T
Gi = Gi(x,t),
Gj = Gj(x), j = 2,3,
Φι (x,t) = ux{x,t), Φ2(x)=p(x), Φ3(χ) = α(χ),
GiOM) = Fx(x,t), G2 = -1/a, G3(x) = -2f'(2x)/a,
Fx(x,t) = [f'{t + x)- f'{t - x)]/2, A(l) = {(x, t) I Ο < χ < t < 21 - x}
Β:(Φ) = ± j T Φ 3 (0[<Μ£,
ί + χ-ξ) + Φ1(ξ,ί-χ + ξ)] άξ,
(1.2)
(1.3)
β3(Φ) = -αΰ3(χ)Β2{Φ) -
27>(Φ)/α
+ 2Β2{Φ)ν{Φ),
Τ(Φ) = Γφ3(ξ)φ1(ξ,2χ-ξ)άξ,
Jo
(1.4)
(1.5)
In [1, 2] Volterra integral equation (1.1) is derived from an inverse acoustic
problem and an inverse electromagnetic problem respectively, and inverse
problem are reduced to (1.1).
We set
||φ||2(*) = έΐ|φ,·|| 2 (ϊ),
3=1
ΙΙΦιΙΙ l m , X ) ) =
« e (0,0,
(1.6)
[
\\*i\\l2(0,x)
J 0= Γ
3= 2,3.
The space of vector-functions Φ(χ,ί) = (Φι(χ,ί),Φ 2 (χ),Φ 3 (χ)) τ with the
norm (1.6) will be referred as L2(l). We define the operator
ν(Φ) = α(χ,ί) + Β(Φ), (x,t)eA(l).
Lemma 1.1.
(1.7)
If G E L2(l), then
V : L2(l) —» L2(l).
The proof of the Lemma 1.1 will be given in Appendix 1.
(1.8)
Well-posedness
W e d e f i n e k-norms
of
one-dimensional
59
a s f o l l o w s (k > 0 ) :
\ m l ( x ) ^
\\*ΛΗ
||Φ,·|| 1 ( χ ) =
χ
Ι
ο Χ
\\^ι\\
} = [
Γ φ
2
*
φ
2 {
ι)(0^άξ,
Z e
ι ( Μ ) Λ ,
^ ξ ) β - ^ ά ξ ,
Χ e
(0,0,
(0,0,
χ e (0,0,
i =
2,3,
JO
Ό
G e L2{1), ε > 0 , k >
and the ball for
Π ( σ , ε , k)
w h e r e ||Φ||2(α;) =
L e m m a
¿
j=ι
1.2.
0
= { Φ e L2(1)
I HG -
Φ||*(0 <
ε},
||Φ,·|β(*).
consider the operator
Leí us
satisfy
ν(Φ)
dehne by
(1.7). Leí
1612 + 2l + 3/2<k
k
(1.9)
and
Gl = \\G\\2{l) <e~2kl /2,
ε < e~2kl / 2 .
(1.10)
(1.11)
2
Then
V(U{G,e,k)) cU{G,e,k).
(1.12)
T h e p r o o f is g i v e n i n A p p e n d i x 2.
L e m m a
M2{l,ct) = 2 4 / α
If G* a n d ε
on n(G,ε,k)
Proof.
Let G e L2(l) and α φ
1.3.
2
+ 5 Ζ + 4 α 2 + 16,
0, Ζ > 0,
k e (Ms(l, α),
Μ3(Ζ,α) = max{M2(Z,a),
oo),
16Í2+2Z+3/2}.
satisfy ( 1 . 1 0 ) and ( 1 . 1 1 ) , then V is a contraction
with contraction constant q2 = Mz{l,a)/k.
operator
> 0
Let φ ί ^ , φ ί 2 )
G Π ( G , e , ib).
We
denote Φ =
Φ ^
estimate
Τ
ω
= η φ ( % ) - ^ ( φ (
(Φ2 = ^ | | Φ ω | | 2 ( Ζ ) ,
.7=1,2
2
) )
ω
,
Gì =
i =
1,2,3
||G|| 2 (0).
-
φ(2)
and
60
S. I. Kabanikhin,
Κ. T. Iskakov, and M.
Yamamoto
T h e n we have
= \[
^(ΟΕΦΐ1' (ξ,ί + Χ~ξ)+
φ5 2) ( ο [ %
+
Φ?5 ( ξ , t - Χ + ξ)] άξ
t + χ - ξ ) + Φ Χ ( ξ , t - χ+«e)]
de,
and
11/2
\r(l)(x,t)\<l[¡QX\m)\2e-2^
χ [ jT[Φ?*
άξ]
I 1/2
( ξ , t + χ - ξ ) + Φ ? > ( ξ , t - χ + Ο ] 2 e 2 f c i de]
;fi^2)(e)i2e2^de]1/2
(e, t + χ - e ) + Φ ι (e, t - χ + e ) ] 2 e- 2 f c « d e ] 1 / 2 ,
"
ι τ α λ ο ί
2
<\\\φ3\\ΐ(χ)1\φ{ίί)(ξ,ί+χ-ξ)+φ?(ξ,ί-χ+ξ)]2^^
+ \ i i * i 2 ) i i 2 - f c ( * ) f*[*i(e,
de
t + χ - e) + * i ( e , * - χ+e)]2
ι ι τ ( : ) ι ι ^ ( χ ) <2||φ 3 |Ι2(«) f e 2 f c Ç
Jo
e - 2 < * de,
l i e f e r n
< ^ Φ 2 β 2 ^ ( | | Φ 3 | | Α ( χ ) + ||Φι||Λ(χ)),
χ e (0,1).
(1.13)
J* φ?*«)*^)
de.
N o w we consider T(2) :
T(2) = β2(Φ(ι))-β2(Φ(2))
=
jT
di-\
Well-posedness of one-dimensional
61
Hence
,1/2
1
+
2
J* |Φ«(0|2 e2« άξ]1/2 [ζ |Φ2(0|2 e-2^ de
< \ IWkWII^II-tM + \
ll-tWIftlUW
and
\r{2)(x)\e-kx < \
e-to(||«slU(®)||®f,ll-*(®) + | | Φ ^ ( χ ) | | Φ 2 | | ( χ ) ) ,
< ^ ^ β » * ( | | Φ 2 | | * ( ® ) + ||Φ8||*(®)),
(1.14)
Let us estimate T( 3 ).
TfoW < 4α2\03(χ)\2Ύ22](χ) + 8Y2(x)/a2
+ 8Τ^2)(χ)Ρ2(Φ(1)) + 8 Y 2 { x ) B f a W ) ,
(1.15)
Y2(x) <2 Γ \Φ3(ξ)\2άξ Γ\Φ?(ξ,2χ-ξ)\2άξ
Jo
Jo
2
2
+ 2 Γ \*?Ηξ)\ άξ Γ \Φι{ξ,2χ - ξ)\ άξ,
Jο
Jο
ν2(Φ^)
<
Γ|Φ^(0\2άξ
Jo
Γ \Φ?(ξ,2χ-ξ)\2άξ
Jo
<ΦΐΓ\Φ^(ξ,2χ-ξ)\2άξ,
Jo
B¡{Φ®)
< Φ*/4.
(1.16)
(1.17)
(1.18)
62
S. I. Kabanikhin, Κ. T. Iskakov, and M. Yamamoto
Therefore
T¡3)(x) < a 2 |G 3 (*)| 2 l *Wkx
+ 2 ( A + 2φή
[J*
( H a l i t e ) + libali\{x))
|Φ3(0|2άξζ
Φΐζ\Φ1(ξ,2χ-ξ)\2άξ]
+
+ I *Wkx
\Φ?(ξ,2χ-ξ)\2άξ
|Φ(ι%,2ζ
(||Φ2|ΙΪ(®) + ||Φ3||
-
ξ)\2άξ
< I e2kx [α 2 φ2|0 ( 3)(χ)| 2 (||φ 2 ||1(χ) + ΙΙΦβΙΙΚ*))]
+ I e2kx ( Α + 2φή [||Φ3||2(®) J* |·<%
+ ΦΙ J* e'21* ΙΦ^χ
2χ - £)|2 άξ
- ξ)\2 άξ]
+ ì e2** [2Φί(||Φ 2 ||ϊ(®) + ||Φ 3 ||1(χ)) J* \Φ?(ξ,2χ
- ξ)| 2 ].
(1.19)
We multiply (1.19) by e~2kx and integrate from 0 to 2; with respect to χ
l|T(3)||2(s) < \ [ * W
+
ïκ ( à
+ 2 Φ
*) [ [
|G(s)(0I 2 [II®2|IÏ(0 + ΙΙΦ3ΙΙ 1(0} άξ
"*s||*(í)
Si
2r
-c)|2
άζάξ+φ
*ιΐφιΐι*ω
+ I φι j T [||Φ 2 ||1(0 + Il®sll2(0] Si Ι φ ι 1 ) ( C . - ΟΙ 2 dζ άξ.
(1.20)
After summation of (1.13), (1-14), (1.20) we obtain
||τ(φ(1))-τ(φ(2))||2(χ) < ^ [ « m l l ^ l l í W + j f ' é w ü j í O l l ^ l l í í O d e ] ,
χ €(0,1).
(1.21)
Well-posedness
of
63
one-dimensional
Here
ω(ο) = (8/α 2 + 2φ4)φ2,
ω(ί)
=
"(3) = \φ*
+ α2\03(ξ)\2Φΐ
W(1)
(C,2£ - C)|2dC,
+ 2ΦΙJ*
+ ¿»lö.iöl2«* +
= φ^
Χ - ΟΙ 2 άζ.
+ 2Φί)
Since (1.21) is valid for all χ € (0,/), we can obtain an upper estimate
||Τ(φ(1)) _ Τ ( φ ( 2 ) ) | | 2 ( 0 < 1 ^||φ(1) _φ(2)||2 ( / )
(L22)
with
ri
3
ω* =w(o)+ /
J0
3=1
We recall that Φ2 = max j = i i 2 | | Φ ω | | 2 ( 0 and G* = ||G|| 2 (/). Therefore
ω* < Mi (J, α, Φ,, G,) = Φ 2 (12/α 2 + 5//2 + 2a 2 G 2 + 4Φ?).
(1.23)
Therefore the condition for the operator V to be a contraction operator is
k e (Μι(/,α,Φ„σ,),οο).
(1.24)
If Φ G n(G, ε, fc) then
\ m
k
( l ) < \ \ G M l ) + e<G.
+ e
and hence
Φ* < e 2W ||Φ|β(ί) < e2*'
(<G* + e) 2 .
(1.25)
Due to the conditions (1.10), (1.11),
(G* + ε) 2 < 2G2 + 2ε2 < 2 e~2fcí
(1.26)
Φ2 < 2,
(1.27)
G 2 < 1.
(1.28)
and hence
Therefore we can estimate
Μ ^ / , α , Φ , , ΰ * ) < M 2 { l , a ) = 2(12/a 2 + 5//2 + 2a 2 + 8)
which completes the proof.
•
(1.29)
64
S. I. Kabanikhin, Κ. T. Iskakov, and M. Yamamoto
Theorem 1.1 [well-posedness for small data]. Let G G £<2(0> ' > 0,
ke{M3{l,a),
2
oo),
2
M3(l, a) = max{24/a + 51 + 4a + 16, 16l2 + 21 + 3/2},
2
2kl
(1.30)
ε e (0,e~ /2).
(1.31)
||G|| 2 (0 < ε 2 ,
(1.32)
If
then the system (1.1) has the unique solution Φ G
continuously on data G.
which depends
Proof. According to Lemma 1.3 and the Banach fixed point theorem,
the operator V has the unique fixed point Φ G Π((7, ε, k) which solves the
system
Φ(χ,t) = ν(Φ)(α;,t),
Let us prove the stability. Let
(x,t) G Δ(/).
(1.33)
j = 1,2 solve the system
ΰ)
= ϋ (χ,ή+Β(Φω).
Φ^(χ,ί)
(1.34)
We denote
Φ(χ,ή = φΜ(χ,ή
- φ(2)(χ,ή,
G(x,t) = G^(x,t)
-
G®(x,t)
and obtain after the subtraction
Φ(χ, t) = G{x, t) + Β{Φ^) - β(φ( 2 )).
(1.35)
According to Lemma 1.3
ll«IU(0<IIG|U(0 + 9ll«IU(0,
and, since q G (0,1), we obtain
ll*IU(0 ^ i - ^ - I I G M <7^-11011.
1 -q
(1.36)
1-q
\\m)<e2kimk(i),
Using the property of the /c-norm:
we obtain from (1.36)
II^IIW ^ Τ Γ α
e2kl
II^IKO-
(1-37)
Well-posedness of
2.
65
one-dimensional
LOCAL W E L L - P O S E D N E S S
Theorem 2.1 [local well-posedness]. Let G G ¿2(^0)) ¡o > 0, α / 0.
Then there exists
G (0, IQ) such that for every I € (0, /*) the solution Φ of
the system (1.1) exists and is unique in 1*2(1) and depends continuously on
data G G ¿2(0 (G is the restriction of G on Δ(/)).
Proof. For l e (0,lo) we define the operator V by (1.7). Due to
Lemma 1.1, we see that V : ¿ 2 O —> ¿2 (0· Let us examine estimate
(3.11) obtained in Appendix 2:
livm-Giiii^il+^iiGim+^+Mi
(2.1)
We suppose that Φ e II(G, ε, k), therefore
Sit < (Gfc + ε) 2 < 2G¿ + 2ε2 < 2G2 + 2ε 2 ,
(2.2)
φ 2 < β 2 * ' φ £ < 2 β 2 * ' ( £ 2 + ε 2 ).
(2.3)
Then it follows from (2.1)
||ν(Φ) — G\\l < ^ e2kl 4(G 2 + ε 2 )
§ + ^
+ ^
+ 2Z2e2W(G2 +
£
2)
(2.4)
If we put
(2.5)
I = 1/k,
then
IIν(Φ) — G\\l < ^ e 2 (G 2 + ε 2 )
2k
Therefore for every pair (G*, ε) we choose /ci > 0 such that for every A; G
(fci,oo)
T M - 1
+
+
^
+
+
+
which means that Φ G Π(ΰ,ε, k) implies ν(Φ) G I1(G,ε, k).
< i2
(2.7)
66
S. I. Kabanikhin,
Κ. T. Iskakov, and M.
Yamamoto
Taking into account (1.22) and (1.23) we obtain
||7(®W) - ν(Φ ( 2 ) )|β < Χ
2
ο / Λ Ο
ο μ ι , Μ ^
+
- r ί ο \ ι· ο Γ 1 6
< I e2(G2 + ε 2 ) | | Φ ^ - φ( 2 )|| 2 g
It is clear that we can choose
we have q(k) G (0,1). Here
+ Φι) ||Φ(1) - φ ( 2 ) ||ι
f +
5
+ Α + 2a 2 G 2 + 4e 4 (G 2 + ε 2 )].
(2.8)
£ {ki, oo) such that for every k G (k?, oo),
q(k) = I e 2 (G 2 + ε 2 )
+ 2a 2 G 2 + 4e 4 (G 2 + ε 2 )
+ |
Therefore we can put
Z* = min(Zo,
(2.9)
l/k2).
Then for every l G (0,1*), the operator V has exactly one fixed point in
11(0, ε, A;). The stability can be proved by the same arguments as in the
proof of Theorem 1.1.
•
Appendix 1. PROOF OF LEMMA 1.1
Let Φ{χ,ί) G ¿2(0 and Φ* = ||Φ||(Ζ). Then we estimate every component of
ν(Φ):
+ ^[Ι*\ΦΛ(ξ)\2άξ]1/2
\ν(Φ){1)\<\0{φ,ί)\
+ χ-ξ)\2άξ]1/2+
χ ([ζ\Φι(ξ,ί
ν(Φ)2{1)(x,t)
[£\Φ1(ξ,ί-χ
+
ξ)\2άξ]1/2),
<2\G{1](x,t)\2
+ Γ | φ 3 ( 0 1 2 ^ Γ[|Φι«,ί + χ-ξ)|2 + |Φι(ξ,ί-χ + Ο Ι 2 ] ^
Jο
Λ'^Ι
/
Jχ
(3-1)
Jο
2J
ρ/,1
ν(Φ)2{ι)(χ,ί)άί<2
Χ
\G(1)(x,t)\2dt
Jχ
+ 2||Φ3||2(χ)^||Φι||^(0^·
(3.2)
67
Well-posedness of one-dimensional
We recall that ||Φι||^)(χ) = ¡ * l ~ x Φ ι ( ζ , ί ) dt and denote G* = ||G||(0· Then
we obtain from (3.2)
Ι ΐ η Φ ) ( ΐ ) Ι Ι Ι ( 0 < 2 | | σ ι ΐ ϋ 2 ( 0 + 2Φ^
(3.3)
I ^ ) ( 2 ) | 2 ( X ) < \ Ι ο Χ \ Φ 3 ( ξ ) \ 2 ά ξ Ι ο Χ \ Μ Ο \ 2 ά ξ + 2\0 2 (χ)\ 2 ,
Ι1^(φ)(2)||2(0 < 2||G2||2(0 + Φ*/2.
(3.4)
(3.5)
| ν ( φ ) ( 3 ) ( ® ) | < |G? 3 («)|
+
\ |α| |G3(z)| [ j T |φ 3 (01 2 de] 1 / 2 [J*
_2_ γ Γ ^ , . „ 20 . j V1/2
2r Γ , .
^ΊΦ3(οι
+
,
|Φ2(£)|2
dt]1/2
o .ji/2
- ο ι 2 ¿e]:
'^I^3(oi2de]1/2[^I|$2(e)i2¿e]1/2
χ [^V3(e)i2de]1/2[^i$i(e,2o;-e)i2de]1/2.
||ν(Φ)(3)||2(/) < 4||G3||2(0 + a^||G (3) || 2 (Z) Η- 16Φ?/α 2 + 4Φ*.
(3.6)
(3.7)
A p p e n d i x 2. P R O O F O F L E M M A 1.2
Let Φ e n(G,e,fc)). Let us prove that ν ( Φ ) G Π (G, ε, Λ))
Jo
Γ
Jχ
V(<%)-Gi)20M)dí
ρ21—χ ρχ
<2||Φ3||2(ζ) /
/ [|$1(^i + a ; - e ) | 2 4 - ^ i ( e , i - x + e)|2]e-2feçe2^de
./χ
.70
ri
οφ2
ρ2Α:ι_ι
Ι|ν(Φ)(ΐ) ~ Gillfc <
φ* = ||φ||(0,
G . = IIGIK0,
2Φ 2
¿||Φι||1,
* * = ||φ||*(0,
(3.8)
G . = l|G|U(0,
68
S. I. Kabanikhin, Κ. T. Iskakov, and M. Yamamoto
\ν{Φ\2)-ΰ2\<1-^\Φ3{ξ)Φ2{ξ)\άξ
j\v{Φ)(2) - G2)2(£)e-2^ άξ < g ||Φ
\\ν(Φ){2)-02\\1<^\\Φ
(3.9) 2\\1
|^(Φ)( 3 ) - Gal <
a\\G3(x)\[J*
|Φ3(0|2 άξ]Φ
Ιp2kx
ο
Ia2kx
IIν(Φ)(3) -
Galli
<
φ2
G
Γ
[J*\Φ2(ξ)\2 rfç] ^
ν Γχ
ί]
Γ fx
τ 1/2
ο
in;
Il®sll2II«2IIg3||2(0 + %
(3.10)
+
œ
By the definition of Φ& = ||Φ||* we obtain from (3.8)-3.10
Γ5
6/
Φί
(3.11)
Therefore since Φ 6 II(G, ε, fc), we have Κ(Φ) G II(G,e, k) provided
Φ», G*, k satisfy the condition
m
t
6/
l i + έΐο|·(0 + 5 +
r5
+ «)·<«.
(3.12)
Taking into account the property of fc-norm
(3.13)
\\a\\l(x)<\\af(x)<e2k*\\a\\l(x)
we conclude that the condition
< ε
(3.14)
Well-posedness of one-dimensional
69
will be justified if
ε2 < e~2kl /2,
G2 < e -2kl
(3.15)
16Z2 + 21 + 3/2 < k.
(3.17)
(3.16)
Thus the proof of Lemma 1.2 is complete.
Acknowledgement
The first author appreciates a very kind support of all Faculty Members
of Graduate School of Mathematics, University of Tokyo, and especially of
Professor Masahiro Yamamoto during their investigations of inverse acoustic
problem.
REFERENCES
1. S.I. Kabanikhin, Κ.T. Iskakov, and M. Yamamoto, iïi-conditional
stability with explicit Lipschitz constant for a one-dimensional inverse
acoustic problem. J. Inv. Ill-Posed Problems (2001) 9, No. 3, 249-267.
2. V. G. Romanov, S.I. Kabanikhin, Inverse Problems for
Equations. VSP, Utrecht, 1994.
Maxwell's
3. J. S. Azamatov and S. I. Kabanikhin, Nonlinear operator equations.
¿2-theory. J. Inv. Ill-Posed Problems (1999) 7, No. 6, 497-529.
4. S. I. Kabanikhin and Κ. T. Iskakov, Justification of the steepest descent method for the integral statement of an inverse problem for a
hyperbolic equation. Sib. Math. J. (2001) 42, No. 3, 478-494.
5. V. G. Romanov and M. Yamamoto, Multidimensional inverse hyperbolic problem with impulse input and a single boundary measurement.
J. Inv. Ill-Posed Problems (1999) 7, No. 6, 573-588.
Ill-Posed and Non-Classical Problems of
Mathematical Physics and Analysis, Samarkand, 2000, pp. 71-80
M.M. Lavrent'ev and S.I. Kabanikhin (Eds)
© VSP 2002
Fourier series in Banach spaces
DJ. KHADJIEV* and A. ÇAVU^
Abstract — Let Τ be the group {elt | 0 < t < 2π}. We consider Τ with its
euclidean topology. Our main results are as follows: 1) Theorem 1 which establishes a connection between spectrums of a continuous linear representation of Τ
in a reflexive Banach space and its conjugate linear representation; 2) Theorem 11
which gives a description of all left (right) simple actions of Τ in Banach algebras;
3) Theorem 18 which establishes an analog of the theorem on integral for continuous
linear representations of Τ in Banach spaces.
A part of results of the present paper was announced in an earlier our
paper [1]. Further we use notions and notations of the book [2] and the
paper [3].
Let Η be a complex Banach space. We denote the group of all invertible
bounded linear operators on Η by GL(H).
A homomorphism a : Τ —y GL(H) is called a linear representation of Τ
in a Banach space H.
Let α be a linear representation of Τ in a Banach space H.
Definition 1. A point χ Ε Η is called a continuity point of a
if limt-^o a(t)x = χ. A linear representation a is called continuous if
l i m ^ o a(t)x = χ for all χ 6 Η.
Every continuous lineax representation of Τ is equivalent to an isometric
continuous linear representation [2, p. 83].
'National University of Uzbekistan, Tashkent, Uzbekistan
^Karadeniz Technical University, Trabzon, Turkey. E-mails: haciyev@risc01.ktu.edu.tr,
cavus@jbsd.ktu.edu.tr
72
Dj. Khadjiev and A. Çavu§
Let α be a continuous linear representation. Then there exists the Riemann's integral:
Fn{x)
=
/
Jo
e~inta(t)xdt
for every η G Ζ. It is easily seen that there exists a positive constant L such
that ||.Fn(z)|| < L\\x\\ for ail χ Ε Η and η e Ζ. We put Spec (χ) = { η Ε Ζ \
Fn(x)
φ 0}, Spec (Η) = U x € H Spec(x)
and
Ηη = {χ e Η \ a(t)x
=
eintx,
Vi Ε Τ } . H n is a closed T-invariant linear subspace of Η. Let Η ' be the
conjugate linear space of H. Denote the conjugate linear operator of a(t)
by a'(t).
Then a' : Τ
GL(H')
is a linear representation of Τ in H'.
Proposition 1. Let a be a continuous isometric linear representation
of Τ in a Banach space H. Then:
1) Fn(a(t)x)
= a(t)Fn(x)
= eintFn(x)
for allt eT,n
G Ζ and χ G H;
2) Fl = Fn for all η e Ζ]
3) ll^nll = lfor all η G Z\
4) ImFn = {yeH\y
= Fn(x),
χ G H} = Hn for all η G Z;
5) Fno Fm = O for ail η φ m, η, m G Z;
6) The subspaces {Hn \ η G Z} are linear independent.
Let us put s n ( x ) : = Σ ^ - η ^ ( ^ ) and ση(χ)
:-
(s0(x)
+ si(z) + · · · +
Proposition 2. Let α be a continuous isometric linear representation
of Τ in a Banach space H. Then:
1) ση(χ)
=
[ Kn(t)a(t)xdt
for all χ e Η and η e Ν, where the
2π JT
function Kn(t) is Fejers kernel and has the form Kn(t) = sin 2 ((n +
l)i/2)/[(n + 1) sin2 (i/2)]
2) sup ||ση(χ)|| < ||x|| for all χ G H.
Theorem 1. Let α be a continuous linear representation of Τ in a
Banach space H. Then limn_>oo cr„(x) = χ for all χ G H.
Theorem 1 is known for homogeneous Banach spaces of functions on Τ
[4, p. 14],
Fourier series in Banach spaces
73
Corollary 1. Let a be a continuous isometric linear representation of Τ
in a Banach space H.
1 ) if Fn(x) = 0 for aline
Ζ then χ = 0;
2) if Fn(x) = Fn(y) for all η e Ζ then χ = y.
This corollary is a generalization of the uniqueness theorem for Fourier
series in Ll(T) [4, p. 13].
Theorem 2. Let α be a continuous linear representation of Τ in a
Banach space H. Then limn_>.oo Fn(x) = 0 for all χ Ε Η.
This theorem is a generalization of Riemann—Lebesque lemma in Ll(T)
[4, p. 13],
Theorem 3. Let α be a continuous linear representation of Τ in a
reßexive Banach space H. Then:
1) a' is a continuous linear representation;
2) Spec (H) = Spec (if');
3) dime Hn = dim c {H' n ) for all η Ε Ζ.
Remark 1. Theorem 3 is not true if Η is not reflexive. For example,
let Η = If1 (Τ), (a(t)f)(x) = f(x +1). Then a' is not continuous.
Let V := {(V„, ||· ||„) | η Ε Ζ} be a family of complex Banach spaces.
We shall consider sequences χ = {χ η I η G Ζ}, where x n
A sequence χ = {xn | η G Ζ} is called bounded, if there exists M > 0
such that ||arn||n < M for all η Ε Ζ. Denote the set of all bounded sequences
by m{V}. m{V} is a complex linear space with respect to the following
operations:
{Zn} + {yn} = {Xn+Vn},
λ{ζ η } = {λχ η },
λ G C.
We define a norm in m{V} by
IMI = nez
sup||z n || n .
Then m{ V} is a Banach space. We define a linear representation β of Τ
in m{V} by ß{t)x = {eintxn \ η G Ζ}.
74
Dj. Khadjiev and A. Çavu§
Denote the set of all χ G τ η { Ϋ } such that lim|n|_>+00 ||xn||n = 0 by
Theorem 4. mo{V} is a closed Τ — invariant linear subspace of m{V},
and χ E m{V} is a continuity point of the linear representation β if and only
if χ 6 77lo{v } .
Prom the Proposition 1 we have supnG^ ||.Fn(a;)|| < ||x|| for all χ Ε Η .
Therefore, a description of all continuous isometric linear representations
of Τ in a Banach space, satisfying the condition sup n e Z ||Fn(x)|| = ||χ|| for
all χ G H, is an interesting problem.
The following theorem is a solution of this problem.
Theorem 5. Let α be a continuous isometric linear representation of Τ
in a Banach space Η and Hn = {χ Ε Η \ a(t)x = emtx, W G Τ}. Then:
1) The mapping F : H
m{H}, where F(x) = {Fn(x) \ η G Ζ}, is an
injective bounded linear operator and F(H) = mo{H};
2) F(H)
= m0{H}
iff sup n e z ||Fn(i)|| = ||x|| for all χ G H.
Let α be a continuous linear representaion in a Banach space H. We
consider the following formal series:
+00
Σ
Xn'
w
- 0 0
where xn G Hn, η G Ζ. For each integer number η > 0 let
η
9n=
Xk'
Σ
k=—n
ση =
So + Si Η
h sn
n+ 1
be the n-th partial sums and n-th Cesaro's means of the series (1).
Definition 2. We say that the series (1) is a Fourier series of the point
χ G H if Fn(x) = xn for all η G Ζ.
Proposition 3. Let α :T —» GL(H) be a continuous linear representation of Τ in a Banach space H. Then the series (1) is a Fourier series of an
χ G H if and only if the sequence {an}is convergent in H.
Theorem 6. Let α be a continuous linear representation in a reñexive
Banach space H. Then the series (1) is a Fourier series of a point χ Ε Η if
and only if the sequence {ση} of the series (1) is bounded.
Fourier series in Banach spaces
75
Remark 2. This theorem is known for LP, 1 < ρ < oo, and symmetric
functional spaces [7].
Let α be a continuous linear representation of Τ in a Banach space H.
Let us denote the set of all sequences χ — {xn \ η G Ζ}, such that x n G
Hn = Fn(H), ne Z, and sup n £ Z ||σ η (ί)|| < +oo, by Ησ, and define
||x|| := sup \\ση(£)|| <+oo.
nez
(2)
Theorem 7. Let a be a continuous linear representation of Τ in a
reñexive Banach space H. Then
i. Ησ is a Banach space with respect to the norm (2);
ii. The mapping F : H -»• Ησ, F (χ) := {Fn(x) \ η G Ζ}, is an isometric
isomorphism.
Let α be a continuous linear representation of Τ in a Banach space
H. Let us denote the set of all sequences χ = {xn | η G Ζ}, such that
xn G Hn = Fn(H), η G Ζ, and sup n € Z ||s n (í)|| <
by Hs, and define
||x|| := sup ||s n (í)|| < +00·
nez
(3)
Proposition 4. Let α be a continuous linear representation of Τ in
Banach space H. Then Hs is a Banach space with respect to the norm (3).
Definition 3. A point a; G if is an s-point of α if lim^oo sn (χ) = χ,
where s„(x) = E L - n ^ i 1 ) ·
Theorem 8. Let α be a continuous isometric linear representation of
Τ in a reñexive Banach space H. If the sequence {s n } of the series (1) is
bounded, then there exists an s-point χ Ε Η such that Fn(x) = xn for all
η G Ζ.
Theorem 9. Let α : Τ —> GL(H) be a continuous linear representation
and
in a Banach space Η, χ G H, p(x,r) := Ση=-οοΓ^ρη(χ)
0 < r < 1.
Then
1 Γπ
= — / Pr(t)a(t)xdt, where Pr{t) := (1 - 2rcosí + r 2 )/
2π
(1 — r2) is the Poisson's kernel;
1) p(x,r)
2) limr_».i p(x, r) — x.
76
Dj. Khadjiev and A. Çavu§
Proposition 5. Let a : Τ ->· GL(H) be a continuous isometric linear
representation of Τ in a Banach space H. Then \\p(x, r) || < ||x|| for all χ Ε Η.
Proposition 6. Let α : Τ -» GL(H) be a continuous linear representation of Τ in a Banach space Η, and {yn} be a convergent sequence in H. If
limn^oo yn = y then Spec (y) C U£Li Spec (y n ).
Proposition 7. Let α : Τ
GL(H) be a continuous linear representation of Τ in a Banach space Η, χ Ε Η \ {0}, and [χ] be the smallest closed
α(T)-invariant subspace of Η containing x.Then Spec (x) — Spec [x].
Theorem 10. Let α : Τ —>• GL(H) be a continuous linear representation
in a Banach space H, and χ G Η \ {0}. Suppose that dime Hn < 1 for all
η e Ζ. Then [χ] = Η if and only if Spec (x) = Spec (H).
Remark 3. Theorem 10 is an analog of the approximation theorem of
Wiener [8, Ch. II],
Theorem 11. Let α :T
GL(H) be a continuous linear representation
in a Banach space Η. Suppose that dime Hn < +oo for all η 6 Ζ. Then a
closed subset A C if is compact if and only if:
i) A is bounded, i.e., there exists a positive constant Κ > 0 such that
||x|| < Κ for all χ e A;
ii) for any ε > 0 there exists a natural number Ν(ε) such that ||σ η (χ) —
x|| < ε for all χ E A and η > Ν(ε).
Theorem 12. Let α :T —» GL(H) be a continuous linear representation
in a Banach space H. Suppose that dime Hn < +oo for all η Ε Ζ. Then a
closed subset A C Η is compact if and only if:
i) A is bounded;
ii) for any ε > 0 there exists a positive δ(ε) such that ||α(ί)(χ) — x|| < ε
for all χ E A and |t| < δ(ε).
Remark 4. For the space Η = Ι^([0,2π]), 1 < p < +oo, the sufficiency
of the Theorem 12 due to Riesz [9, p. 69].
Theorem 13. Let α : Τ —» GL(H) be a continuous linear representation
of Τ in a Banach space H. Suppose that dime Hn < +oo for all η Ε Ζ. Then
a closed subset A C Η is compact if and only if for any ε > 0 there exists
a natural number Ν(ε) such that ||^γση(χ) — x|| < ε for all χ E A and
η > Ν(ε).
Fourier series in Banach spaces
77
Theorem 14. Let a : Τ
GL(H) be a continuous unitary linear
representation of Τ in a Hilbert space H. Then:
Ο Π Ξ - ο ο I I M I I 2 = IMI2
for
a" * e H;
ii) ||βη|| = 1 for all η G Ν;
iii) limn_>.+00 s n (x) = χ for all χ e Η.
Definition 4. Let α : Τ
GL(H) be a continuous linear representation
of Τ in a Banach space H ,V be a linear subspace of H, and χ E V. χ is
said to be a(T)-invariant if a(t)x = χ for all t G T.
Let us denote the set of all a(T)-invariant elements of V by VT.
Definition 5. Let a : Τ -» GL(H) be a continuous linear representation
of Τ in a Banach space Η and V be a linear subspace of Η. V is said to be
a(r)-invariant if a(T)V C V for all t G T.
Proposition 8. Let a : Τ
GL(H) be a continuous linear representation of Τ in a Banach space Η and {VT \ τ G 5} be a family of α (T)-invariant
subspacesofH.
Then (£T
eS
FTT) = (Eres^)T
=
(Σ^Τ^Γ·
This proposition is a topological analog of the statement (d) of the theorem in [5, Ch. II, 3.2].
Let H be a Banach algebra and Aut(H) be the group of all bounded
automorphisms of H.
Definition 6. A homomorphism α : Τ —> Aut(H) is called an action
of Τ in a Banach algebra #.Let HT = {x G H | a(t)x = x, Vi G Τ} be
the set of all Τ — invariant elements of H. It is well-known that HT is a
closed subalgebra of H. Denote the complete lattice of all closed left (right)
ideals of Η by L¡(H)(Lr(H)) and the complete lattice of all T-invariant
closed left (right) ideals of Η by LJ(H)(LJ(H)),
respectively. Denote the
complete lattice of all closed left (right) ideals of the subalgebra HT by
Li(HT){Lr{HT)).
For I G Li{HT) put Ie := ΊΠ. Ie is the smallest closed
left ideal of H containing I and T-invariant. For A G Li{H) put A° := ΑΠΗ
and I e c := ( I e ) c .
Proposition 9. Let a be a continuous action of Τ in a Banach algebra H, and FQ : Η
Η be the lineax operator defìned by FQ(X) :=
— / a(t)xdt.
Then Fo(ax) = aFo(x) and F0(xa)
2π JT
α G H T , x G H.
= Fo{x)a for all
78
Dj. Khadjiev and A. Çavu§
This proposition shows that FQ is an analog of Reynold's operator in
invariant theory [6, p. 27].
Theorem 15. Let a : Τ -)• Aut(H)
Banach algebra H. Then
be a continuous action of Τ in a
i) IeC = I for all I G L ¡ ( # T ) ;
ii) the mapping e : Li(HT)
-> Lj(H)
I—íIe
is injective.
Remark 5. An algebraic version of Theorem 15 was obtained for actions
of reductive groups in polynomial rings [10, 11].
Let M(H) be the set of all maximal ideals of a commutative Banach
algebra H.
Proposition 10. Let α be a continuous action of Τ in a commutative
Banach algebra H. Then:
i) if I G M (Η) then Ie G
ii) the mapping c : M(H)
M(HT);
—>• M(HT)
is surjective.
Definition 7. An action α : Τ -¥ Aut(H) of Τ in a Banach algebra Η
is called left (right) simple if Lj(H) = {0,H} {Lj(H) = {0,-H"}).
Theorem 16. Let α : Τ —• Aut(H) be a continuous action of Τ in a
Banach algebra H. Then α is left simple if and only if:
i) Spec (Η) is a subgroup of Z;
ii) dime Hn < 1 for all η G Ζ;
iii) HnHm = Hn+m
for all m, η G Spec (H).
Definition 8. Let α : Τ -ï GL(H) be a continuous linear representation in a Banach space Η, χ G H, and to G T. If the limit
dfQx := limf_>o(a(i):E — a(to)x)/(t — to) exits then χ is said to be differentiate at to ζ Τ with respect to a, and dfQx is called the derivative of χ
at ίο € Τ with respect to a.
Proposition 11. Let
tation in a Banach space
at to Ε Τ with respect to
respect to a. Furthermore
a :Τ
GL(H) be a continuous linear represenΗ, χ G H, and to G T. Then χ is differentiate
α if and only if χ is differentiable at 0 G Τ with
dfQx = α(ίο)ά®χ.
Fourier series in Banach spaces
79
Definition 9. Let D(a) be the set of all points of i/,differentiate at
0 G Τ with respect to a.
Theorem 17. Let a : Τ —>• GL(H) be a continuous linear representation
in a Banach space H. Then
1) D(ot) is dense in H;
2) D(a) — Η if and only if Spec (H) is a fìnite set;
3) D{a) is a linear subspace of H;
4) Fn(d%x) = in Fn (x) for aline
Ζ and χ G -D(a).
Theorem 18. Let α :T —>· GL(H) be a continuous linear representation
in a Banach space Η and α Ε Η. Then the differential equation dfix = α
has a solution if and only if Fo(a) = 0.
Remark 6.
gral [12,ch. VI].
This theorem is an analog of the theorem about inte-
REFERENCES
1. Dj. Khadjiev and A. Çavu§, The imbedding theorem for continuous
linear representations of the rotation group of the circle in Banach
spaces. Dokl. Acad. Nauk Resp. Uzbekistan (2000) 7, 8-11.
2. Y. I. Lyubich, Introduction to the Theory of Banach
of Groups. Birkhauser Verlag, Berlin, 1988.
Representations
3. Dj. Khadjiev and R. G. Aripov, Linear representations of the rotation
group of the circle in local convex spaces. Dokl. Acad. Nauk Resp.
Uzbekistan (1997) 9, 8-11.
4. Y. Katznelson, An Introduction to Harmonic Analysis. Dover Pubi.,
New York, 1976.
5. T. H. Kraft, Geometrische Methoden in der Invarianten-Theorie.
Friedr. Viewag & Sohn Braunschweig, Wiesbaden, 1985.
6. D. Mumford, J. Fogarty, and F. Kirwan, Geometrie Invariant Theory.
Springer-Verlag, Berlin—New York, 1994.
7. Dj. Khadjiev and U. Karakbaev, The description of types of Fourier
series in symmetric spaces. Dokl. Acad. Nauk Resp. Uzbekistan (1991)
7, 11-13.
80
Dj. Khadjiev and A. Çavu§
8. Ν. Wiener, The Fourier Integral and Certain of its Applications. Dover
Pubi, New York, 1933.
9. L. A. Lusternik and V. I. Sobolev, Elements of Functional Analysis.
Gordan and Breach, Science Pubi., New York, 1968.
10. M. Nagata, Invariants of a group in an affine ring. J. Math. Kyoto
Univ. (1964) 3, No. 3, 369-377.
11. Dj. Khadjiev, About properties of extension and contraction operations of ideals of fixed algebras of reductive groups in a positive characteristic. Fund. Anal. Appi. (1991) 18, No. 4, 273-280.
12. Β. M. Levitan and V. V. Jikov, Almost Periodic Functions and Differential Equations. Moscow State Univ. Pubi., Moscow, 1978.
Hi-Posed, and Non-Classical Problems of
Mathematical Physics and Analysis, Samarkand, 2000, pp. 81-101
M. M. Lavrent'ev and S. I. Kabanikhin (Eds)
© VSP 2002
Ill-posed and inverse problems
for hyperbolic equations
M. M. LAVRENT'EV*
1. "The notions of ill-posedness for boundary-value problems for equations of mathematical physics and ill-posed problems were introduced in
the beginning of 20th century by the French mathematician J. Hadamard
[23, 24]. Further, these notions were revised and set in most textbooks of
mathematical physics [5, 13, 14].
At first it had seemed that ill-posed problems do not describe any real
physical phenomena. This point of view was universally adopted for years.
The first who begin to consider the concept of ill-posed problems was
A.N. Tikhonov [16]. Starting in 60th, the theory of ill-posed problems
becomes of an interest of scientists. It had turned out that this theory is
connected with a lot of various applied problems.
A considerable contribution to the theory of ill-posed problems was done
by A. N. Tikhonov and his disciples (Moscow school), M. M. Lavrent'ev and
his disciples (Novosibirsk school), V. K. Ivanov and his followers (Ekaterinburg school) [3, 6, 7, 17, 18]. The groups of scientists who develop the
theory of ill-posed problems are in republics of Soviet Union and in other
countries [4, 9, 15, 21, 22, 26, 27].
In this report we shall tell about some directions in the ill-posed problem
theory which are poorly known.
The classical ill-posed boundary-value problems for differential equations
are the Cauchy problems. First of all, it is the Cauchy problem for the
Laplace equation and the Cauchy problem for the heat conductivity equation with inverse time. The Cauchy problems for hyperbolic and parabolic
* Sob oleν Institute of Mathematics, Siberian Branch of Russian Academy of Science,
Novosibirsk
82
M. M. Lavrent'ev
equations with the data given in time-like manifolds axe ill-posed problems
also. All these problems are connected with the problems of interpreting
the data of physical measurements.
2. We consider the Dirichlet problem for the simplest hyperbolic equation — the d'Alembert equation. We need find the solution u(x,t) of the
equation
d2u _ d2u
( }
in the rectangle
0 < a; < 1,
0
<t<l,
satisfying the boundary conditions
u(0,t) = u(l,t)
u(x, 0) = 0,
= 0,
u(x,l)=f{x).
(2)
In order that the solution of this problem be unique it is necessary and
sufficient that the number I be irrational.
3. The Dirichlet problem for the d'Alembert equation was considered
by several authors. Certain results on this problem can be found in [25]. In
our paper the d'Alembert equation is considered in the form
Λ
dxdy
=0
(3,
for the domains convex in the directions of the i-axis and y-axis. This means
that the boundary of such domains is a Gordan curve which has no more
than two common points with each curve parallel to x-axis or y-axis.
The author of [25] connects the research of the Dirichlet problem with the
mappings of the domains of the (x, y)-plane onto certain canonical domains
of the (u, υ)-plane of the form
u = φ(χ),
υ = ip(y),
(4)
where φ(χ) and ifi(y) are continuous monotone functions.
If the functions φ(χ), il>{y) are continuously differentiate, the solutions
of the d'Alembert equation are invariant with respect to these mappings.
The analogous situation holds true in the case of the Dirichlet problem
for the Laplace equation and for conformai mappings.
In this paper no applied problems connected with this problem are mentioned.
Ill-Posed
and inverse problems for hyperbolic
equations
83
4. We observe the two scientific directions connected with mathematical description of physical phenomena, where the Dirichlet problem for the
d'Alembert equation and the mappings of form (4) are used.
The first direction is the research of boundary-value problems for the
system of equations which describes the small vibrations of rotating liquid.
The first settings and results on this problem belong to S. L. Sobolev. However, the first steps here were connected with the defense theme; as a result,
the first publication on this subject belongs to Sobolev's disciple R. A. Aleksandryan [1]. Further, a lot of works of S.L. Sobolev and his followers had
appeared. The most of these results can be found in the monograph of S. L.
Sobolev and T.I. Zelenyak [2].
The second direction is connected with gas dynamics. First settings
belong to M. A. Lavrent'ev. The first publications in this direction belong
to Β. V. Shabat and M. M. Lavrent'ev [8,19]. Further development was done
by M. A. Lavrent'ev and B.V. Shabat [20, 10].
It is known, that the two-dimensional stationary flow of ideal gas is
described by the system of elliptic equations; the supersonicflowis described
by the hyperbolic system.
The simplest system of elliptic equations is the Cauchy—Riemann system
du _dv
du _
dv
dx
dy '
dy
dx
The solutions of the Cauchy—Riemann system realize the conformai
mappings.
The simplest system of hyperbolic equations is the d'Alembert system
£ = 0 ,
dy
£ = 0 .
ox
The solutions of this system are the mappings of form (4). In [10], the
mappings of this form are called the /i-conformal mappings. These mappings
we shall call by /i-mappings.
5. We shall cite now the results of the work [10] on existence of hmappings of the two classes of domains onto the canonical domains.
The mappings of the half-plane type. Let f(x)
be a continuously differen-
tiable function such that
0 < α < f'(x)
where a, b are certain constants.
< b,
/(0) = 0,
84
M. M. Lavrent'ev
We consider the following domain in the plane (x, y):
D = {(x,y)\y>
f(x)}.
Theorem 1. There exists an h-mapping of the domain D onto the
half-plane
Do = {(«,i>) I ν > u}
satisfying the condition
u(0,0) = v(0,0) = 0 ,
i.e., leaving the origin immovable (Figure 1 ).
The mappings of the strip type. Let fi(x), fii^) be continuously differentiable functions satisfying the conditions
0 < α < f'k{x) <6,
k = 1,2,
f2{x) > fi(x) + c,
where o, b, c are certain constants, c > 0.
We consider the domain
D = {{x,y) I h{x) < y < f2{x)}.
Theorem 2. There exists an h-mapping of the domain D on the strip
D0 = {(u,t>) I u < ν < u + 1}
satisfying the condition
u(0,0) =v(0,0) = 0
(see Figure 2).
ΠΙ-Posed, and inverse problems for hyperbolic equations
7
χ
85
u
Figure 2
6. Mappings of bounded domains.
We shall give here some definitions and the results from [25] relatively hmappings of bounded domains.
Let C be a Gordan curve convex relatively the x- and y-axes. The point
Ρ for which the lines parallel to the z-axis or to y-axis and coming through
this point have no more points common with this curve is called the vertex
of the curve C. The curve C may have one, two, three or four vertices.
Let Ρ be a point in the curve C with the coordinates (x, y):
P(x,v)eC.
We denote by AP the point on the curve C with the coordinates {x\,y) and
by Τ the transformation of the curve C on itself:
TP -
BAP.
The vertices C are immovable points of the mappings A or B. The sequence
of points P , AP, TP, ..., TkP and the segments connecting these points we
call Λ-polygon defined by the point P. If there exists such η that T n P = P,
then the minimal η we call by the period of the point P. The point Ρ in
this case is called the P-periodic point of the curve C. The mapping Τ we
shall call even if it conserves the positive direction on C.
For the curves C, relatively the transposition T, the four cases are possible.
1. All the points of C are periodic. The curve C, in this case, we shall
call periodic.
2. The curve C contains both periodic and nonperiodic points. Such
curve we shall call half-periodic.
86
M. M. Lavrent'ev
3. There are no periodic points in the curve C. There exist no points Ρ
such that the set P, TP, ..., TkP, ... is everywhere dense in C. We
call the curve C in this case intransitive.
4. There are no periodic points in the curve C, but there exists a point
P, so that the set P, TP, ..., TkP, ... is everywhere dense in C. In
this case, we shall call the curve C-transitive.
In the case 1 all the points have the common period n. If Ρ is an arbitrary
point, then the points P, TP, ..., Tn~lP divide C o n n nonoverlapping arcs.
In the case 2 all periodic points have the common period n. The set F
of all periodic points is closed. The additional set C \ F is a countable set
of arcs with the ends belonging to F. Each of these arcs is invariable under
the action of the operator Tn.
We consider now the case 3. Let Q be an arbitrary point from C and
σ be the set of limit points of the set Q, TQ, ..., TkQ,
The set σ is
perfect, nowhere dense, and is independent on the point Q.
In the case 4 the transformation Τ is equivalent topologically to rotation
of a circle. This means that there exists a real ξ and a continuous mapping
t = f(P)
of the points Ρ of the curve C on the points e2mt of a unit circle in the
complex plane so that
f(TP)=t
+ £.
The constant ξ is irrational and is defined uniquely by the curve C. This
constant is called by the modulus of the curve C.
Theorem 3. Let C\, C2 be transitive curves with the common modulus
ξ; Di, D2 be the domains bounded by these curves. Then there exists the
h-mapping of the domain D\ onto the domain
Corollary 1. Let C be a transitive curve with the modulus ξ and D
be the domain bounded by C. Then there exists the h-mapping of the
domain D onto the rectangle with the sides
y = -X + 1
xe[0,l],
y = x-1
y=χ+1
®e[M],
χ G [Ο,η — 1],
y — ~x + 2n — 1 χΕ[η-\η],
(see Figure 3).
η=
1/(1-ξ)
Ill-Posed and inverse problems for hyperbolic
y '
equations
87
k
s
\
\
<
\
λ
1
κ
b
ì
1
χ
ri
Figure 3
7. Mappings of domains with two vertices.
It is easy to see that the domain which has two vertices can be transformed
to the domain
D = {(x,y)
I h(x)
<y<
f2(x),
χ G (0,1)}
with the aid of the ^-mapping. The functions fk{x), k = 1,2 satisfy the
conditions:
1) Λ(0) = ο, Λ(1) = 1;
2) the functions fk(x) are continuous and monotonically increase;
3) / i ( ¡ r ) < / 2 ( a O , 0 < x < l .
The domains of such type we shall call by the domains of the gap type
(see Figure 4).
M. M. Lavrent 'ev
88
Theorem 4. For each domain D of the gap type, there exists the hmapping of this domain onto the domain
D0 = {(x,y) I χ2 < y < χ, χ e (0,1)}
(see Figure 5).
Proof. Let h be the /i-mapping realized by the functions
χι = φι (χ) = f2(x),
yi=y.
This mapping transfers the domain D into the domain
D\ = {(&i,yi) I h(x\) <yi<
h(xi)
®ι},
f2[f2(x)]=x.
= fl[f~2(Xl)],
Consider now the /i-mapping of the domain D\ realized by the functions
xi = Ψ2{Χ\),
V2 = <^2(yi),
where the function ψ2{χ{) is defined from the relations:
φ2{Η[ψ2{Χ2)]} = ®2>
Ψ2[φ2{χΐ)] = Χ\·
Evidently, the /i-mapping consisted from these two h-mappings applied consequently transfers the initial domain D into the domain Do defined in the
theorem formulation.
•
The similar result may be obtained for the domains which have three
vertices.
8. Mappings of the regular domains.
Let D be a domain convex relatively the x-axis and the y-axis, which have
four vertices.
Denote the coordinates of these vertices by (αι,&ι), (02,62), (03,63))
(04,64). Let
αϊ < α2 < a 3 < 04,
62 < h < 64 < 63.
We call the domain regular if
Û2 = θ3)
h = h·
It is easy to see in this case that there exists the /i-mapping of the domain
D onto the domain, where
(see Figure 7).
01 = —1;
02 = 03 = 0;
04 = 1,
62 = - 1 ;
h = 64 = 0;
63 = 1.
Ill-Posed and inverse problems for hyperbolic equations
89
So we have supposed that D is the domain bounded by the curve C
consisted from the four parts C — C\ U
U C3 U C4:
Ci = {(x,y)\y
C2 = {(x,y)\y
= fi(x), χ e [0,1], /ι(0) = 1; / i ( l ) = 0 } ;
= f2(x), χ e [0,1], / 2 ( 0) = - 1 ; / 2 ( 1 ) = 0 } ;
c 3 = {(*,?) I y = Mx), χ e [ - 1 , 0 ] , / 3 ( - i ) = o; / 3 (o) = - i } ;
CA = {(x,y)\y = U(x), χ 6 [—1,0], / 4 ( - l ) = 0; / 4 (0) = 1}.
All the functions fk{x) are monotone and continuous.
We consider the /i-mapping of the domain D so that
χΐ = φι{χ),
y1=xl)l{y),
where the functions φ 1 and ψ 1 are defined as follows:
1°. φι{χ) = φ\(χ) =x,xe
-x + 1;
[0,1]; ψ1 (y) = φ\(ν), y E [0,1]; ^\[fi(x)]
2°. φΗν) = ^ ( v ) , y e [-1,0];
3°. φ\χ)
=
= X - ι, X e [0,1];
= φΙ(χ), χ G [-1,0]; φ\(ν) = / 3 (x), y = / 3 (x); ^ [ / 3 ( x ) ] =
-<PÜX)
-
1
=
Λ(®);
4°. Φ1 (y) = Φ\{ν) = vì(y),
ΨΐΙί(χ)} = / 4 [νΊ(χ)]·
y 6 [0,1]; ^ ( y ) = /4[</>1(χ)], y = /(χ);
This mapping transfers the domain D into the domain DQ of the plane
(xi,yi) bounded by the three sides of the square and the curve yi = f ( x i )
(see Figure 8).
M. M. Lavrent 'ev
90
Consider now the /i-mapping transferring the domain DQ into the domain D\ of the plane (U, Υ) of the type similar to DQ. We mean the domains
bounded by three sides the square and by the certain curve
υ = F(u),
F(-1)=0,
F{0) = 1,
u 6 [-1,0],
F'(u)>0
η € (-1,0).
Let this mapping be defined by the functions
η = φ(χι),
v = i¡>(yi).
Since the mapping transfers DO into D\, the functions are connected by the
relations
φ(1 - χ) = - φ { χ ) + 1,
φ{χ
-
1) = φ { χ ) -
1,
φ{χ)
s e [0,1],
= - φ ( - χ -
1) -
1,
x €
[-1,0].
Thus, the values of the functions φ{χ), φ{χ) in the segment [—1,1] are
defined by the values of the function ψ{χ) in the segment [0,1].
We suppose now that the functions f(x) and F(u) which define the parts
of boundaries of the domains DO and D\ satisfy the inequalities
f{x) >x-l,
χ G (-1,0),
F(u) >u-
1,
u E (-1,0).
The functions f(x), F(u), φ(χ) are connected by the functional relation
f[tp{l-x)-i\
=
F[<p(l-x)-l].
Now, we consider that the functions f(x), F(u) are given and this relation is a certain functional equation relatively the function φ(χ). It is easy
Ill-Posed and inverse problems for hyperbolic equations
91
to show that if the conditions I o and 2° hold, this equation has a unique
solution satisfying the conditions
φ(0)
= 1,
φ{1) — 0,
φ'{χ)
< 0,
a; G ( 0 , 1 ) .
The function F(u), for example, may be taken as follows:
F{u) = Λ/1 + u;
U G [—1,0].
Thus, we have established the following theorem.
Theorem 5. For each regular domain D of the plane (x, y), for which
the function f(x) satisfies the inequality (5), there exists the h-mapping of
the domain D onto the domain D1 of the plane (u, ν) bounded by the three
sides of the square and the part of the parabola
ν — yl + u.
Thus, we have established the theorems on existence of /i-mappings onto
the canonical domains for the two classes of unbounded domains and three
classes of bounded domains convex with respect to the χ and y-axes. It seems
to be interesting to research the possibility of /i-mapping of an arbitrary
domain on the canonical domains.
9. In Section 6 we had classified the Gordan curves bounded convex
domains given in paper [25]. It is easy to see that the curves bounded the
domains of gap-type are half-periodic; the periodic points with the period 1
are the two vertices of the curve. The curves restricted the regular domains
may be either periodic with the period equal to 2 or half-periodic. In the first
case there exists an /i-mapping of the domain onto the square. In the second
case, the period for periodic points is equal to 2; the vertices of the domain
are periodic points. If the domain satisfies the conditions of Theorem 5, the
periodic points are only the vertices.
10. We pass now to the Dirichlet problem for bounded convex domains.
We restrict ourselves by the problems for the classes of domains considered
in Theorems 3, 4.
So, let, at first, D be a domain defined in Theorem 3. Let gi{x), 52(2)
be continuous functions defined in the segment [0,1] and such that
i?i(0) = <?2(0),
5l(l) = 52(1)·
We need to find the solution of the d'Alembert equation
dxdy
92
M. M. Lavrent'ev
satisfying the conditions
u(x,fi(x))
=gi(x),
u(x,f2(x))
=92{x)·
(6)
We shall consider the generalized solutions of the equation; i.e., each function
u(x,y) =
φ(χ)+ψ(ρ),
where φ(χ) and ip(y) are continuous, we shall consider as the solution of this
equation.
Theorem 6. Solution of problem (3), (6) exists and is unique.
Proof. Solution of the problem formulated above is equivalent to solution of the following system of functional equations relatively the functions
φ and φ:
φ[ΐ2(s)] - Ψ[Μχ)] = 92{x) - 9ι(χ).
(8)
From the conditions imposed on the functions f\(x), fi{x) it follows
that the solution of functional equation (8) relatively the function ψ exists
and is unique. Hence the statement of theorem follows. Evidently, in the
space C of continuous functions, the solutions of system (7) depend on the
functions gi(x), 92{x) continuously. Thus, the Dirichlet problem is wellposed in classical sense for the d'Alembert equation in the domains of gap
type.
•
We consider now the Dirichlet problem for the domain bounded by transitive curve. By Theorem 3, this problem is equivalent to solution of the
d'Alembert equation in form (1) satisfying boundary-value conditions (2)
for / = £/(l — £), where ξ is the modulus of the curve.
It is known that the solution of equation (1) with boundary conditions
(2) may be represented in the form
00
where
Ill-Posed and inverse problems for hyperbolic
equations
93
Since the number I is irrational, we have
sin knl φ 0.
We shall give now one of the notions of ill-posed problem theory.
If, for the boundary-value problem, there exists a pair of functional
spaces in the definition of the norms, where a finite number of derivatives is
used and the problem is well-posed in this pair of spaces, then the problem
is called weakly ill-posed. If there are no such spaces, then the problem is
strongly ill-posed.
The Cauchy problem for the Laplace equation and the other Cauchy
problems mentioned in the beginning of this paper axe strongly ill-posed.
The classical weakly ill-posed problem is the problem of differentiation.
Since the number sin knl may attain the extremely small values, the
Dirichlet problem will be ill-posed in the pairs L2 -> ¿2 and C —> C. The
type of instability of this problem depends on the properties of the number I.
As it was noted in [25], if the number I is the Liouville number, i.e., for each
n, there exist such integers ρ and q that
\l-p/q\<l/qn,
the problem will be strongly ill-posed. If there exist such numbers c > 0,
σ > 0 that for each k the inequality
I sin kirl\ >
Ck~a
holds, then the problem will be weakly ill-posed and the inequality
I M | L
2
<
C\\f\\w?,
m>
σ
holds. The analogous condition is cited in [7] relatively the following problem
for the Laplace equation:
find the function u(x,y) which solves the Laplace equation
Au = 0
and satisfies the conditions
u( 0, y) = ΐί(π, y) = 0,
it(f, y) = f(y),
0 < χ < π,
0 < y < o.
11. The inverse problems of tomography are the mathematical problems which deals with interpreting the tomography data. In medicine and in
94
M. M. Lavrent 'ev
industry the most frequently encountered is X-ray tomography. Interpreting tomography data in medicine tomography is connected with the Radon
transformation. The corresponded mathematical problem is as follows.
Let u(x, y) be a continuous finite function with the carrier disposed in
a finite domain D. We may suppose that D is the unit disk: D = {(x,y) \
χ2 + y2 < 1}. The Radon transformation of the function u(x,y) is the
following function
(9)
Equality (9) is considered as a linear operator equation relatively the function u(x,y).
We need to find the function u(x,y) given the function
f(x,y,a).
In the mathematical model of X-ray tomography the function u(x, y)
is connected with the coefficient of absorption of X-rays. In medicine tomography it is appropriate to consider the desired function as an arbitrary
continuous or piecewise continuous function since the distributions of absorption coefficients may rather differ for various people. Moreover, even if
we take the concrete man, then the distribution of the absorption coefficient
may vary essentially on whether this man stays or lies.
In the industrial tomography, another situation is of an interest. The
distribution of the absorption coefficient mostly has the standard form. The
goal of tomography research is to find the deviation from the norm, i.e.,
the defects of the product. If the product is obtained by means of casting,
then inside this product there may be cavities and gaps. Thus, we obtain
the following mathematical problem: in equation (9), the function u(x,y) is
equal to unity in a certain domain Do C D and outside Do, it is equal to
zero:
{x,y) Ε D0 CD,
{x,y)éD0.
The boundary of Do is a piecewise smooth curve. Since, in this setting, the
function u(x,y) is defined by two functions dependent on the one variable,
it is natural to suggest that the solution of equation (9) can be obtained by
the information which is less complete than the information usually set in
the general case relatively the right-hand side f(x,y,a).
We consider the
problems connected with the solution of (7) in the case when the values of
f(x, y, a) are known for two values of the parameter a. So, we suppose that
the two functions are known
fi(x,y)
(see Figure 9).
= f(x,y,
0),
f2 (x,y) =
f{x,y,a)
Ill-Posed, and inverse problems for hyperbolic equations
95
Figure 9
The inverse problem formulated below is analogous to the well-known
inverse problem of potential theory which came to the attention of many
scientists [11, 6, 12].
Let D be a domain considered in Theorem 3, Section 5:
D = {(x,y) I fi{x)<y<f2(x),
2 e (0,1)},
where the functions fk{x) satisfy conditions 1), 2), 3) mentioned in the
formulation of Theorem 4.
Evidently, that if the functions <¿>fc(y), k — 1,2, are inverse relatively the
functions fk(x), i- e.,
<Pk[fk{x)] = X,
then
D = {(x,y) I ψ2{ν) <x<
<pi{y), y e (0,1)}.
Denote
u(x) = f2{x) - fi{x),
v(y) = <pi(y) - f2(y)·
Inverse problem. Find the functions fk(x) or φ^(y), the functions
u(x), v(y) given.
Theorem 6. The solution to the inverse problem is unique.
Proof. Let XQ be an arbitrary point in the interval (0,1):
0<
XQ
<
1.
Denote
h(xo) = yo,
xi=yo-v(y0),
96
M. M.
Xk+i =Xk~
Vk+1 =
Lavrent'ev
v{yk),
(10)
yk~u(xk+1).
Evidently, we have
zfc+i < Xk,
Vk+1 < yk;
lim xn — lim yn - 0.
Relations (10) may be considered as the functional equation relatively the
value
/i(®o) = yoThe solution of this functional equation is unique, as it is easy to show,
whence the statement of theorem follows.
•
12. Inverse problem for regular domains.
We confine ourselves by the inverse problems considered for the domains
satisfying the conditions of Theorem 5.
Let φι(χ), <P2(x), Vi(y)> Ψϊίν) be continuously differentiate functions
defined in the segments [01,02], [¿»1, £>3] so that
1° φι(χ) < φ2{χ), χ e (αϊ, o 3 ); Vi (y) < V2(y), y e (h ,63);
2o <¿>i(°i) = ^2(01), Vi(03) = ^2(03); Vi(61) = thipi), Vi (63) =
);
3° φ\{χ) < 0, χ G (01,02); ψ\{χ) > 0, χ G (02,03); φΙ{χ) > 0, χ e
(01,02); ψ\{χ) < ο, χ g (02,03); vì(y) > o, y e (61,62); Vì(y) < °>
y e (62,63); V2(y) < 0, y g (61,62); ψ\{ν) >0,ye
(62,63);
4° ψι[Φι{ν)] = y, φ2[ψ2^)] = y-
Denote by D the domain
D = {{χ,y)
I φι{χ)
< y < <P2(x)} = {(x,y)
I Vi (y) < x <
*h(v)}·
We suppose also that the domain D satisfies the condition of Theorem 5
and denote by u(x) and v(y) the following functions
u(x) = φ2(χ) - φι{χ),
v(y) = φ2 (y) - Vi (y)·
(H)
El-Posed, and inverse problems for hyperbolic
('
97
equations
i)
(^4,2/4) (a,b)
Figure 10
Inverse problem. Find the functions φι(χ), ψ2(χ), Ψι(ν), Ψ2Ü/)> given
the functions u(x), v(y).
The points (01,62), (02,63), (03,62), (02,61) axe the vertices of the domain D.
Prom (11) it follows that
63 = 6i + u(a 2 ),
It is easy to see that the numbers
the functions u(x), v(y).
Really, we have
α3 = α ι + υ ( 6 2 ) .
Λ; = 1,2,3 are defined uniquely by
u(x) = 0, χ < οι, χ > 03,
u(x) > 0, χ G (01, 03),
«(02) = maxii(x),
v{y) = 0, y < 61, y > 63,
v(y) > 0, y e (61, 63),
u(6 2 ) = maxv(y).
Let (xo,yo) be a certain point in the boundary of D. Let, for example,
νο = ψι{χο),
αι<χ0<α2.
Then the points (xk,Vk) defined by the equalities
X4k+1 = XAk +v{V4k),
VAk+l = y4fc)
x4k+2
J/4fc+2 = Vik+1 + U(x4k+l),
= ^4ik+l,
®4fc+3 = XAk+2 ®4(fc+l) = ®4Jfc+3,
v{y4k+2)
V4k+3 = V4k+2,
3/4(Jfc+l) = V4k+3 - u(xAk+3)
(see Figure 10) lie also in the boundary of the domain D.
(12)
98
M. M.
Lavrent'ev
From the supposition that the domain D satisfies the condition of Theorem 5 it follows that
lim X2k = o 2 ,
K—foo
lim y2k = h,
lim y2fc+i = h«-+00
fc—foo
(13)
Equalities (12), (13) are the equations for determining φι(χο). It is easy
to see (taking into account the properties of the functions <Pk{x)) that the
solution of this equation is unique.
So, we had established the following theorem.
Theorem 7. Solution of the inverse problem formulated above is unique.
13. Inverse problem for the domains with the a transitive
boundary.
Let D be a bounded domain convex relatively the axes (x) and (y) whose
boundary is continuously differentiable. Denote the vertices of this domain
by (αι,δι), (®2,6 2 ), (03,63), (04,64), αϊ < α 2 < α 3 < α 4 ; 6 2 < h < 64 <
63. Then there exist the functions fi(x), f2(x), 0i(y), 02 (y) continuously
differentiable in the intervals χ G (01,04), y G (62,63) such that
D = {(χ, y) I fi {χ) <y<
= {(®,ν) 19i(y) <χ<
/2(χ),
χ G (αι,α4)}
92(y), y e (62,63)};
1) Λ (01) = Λ (αϊ), /i(t»4) = / 2 Μ ; 0ι(6 2 ) = 02(62), 01(63) = 02(63);
2) fi{x)
= ¡2(x), χ e (01,04); 9i(y) = 0 2 ( y ) , y e (62,63);
3) f{{x) <0,xe
(αι,α 2 ); f[(x)
?2{χ) < Ο, χ G (03,04);
> Ο, χ G (α 2 ,α 4 ); f^{x) > Ο, χ G ( o i , o 3 ) ;
4) 9i[h(x)] = χ, χ G (01,02); gi[f2(x)] = χ, χ e (02,03); 92[h{x)] = χ,
a; G (02,04); ff2[/2(a;)] = x, x € (03,04);
We denote by it(®), υ (y) the functions
«(®) = fi(x) - fi(x),
v{y) = 02(y) - 01 (y).
(14)
Inverse problem. Given the functions u(x), v(y), find the functions
fi(x), /¡(x), 91 (y), 02(y)·
We assume, in addition, that the curve C that bounded the domain D
is transitive and the vertices of C axe known.
Ill-Posed and inverse problems for hyperbolic
equations
99
Theorem 8. The solution of this inverse problem is unique.
Proof. We consider the left vertex (αι,δι) of the curve C. By (14), the
point (a;i,6i) G C, where x\ = αχ + υ ( 6 ι ) . Further, we have (xi,yi) G C,
where yi — b\ + it(xi).
Consider the sequence of points ( x k , y k ) , {xk+i,Uk), A; = 1 , 2 , . . . defined
as follows:
Xk+1 = xk±v(yk),
Vk+i =
^
yk±u{xk+i).
The signs before v(yk), u(xk) in (15) are defined dependently on the mutual
disposition of the points {xk,yk), {xk+hVk) and the vertices of the curve C.
Since, by our assumption, the curve C is transitive, the points (xk,yk),
{xk+i,yk) form the everywhere dense set in the curve C, whence the theorem
follows.
•
REFERENCES
1. R. A. Aleksandryan, On a Sobolev problem for special equations with
partial derivatives of the fourth order. Dokl. Akad. Nauk SSSR (1950),
73, No. 4 (in Russian).
2. T. I. Zelenyak, Selected Questions of Qualitative Theory of Equations
with Partial Derivatives. Novosibirsk Gos. University, 1970 (in Russian).
3. V. K. Imanaliev, P. S. Pankov and O. G. Brender, The method of subsequent differentiating for construction of generalized solution of the
integral Fredholm equation of the first kind. In: Research of IntegroDifferential Equations Akad. Nauk KSSR, Frunze, 1983 (in Russian).
4. R. Courant, Equations
(in Russian).
with Partial Derivatives.
Mir, Moscow, 1964
5. M. M. Lavrent'ev, On Certain Ill-Posed Problems of Mathematical
Physics. Sib. Otdel Akad. Nauk SSSR, 1962 (in Russian).
6. M. M. Lavrent'ev, V. G. Romanov and S. P. Shishatskii, ΠΙ-Posed Problems of Mathematical Physics and Analysis. Nauka, Moscow, 1980 (in
Russian).
7. M. M. Lavrent'ev, On a boundary-value problem for hyperbolic system. In: Matem. Zbornik (1956), 38(60) (in Russian).
100
M. M. Lavrent'ev
8. R. Lattes and J.-L. Lions, Quasiinversion Method and Its Applications. Mir, Moscow, 1970 (in Russian).
9. M. A. Lavrent'ev and Β. V. Shabat, Problems of Hydrodynamics and
Their Mathematical Models. Nauka, Moscow, 1977 (in Russian).
10. P. S. Novikov, On uniqueness of inverse problem of potential theory.
Dokl. Akad. Nauk SSSR (1938), 18, No. 3 (in Russian).
11. A.I. Prilepko, On inverse problems of potential theory. Differential
Equations (1967) 3, No. 1 (in Russian).
12. I. G. Petrovskii, Lections on Equations with Partial Derivatives. Fizmatgiz, Moscow, 1961 (in Russian).
13. S. L. Sobolev, Equations of Mathematical Physics. Nauka, Moscow,
1966 (in Russian).
14. V. H. Strakhov, On solution of ill-posed problems of magneto and
gravimetry represented by integral equations of convolution type. Izv.
Akad. Nauk SSSR. Fizika Zemli (1967) 4 (in Russian).
15. A. N. Tikhonov, On stability of inverse problems. Dokl. Akad. Nauk
SSSR (1943) 39, No. 5 (in Russian).
16. A. N. Tikhonov and V. Ya. Arsenin, Solutions of Ill-Posed Problems.
Wiley, New York, 1977.
17. A. N. Tikhonov, On solution of ill-posed problems and regularization
method. Dokl. Akad. Nauk SSSR (1963) 153, No. 1 (in Russian).
18. Β. V. Shabat, On the analog of Riemann's theorem for hyperbolic system of differential equations. Uspekhi Math. Nauk (1956) 11, 3(69) (in
Russian).
19. B.V. Shabat, On hyperbolic quasiconformal mappings. In: Certain
Problems of Mathematics and Mechanics. Nauka, Leningrad, 1970 (in
Russian).
20. Sh. Yarmuhamedov, On analytic extension of holomorphic vector by
its values on the part of boundary. Izv. Akad. Nauk Uz. SSR, Ser.
Fiz.-Mat. (1980) , No. 6 (in Russian).
21. J. Douglas, Numerical Method for Analytic Continuation, Boundary
Problems Different Equations. Madison Univ., Wisconsin Press, 1960.
Ill-Posed and inverse problems for hyperbolic equations
101
22. J. Hadamard, Sur les problèmes aux derivees partielles et leur signification physique. Bull. Univ. Princeton (1902) 13.
23. J. Hadamard, Le Probleme de Cauchy et les Equations aux Derivees
Partielles Lineaires Hyperboliques. Paris, Herman, 1932.
24. F. John, The Dirichlet problem for a hyperbolic equation. Amer.
J. Math. (1941) 63.
25. C. Pucci, Discussione del problema di Cauchy pur le equazioni di tipo
elliptico. Ann. Mat Pura ed Appi. (1958).
26. M. Yamamoto, Conditional stability in determination of force terms
of heat equations in rectangle. Math, and Comp. Modelling (1993) 18.
Ill-Posed and Non-Classical Problems of
Mathematical Physics and Analysis, Samarkand,
M.M. Lavrent'ev and S.I. Kabanikhin (Eds)
© VSP 2002
2000, pp. 103-124
Systems of linear integral equations of Volterra type
with singular and super-singular kernels
N. RAJABOV
Abstract — In this paper linear systems of integral equations of Volterra type
with the left and right fixed singular and super-singular points are researched.
Depending on disposition of singular points, on the singularity order, and on the
roots of the correspondent characteristic equations, the general representation of the
solution set is obtained. The cases when the correspondent homogeneous systems of
integral equations have nontrivial solutions and when there exist no solutions except
the trivial solution are considered. In the case when the homogeneous system has
the trivial solution only, the correspondent inhomogeneous system has a unique
solution. In the other cases the general solutions of the inhomogeneous systems
depend on several arbitrary constants. In some particular cases the solutions of
these systems of integral equations are found explicitly. The behaviour of solutions
in the neighborhood of singular points is investigated. Note that the problem of
determining the collection of continuous solutions of systems of ordinary differential
equations with fixed boundary singular and super-singular points is reduced to
consideration of such systems of integral equations [1, 2].
1.
THE CASE W H E N THE KERNELS ARE
T H E F U N C T I O N S OF VARIABLE I N T E G R A T I O N
Let Γ = {α < χ < 6} C Ä 1 . We consider the following system of integral
equations
where ajk(x), f j ( x ) , (1 < j, k < η) are known continuous functions of the
points from Γ, η € Ν , and a = const > 0.
104
Ν. Rajabov
We shall seek the solution of system (1.1) in the class of functions continuous in Γ : Vj(x), (1 < j < n), which have the proximity order in the
point χ — a higher than a — 1:
Vj{x) = o[(x — a) 7 ; ],
jj > a — 1
for
χ
a.
The scalar equation of type (1.1) was investigated in [3, 7].
In this section the general solution of system (1.1) is obtained depending
on α (α < 1; a — 1; a > 1) and the roots of characteristic equation
Δ ( λ ) = det ||α,·*(α) -
= 0.
(1.2)
In the case when a,jk(x) = ajk — const, the solution of system (1.1) is found
explicitly.
When the roots of characteristic equation (1.2) are real and distinct the
following statements hold.
T h e o r e m 1.1. Suppose that α > 1 in system (1.1) and the roots of
characteristic equation (1.2) Xj (1 < j < n) are real and distinct; Xj < 0;
(1 < j < rc)· Let, besides, fj(x), ajk{x) € C(T); (1 < j, k < η) and satisfy
the following conditions for χ -> α:
τι
/ί(ι) = ο [ β χ ρ ( ( ^ λ η , ) ω β ( ι ) ) . ( ι - ο ρ ] ,
71 > α - 1,
(1.3)
ΤΙ
= o e x p ( ( ^ Am)wa(a;)) - ( χ - α ) 7 1 ] ,
(1.4)
771=1
ajk(x)
- ajk(a)
771=1
72 > α — 1,
ωα(χ)
= [(α - 1 ) ( ι - a ) Q _ 1 ] _ 1 ,
1 < j,
k<n.
Then homogeneous system (1.1) has η solutions linearly independent. Inhomogeneous system (1.1) is solvable in the class Ο(Γ) and its general solution
has η arbitrary constants. In this case the general solution of system (1.1)
is expressed via the resolvents of the system of Volterra equations of the
second kind with a weak singularity in the point χ — α:
Vj{x) = T j a ) [ c 1, c 2 , . . . , cn, /i(x), f2{x),..
n
rX
/
+ Σ
k=lJa
where Cj (1 < j <n)
. , fn(x)]
T^(x,t)T¡ca)[cl,c2,...,cnJ1(t),...,fn(t)}dt,
are arbitrary constants.
Systems of linear integral equations
In this representation T^(x,
of integral equations
Vii*) - Σ Γ
l—l J a
K )(
71
** ^
^
Σ
t) are the resolvents of the following system
ß <X' *)*«(*) d ¿
= T a) C
i t l ' C2' ...,Cn,fí(x)t...,
fn(x)]i
1
= ±ckj
k=1
T¡a)[ci,c2,...,
105
±
m=l
) exp
<j<n,
- <,.(*))],
^
Cn, fi(x),...,
'
η
/ n (x)] = Σ °kj exp[afco;Q(a:)]cfc
^^
.
çx
~ ~ I / m ( s ) - α * / exp[Ajk(o; Q (a;)-a; a (<))]/ m (í)(í-a)~ a dí ,
1 < i < η,
where Δ = det ||cfc¿||,
the system
η
are the cofactors of cmk·, ckj are the solutions of
Σ ajk(a)ck - Acj = 0
(1 < j < n)
k=1
correspondent to the values \ = \¡t (1 < k < n).
Theorem 1.2. Let λ = 1 in system (1.1) and the roots of characteristic
equation (1.2) Xj (1 < j < n) be real and distinct; Xj < 0 (1 < j < n).
Let, moreover, fj(x), Q>jk{x) £ C(r) (1 < j,k < n) and satisfy the following
conditions for χ —» α
f j ( x ) = o[(x - α)-Σ™=1 ^m+73j ;
73
>
0)
λτη+74
ajk(x) - ajk(a) = o[(x - a)~
],
74 > 0, 1 < j, k < η.
Then homogeneous system (1.1) has η linearly independent solutions. Inhomogeneous system (1.1) is solvable and its general solution belongs to
the class C(T) and contains η arbitrary constants. General solution of system (1.1) is expressed via the resolvents of the system of integral Volterra
equations of the second kind with the weak singular point χ = α:
yj(x) = ìj X ) [ci, c2, ...,cn,fi{x),f2{x),...,
/„(a;)]
+Σ /
rjk(x,t)Tk[c1,C2,...,CnJ1(t),...,fn(t)}dt,
k=1 Ja
where Cj (1 < j < n) are arbitrary numbers.
Ν. Rajabov
106
Here we denote by Γ j k { x , t ) the resolvents of the following system of
Volterra equations of the second kind with the weak singularity
n
yj(x)~Y^
1=1
px
/ Kjl(x,t)yk{t)àt
= T¡1][ci,c2,...
,cn, fi(x),...
Jn(x)],
Ja
1
fc=l
1, C2, •••,Οη, fl(x),
m= 1
fn(x)] = ^
v
<j<n,
'
ckj(x - a)
Xk
di,
1< j < η
771=1
Now, we shall give the proofs of these theorems.
Proof of Theorem 1.1. Suppose that a > 1 in system (1.1). Adding
and subtracting the numbers aj k {a) from the integrands, we rewrite the
system as follows:
Π
(x)
+
J2a
yj
jk(a)
k=ι
nJJ
( t - a ) - a y k ( t ) d t = F¡a)(x),
1 < j < n,
(1.5)
1 <j<n.
(1.6)
Ja
where
71 /»X
a)
(t-a)-a[ajk(t)-ajk(a)]yk(t)dt,
F¡ (x) = fj(x)-J2
Ja
k=i
Since the functions Fja\x) (1 < j < n) are known, we find at first the
general solution of system (1.5).
In the case when the functions Fja\x) are known (a,j k (x) = a,jk(a)) we
shall call the system (1.5) by the model system or by characteristic system of
integral Volterra equations with the super-singular (a > 0) left fixed point
in the kernel.
Assume that system (1.5) has the solutions, these solutions are different i a t e and F¡a){χ) e C'{Γ) (1 < j < η). Differentiating both sides of (1.5)
we obtain the following linear system of ordinary differential equations with
Systems of linear integral equations
107
the left boundary singular (a = 1) or super-singular (a > 1) point. The
theory of such systems
η
a
y'j(x) + ( x - a ) - J 2 * j k ( a ) y k ( x ) = F ¡ ( x ) ,
fc=l
1<3<η.
(1.7)
is well developed in [1]. It is known [1] that this system has also the solutions
which are unbounded in the point χ — a. We need to find the continuous
solution of this system for which χ = a is the zero of the order higher than
α - 1.
Firstly, as in [1], we find the solution of the homogeneous system
η
y'j{x) + (χ - α)" Y^ajk{a)yk{x)
Ω
= 0.
(1.8)
A;=l
The solution of system (1.8) we shall seek in the form (see [1])
yj(x) = Cj exp [λω α (ζ)],
ωα(χ) = [(α - l)(x - α ) α - 1 ] _ 1
(1 < 3 < η).
(1.9)
Acting similarly as in the theory of ordinary differential equations with
constant coefficients we pass to the following algebraic system
η
a
Σ
ik{a)ck
- Xcj = 0
(1 <j<
n).
(1.10)
k=l
The determinant of this system has the following form
A(X)=det\\ajk(a)-Xôjk\\=0.
(1.11)
Suppose that the roots of algebraic equation (1.11) are real and distinct
where λ;· < 0 (1 < j < n). The following fundamental system of solutions
corresponds to these roots
ykj{x) = ckj exp [λ^α^χ)],
1 < j,
k<n,
where ckj solve system (1.10) for the values λ = Xj (1 < j < ή).
Then the general solution of system (1.8) from the class 6"(Γ) Π C ( r ) is
given by the formula
η
Vj(x) = Σ akiCk
k=ι
exp
[λ*ωα(χ)]'
h < o (1 < k < η),
1 < j < η,
Ν. Rajabov
108
where ck (1 < A; < η) are arbitrary constants. By the conditions of Theorem 1.1, \k < 0 (1 < k < n)·, therefore, yj(x) G C"(r) and in the point
χ = a they vanish.
Using the method of variation of parameters, we obtain that if the solution of inhomogeneous system (1.7) exists, then it has the form
n
^^
rx
ck + ^ — /
βχρ[-λ*ω α (ί)]^(ί)<1ί
Ja
m=ι
1
<j<n,
yj(x) =
k= ι
where Δ — det||c¿j||, Amk are the cofactors of cm£, ck (1 < k < n) are
arbitrary constants.
Integrating by terms the integrals which stand in the right-hand side of
this equality and setting
exp (-\mUa{t))Fm(t)\t=a
—0
we obtain the following equalities
τι
VAX) = Σ
k=1
(1 < m < n),
n
η
eX
-λ* J*
P \.Xk<¿a(x)]ck + J2 Ckj J2
k=1 m=l
(1.12)
A
[FmHx)
exp[XkMx)-ua(t))](t-a)-aF^(t)dt\
= Ί?[α, ...,Cn·, F{ a ) (x),..., F^(x)},
1 <j<n.
(1.13)
These solutions of system (1.5) were obtained in supposition that the
desired functions yj(x) (1 < j < η) and the function Fja\x) (1 < j < n)
from this system axe differentiable and F^a\x) (1 < j < n) satisfy the
conditions of type (1.3). By immediate computation we can verify that the
functions yj(x) (1 < j < n) defined by (1.13) satisfy the system of integral
equations (1.5) for each Fja\x) G C(r) (1 < j < n) satisfying conditions
(1.12).
•
Thus, we have proved the following lemma.
Lemma 1.1. Let in system (1.5) the numbers a,jk(a) (1 < j, k < n)
be such that the roots of characteristic equation (1.2) Χ3 (1 < j < η) are
real and distinct and Xj < 0 (1 < j < n). Let, moreover F^ (x) € C'(T)
(1 < j < n), they vanish in the point χ = α and in the neighborhood of
χ = α the following asymptotic formulas hold
F¡a)(x) = ο [ β χ ρ ( - | λ μ Ω ( χ ) ) ( χ - α ) 5 ] ,
¿>a-l,l<¿<n.
(1.14)
Systems of linear integral equations
109
Then system (1.5) is solvable and its general solution contains η arbitrary
constants and is given by formula (1.13).
Suppose now that the functions Fja\x) (1 < j < η) have form (1.6);
Uj(x) (1 < j < n) are unknown. By immediate computation we can verify
that if the functions f j ( x ) (1 < j < n), a,jk(x) — α^(α) (1 < j, k < n)
satisfy conditions (1.3) and (1.4) of Theorem 1.1 respectively, yj{x) E C(T)
(1 < j < n) and vanish for χ
a, then the functions F*{x) (1 < j < n)
satisfy conditions (1.14) for χ -» a. Act by the integral operator which is
inverse to the integral operator (1.5)
rjia —
(
T
l\
\Tn/
on both sides of equality (1.13). As a result we obtain the following system
of linear integral Volterra equations with the weak singular point t = a
"
ΓX
Kji(x, t)yi(t) dt = T"[cl,...,cn-J1(x),...,
yj(μ - Σ
1—1 Ja
fn(x)},
(1.15)
1 < j < n,
where
Kjl(X,t)
= ±ckj
k=1
¿
m—1
exp[Xk{ujQ{x)
(t-aY
l<k<n,
h<0.
-ωα{ί))],
(1.16)
Solving system (1.15) we find
Vj(x) =T?[ci,...,Cn-,fi{x),...,
fn{x)]
n
rx
+
rjl(x,t)Tla[cl,...,cn-Jl{t),...,fn(t)]dt,
Ja
ί=ι
where Tji(x,t)
(1.17)
(1 < j, I < η) are the resolvents of system (1.15).
Proof of Theorem 1.2. Let a = 1. In this case, for system (1.1), we
obtain the following characteristic system
k=ι
Ja
1
a
),
1
<j<n,
(1.18)
Ν. Rajabov
110
where
ή % ) = Mx) - ± Γ
fe=iJa
1
! < j < n.
dt>
a
(L19)
The solution of homogeneous system (1.18) we shall seek in the form
yj(x) = Cj(x - a)~x
(1
<j<n).
Acting following the scheme of the proof of Theorem 1.1 we see that if the
functions Fj^(x) (1 < j < n) for χ ->· a have the behaviour
F¡1]{x) =ο[(χ-α)Σ?=ιΙλ*Ι+<^
¿>0,
(1 <j<n),
(1.20)
the roots of characteristic equation (1.2) are real and distinct and
< 0
(1 < j < k), then the solution of system (1.18) from the class C(r) is given
by the formula
η
k-1
JL Λ , r ...
•
E ^ M m—1
^
í
Γ * / * - « χ λ, pi1)
z í ) « · ^ *
ξ Τ ^ ο , · . . , ^ ^ ^ ) , . . , ^ ^
]
(l<J<n).
(1.21)
•
Prom these considerations the below lemma follows.
Lemma 1.2. Suppose in equation (1.18) the functions F^fa) G C(r)
vanish for χ —> α and their behaviour is defined in asymptotic formula (1.20).
Suppose also that the roots of equation (1.2) are real and distinct, A^ < 0
(1 < k < n). Then system (1.18) is solvable and its general solution from
the class C(T) contains η arbitrary constants and is given by formula (1.21).
Substituting the values F ^ \ x ) (1 < m < n) into representation (1.21)
and repeating the scheme of the proof of Theorem 1.1 when determining
Pj(x) (1 < j < n), we come to the following system of integral equations
n
px
/
1=1
Kjl{x,t)yl{t)dt
=
T¡1}[ci,...,cnJ1{x),...fn(x)},
J a
1 <j<n,
(1.22)
111
Systems of linear integral equations
where
- ± c
w
fc=l
V
t
%*(««<'» :
(i23)
m=l
Suppose that the functions ami(t) — ami(a) satisfy the conditions of Theorem 1.2 and λ* < 0 (1 < k < ή). Then the kernels Kß(x,t) defined by
formula (1.23) will have the weak singularity in the neighborhood of the
point t = a and will be continuous for each fixed t with respect to x. If the
functions fj(x) (1 < j < n) satisfy the conditions of Theorem 1.2, then the
functions T^ [c\, ...,cn, fi{x),... fn(x)] (1 < J < n) will be continuous for
λ/c < 0 (1 < j < n). Prom the theory of linear systems of Volterra equations
it follows that system (1.22) has a unique solution
yj(x) = r j ^ f c i , ...,Cn,fi(x),...
fn{x)]
Π rtj
+ Σ
Tji(x,t)T¡l\ci,...
,c n , / ι ( ί ) , . . . fn(t)] di,
1=1
l<i<n,
(1.24)
J a
where Γ^(χ,ί) are the resolvents of system (1.22).
Corollary 1.1. In the case when Ojk(x) = const and thefirstconditions
of Theorem 1.1 or Theorem 1.2 hold, then the general solution of system (1.1)
from the class C(Γ) is written explicitly by the formula
=2ja)[ci,...,cn;/i(a:),.../n(x)]l
1 < j < η, α > 1,
where ci, C2,..., Cn are arbitrary constants.
Corollary 1.2. Suppose that üjk(x) = ajk = const, a > 1, the roots
of characteristic equation (1.2) are real and distinct, Xj > 0 (1 < j < n),
fj{x) € C(r) (1 < j < n) and satisfy the following conditions for χ
α:
fj(x) = o[{x — α)ε],
1 < j < η,
ε > α — 1.
Then homogeneous system (1.1) has only the trivial solution and the inhomogeneous system (1.1) has a unique solution. In this case the general
solution of system (1.1) from the class C( Γ) is given explicitly by the formula
yj(x) = Τ ] α ) [ 0 , 0 , . . . , 0; fi (χ),...,
fn(x)],
1
<j<n.
Ν. Rajabov
112
In the case when η = 2, üjk(x) = ajk = const and the conditions of
Theorem 1.1 or of Theorem 1.2 hold, the general solution of system (1.1)
from the class C(r) is given respectively by the formulas
Vj(x)
=
K}\'X2[C1,C2JI(X)J2(X)]
Ξ
CmCmjix - a)~Xm
πι—1
Σ
+ &Öl[cij(c22fl{x) - C2lh{x)) + c2j(cnf2(x)
+ Δ
~ C2lfl(x))]
0 1 J* [ ( ^ ) A l c i J A 1 ( c 2 2 / 1 ( t ) - c 2 1 / 2 ( i ) )
c
+ (—z~)
2j^2{cnf2{x) - C2l/l(œ))
VX GL '
dt
•
,
n
τ—a
for a — 1 and
Vj(x) = Kj^X2[ci,c2,fi(x),f2{x)}
+ A^lcijfaflix)
- Δο 1 J
= Σ CmCmj exp[\mua{x)]
771=1
- C 2 l/ 2 (x)) + c2j(cnf2(x)
I exp [λι(ω α (ι) - ua(t))]\i(c22fi(t)
+exp[\2(uja{x)-uja{t))]-\2c2j(cnf2{t)-c12h{t))
- Cl2/l(x))]
- c 2 i / 2 (x)) · cij
} (t — a)~a di, j = 1,2
for a > 1. Here ci,c 2 are arbitrary constants, Δο = cn-c 22 — ci 2 -c 2 i; cij,c 2 j
(j = 1,2) are the solutions of the algebraic system
2
- Ssj\) = 0,
j = 1,2
s=l
for λ = λι and λ = λ 2 respectively; ôss = 1, Ssj = 0 for s ψ j.
Further, for simplicity, we shall formulate the results for the case η = 2.
In this case η = 2, when in system (1.1) the roots of characteristic
equation (1.2) are real and equal or the roots are complex conjugate, the
following statements hold.
Theorem 1.3. Suppose in system (1.1) α > 1, η = 2 and the roots
of characteristic equation (1.2) are real and equal: λχ = λ 2 = λ < 0. Let,
moreover, the functions fj{x), djk{x) 6 (7(Γ) (1 < j, k < 2) satisfy for
χ —> α the following conditions
f j ( x ) = o[exp (-μμ β (ζ))(χ - a n
75 > 2(a - 1),
76
ajk{x) - ajk (a) = o[exp (-|λ|ωβ(®))(α: - a) ], 75 > 2(a - 1).
(1.5)
(1.6)
113
Systems of linear integral equations
Then homogeneous system (1.1) has two linearly independent solutions. Inhomogeneous system (1.1) is always solvable in the class C{Γ), and its general solution has the two arbitrary constants. The general solution of system (1.1) is expressed by the resolvents of the system of Volterra equations
with the weak singularity in the point t = a:
Vj ) = Mi [ci « c21 h (s), h (®)]
2 χ
+ ¿ í rjk(x, t)Mj[cι, ça, fx(t), f2(t)] dt, 1 < j < 2,
k=lJa
where ci and c2 are arbitrary constants.
We denote here by Γjk(x,t) the resolvents of the following system of
integral equations
Vj(x)
m=l
where Kjm(x,t)
system (1.1).
Kjm(x,
t)ym{t) d t - Mj[ci,c2, fi(x), Λ (a;)], 1 < j < 2,
are the known functions dependent on the coefficients of
Corollary 1.3. Let n — 2, ajk{x) = a,jk = const and condition (1.5) of
Theorem 1.3 hold. Then the general solution of system (1.1) from the class
C'(Γ) Π C{T) is as follows
yj(z) = Mj[c1,c2,f1(x),f2(x)],
j — 1,2
where
Afi[ci, C2, fi{x), h{x)] = exp [λωβ(χ)](οι + ωαο2)
+ [(θ22 - α,\ι)2~ιωα(χ) + 2]fi{x) Λ
αχ2ωα{χ)f2{x)
J
- / exp [λ(ω α (ι) —ωα(ί))](ί — α)-α[(α22 —θιι)(ωα(ί) — 1 — ω α (ί)λ2 _ 1 )/ι(ί)
Ja
+ 012(2 - 2Χωα(ί) - \ua(x))f2(t)] dt,
M2[ci,c2,fi(x),f2(x)]
=
βχρ[λωα(χ)](6ιιθι+ωαθ2)+(6π[(α22-αιι)ωα(α;)+2]
+ 6i 2 (a;)2 _1 (oii - a 22 ))/i(a;) + (&12ΟΦ12 -
2ai2bnua(x))f2(x)
- Í ( t - a ) ~ a exp [λ(ω0(α;) -ω Λ (ί))]{[6π(λ(α 2 2 -αιι)ω α (ί) + 2) fe-ou)
Ja
+ 6ι2(α:)λ2-1(αι1 - α 22 )]/ι(ί) + [2α12 - 2α12λωα(ί) + bl2{x)a12]f2{t)}
dt,
Ν. Rajabov
114
6ιι = (2οι2) Ηα22 - απ),
612(2;) = 611 ωα(χ)+α12,
and Ci,C2 are arbitrary constants.
Theorem 1.4. Let in system (1.1) α > 1, η = 2, the roots of characteristic equation (1.2) be real and equal, i.e. Λι = Λ2 = Λ > 0. Let, moreover,
fj(x), üjk(x) E C(r) fi < j, k < 2) and, for χ
α, condition (1.6) and the
following condition hold
f j ( x ) = o[{x - α)ε],
ε > α - 1,
j = 1,2.
Then homogeneous system (1.1) has the trivial solution only. Inhomogeneous system (1.1) has a unique solution in the class C(Γ):
[
r jm (x,í)Afm[0,0,/i(t),/ 2 (¿)]dí,
1 J 0.
m=l
3 = 1,2.
Here Tjm(x, t) are the resolvents of the following system of integral Volterra
equations with the weak singularity in the kernel
y j
( x ) = Mj[0,0,fi(x),f2(x)]
+ Σ
VÁX) + Σ / Κί™(χ> Qy*»® dt
™
—ι J 0,
m=i
and Kjm(x,t)
tem (1.1).
=
3 = 1,2,
are the known functions dependent on the coefficients of sys-
Theorem 1.5. Let in system (1.1), τι = 2, α > 1, f j ( x ) G C(r),
üjk(x) = ajfc = const. Suppose, moreover, that
1) the roots of characteristic equation (1.2) are complex conjugate, i.e.,
λι = ρ + iq, λ2 = ρ - iq, ρ < 0;
2) the functions fj(x), (j = 1,2) vanish in the neighborhood of the point
χ = a, and their behavior is as follows:
fj(x) = o[exp (-|ρ|ωβ(®))(® - α)77],
ηΊ > α - 1, j = 1,2.
Then homogeneous system (1.1) has the two linearly independent solutions.
Inhomogeneous system (1.1) is solvable in the class C(T) and its solution is
given by the formula
yj{x) = Ej[cuc2,h{x),h{x)},
3 = 1,2,
115
Systems of linear integral equations
where ci, c 2 are arbitrary constants and
Ei[ci,c2,fi{x),f2{x)]
°12
+ fi(x) ~ ~
J
Γ
= exp[-|p|<J a (a;)][ci cos(gu>a(a;))] + c2 sm(qu>a{x))]
( ί - α ) ~ α β χ ρ [ ρ ( ω α ( χ ) -ω α (ί))]{[ρ8ίη[?(ω α (χ) - ω β ( ί ) ) ]
+ qcos[q(u)a(x) - ω α (ί))]/ 2 (ί)] + [ançcos[ç(c<;a(i) - u;a(a;))]
+ (ana 2 2 - αι 2 α 2 ι - p a n ) s i n [ ç ( a ; a ( i ) - w a (a;))]]/i(í)} di,
E2[ci,c2,fi(x),f2(x)]
--^-sm(qua(x))ci
-q~l
=exp[-|p|a; e (ar)] f ( - — — )
L \ a\2 /
+ ^-^-cos(qua{x))
í {t-a)~aexp\p(ua{x)
Ja
+ a2iqcos[q(ua(t)
+ ^P ^
cos(gua(x))
sin(gt<;a(a;))jc2 j
+f2{x)
- u; a (i))]{[po 2 i sin[ç(a> a (a;)-ω) β (ί))
- ω β (®))]]/ι(ί) + [(2pq ~ ?βιι) cos[q(ua(t) - ω α (χ))]
+ ( q 2 - p ( p - an))sin[g(a; a (í) - ua(x))]f2(t)}
dt.
R e m a r k 1.1. The statement analogous to Theorem 1.5 is obtained in
the case when ajk(x) φ const and the condition of Theorem 1.5 hold. We
suppose, moreover, that the functions ajk(x) satisfy the conditions similar
to (1.6) in Theorem 1.3 (for λ = ρ). In this case the general solution of
system (1.1) is expressed via the resolvents of the system of the integral
Volterra equations of the second kind with the weak singularity in the point
t = a and with the right-hand side Ej[ci,c2, fi(x), f2(x)], j — 1,2.
R e m a r k 1.2. The case when a = 1, η = 2 and the roots of characteristic
equation (1.2) are complex conjugate and Re λ = ρ < 0 is studied also. The
statement similar to Theorem 1.5 is obtained.
R e m a r k 1.3. The analogous results hold in the case when η = 2, a > 1
and the roots of characteristic equation (1.2) are complex conjugate and
Re λ = ρ > 0.
Theorems 1.1-1.5 and the statements mentioned in Remarks 1.1-1.3 are
proved on the base of connection of system (1.1) and the system of lineax ordinary differential equations with singular and super-singular coefficients [1]
(see also [2, 3]).
Ν. Rajabov
116
2.
THE GENERAL CASE
In this paper we research also the general linear system of Volterra equations
η çX
Vj(x)
+™Σ—1 /J CL - ar Kkm(x,t)y (t)
a
m
τη—1
dt =
/_,·(*), 1 < ; < η, (2.1)
depending on the order of singularity α (α < 1, a = 1, a > 1) and on the
roots of correspondent characteristic equation
Δ(λ) = det IIK j m (a, a) - XSjm\\ = 0.
(2.2)
We represent system (2.1) as follows:
Vj(x) + Σ Ki™(a>a) Í (* ~ a)~aVm(t) dt = Fjix),
J a,
m=lι
j = 1,2,
(2.3)
where
η ρχ
Fj{x) = fj{x) - Σ / (' - a)-a(Kjm(x,
m=l ι J α
t) - Kjm(a, a))ym{t) dt
(l<j<n).
(2.4)
Suppose that μ;·(1 < j <n) are the roots of characteristic equation (2.2)
Δ(μ) = det \\Kjm(a, a) - ßöjm\\ = 0.
2.1. Let in system (2.1), a = 1, η = 2 and the roots of characteristic
equation (2.2) μj ( j = 1,2) be real and distinct. Suppose, moreover, that
f j ( x ) 6 C(F), Kjm(x,t) € C ( I ) , R = {a < χ < b, a<t<b},
yj(x) e C(F)
and the following condition holds for χ —> a:
^(χ)
= ο[(χ-α)~^+^],
j = 1,2.
(2.5)
If the solution of system (2.3) exists for μj < 0 (j = 1,2) then, following
Theorem 1.1, it is expressed as follows:
yj(x)
= K^[cl,c2,F1(x),F2(x)),
j = 1,2,
(2.6)
where
2
F^x), F2(X)] = £ c*c*i(* - a)~"h
k=l
Δ0 [cij(c22-Fi0c) c21F2(x)) + c 2 j (cnF 2 (x) -
K^[cltc2,
+
-
c12Fi)(x))] -
117
Systems of linear integral equations
- Δ ο 1 J*
[ ( L - i ^ c y ^ c œ F i W - C2iF2(t))
+ C^X*
c2^2{CllF2{t)
\x — a,/
- C 12 FI(Í)) 1
(2.7)
J t —a
where c\, c2 are arbitrary constants.
Now, substituting instead of the functions Fj(x) ( j — 1,2) their expressions from formula (2.4), we obtain the following system of integral Volterra
equations
Vii*) + Σ
Γ M (x,t)ym(t)
-, Ja jm
dt =
K^[c1,c2,f1(x)J2(x)l,
m=l
3 = 1,2,
(2.8)
where
Mjm(x, t) = AQ1[cij{c22(Kim(x,
+ c2j{cn(K2m{x,t)
-
— K\m{a,
í ÍÍ"—~Y
t — a Jt l\x — a/
t)-Kim(a,
a))-C2i(K2m(x,
a)) - ci2(Kim(x,t)
c
- Kim(a,a))}]/(t
ij^i{ c 22(-K"im(e,i) g
- ci2(Kim(s,
Assume that the functions Kjm(x,t)
t —>· χ —>· a
Kjm(x,t)
- a)
KXm{a,a))
χ — a/
-Kim(a,a))
o))}
β2
(
{cu(K2m(s,t)
t)-K2m{a,
t)
c2jß 2
-Kim{a,a))}
ds
s—a
satisfy the following conditions for
- Kjm(a, a) = o[(x - a)ß>(t - α)" 2 ],
(2.9)
ßi > 0, /?2 > \ßj \ — 1, j — 1,2.
Then, system of integral equations (2.8) will be the system of integral
Volterra equations of the second kind whose kernel will have the weak singularity in the point t = a.
If the kernels Kjm(x,t) satisfy condition (2.9) for χ —>· α, t —> a, then
system of integral equations (2.8) has a unique solution which is as follows:
Vj{x) = i^2[Cl,C2,/i(:r),/2(s)]
- £
ι Ja
771=1
l C r [ c i , c 2 , / i ( í ) , / 2 ( í ) ] d í , j = 1,2.
ΓTjm(x,t)
(2.10)
Ν. Rajabov
118
It is easy to see that the solution yj(x) of system (2.1) vanishes in the
point χ = a and its behavior is defined by the asymptotic formula
yj(x)
= o[(x-a)M+M],
¿ = 1,2,
(2.11)
in the case when the functions f j ( x ) vanish in the point χ = a and
f j ( x ) = o[{x - e ) l w l + l M l + i ] ,
δ > 0.
(2.12)
Thus, we have established the following theorem.
Theorem 2.1. Suppose in system (2.3), α = 1, η = 2, f j ( x ) € C ( r ) ,
Kjm(x, t) e C(R) (¿, m = 1,2) and the roots μ-j ( j = 1,2) of characteristic
equation (2.2) are negative and distinct. Let, moreover, the functions f j ( x ) ,
Kjm(x, t) satisfy conditions (2.12) and (2.9) when χ -* α, t -> α. Then,
homogeneous system (2.1) has two linearly independent solutions. Inhomogeneous system (2.1) is solvable in the class C(Γ) and its solution contains
two arbitrary constants. In this case, the general solution of system (2.1) is
given by formula (2.10), where c\, C2 are arbitrary constants.
In the case when α = 1, η = 2, ¿t¿ (¿ = 1,2) are real and distinct, ß j > 0,
the following statement holds.
Theorem 2.2. Suppose in system (2.1), η = 2, α = 1 and the roots of
characteristic equation (2.2) ßj ( j = 1,2) are real and distinct, ßj > 0. Let,
moreover, f j ( x ) 6 C(F), Kjm(x,t)
€ C(S) (j,m = 1,2) and
Kjm{x,t)-Kjm{a,a)
= o{(t-a)e),
ε > 0.
(2.13)
for χ
a, t -» a. Then inhomogeneous system (2.1) has the trivial solution
only. Inhomogeneous system (2.1) has the unique solution which is given by
formula (2.10) for c\= C2 = 0.
In the case when α = 1, η = 2, ßj ( j = 1,2) are real and distinct, μι < 0;
μ2 > 0 the following statement holds.
Theorem 2.3. Suppose in system (2.1) α = 1, η = 2 and the roots of
characteristic equation (2.2) are real and distinct and μι < 0, μ2 > 0. Let,
moreover, f j ( x ) e C(T), Kjm(x,t)
6 C ( I ) (¿,m = 1,2) and
Μχ) = φ - α ) Μ ) ,
Kjm{x,t)
- Kjm{a,a) = o[(x - α)^1],
βι > |μι| - 1
119
Systems of linear integral equations
for χ -4 a, t
a. Then homogeneous system (2.1) has the trivial solution
only. Inhomogeneous system (2.1) is solvable in the class C(Γ) and its
solution has one arbitrary constants. The general solution of system (2.1)
is given by the explicit formula via the resolvent of the following system of
integral Volterra equations of the second kind with the weak singularity in
the point χ = t = a:
2
yj(x)
+ Σ
ΓΜί™(χ>í)j/m(í)dí
=
κ
™_i
m = l J a,
ϊ \ '
μ
2
Μ Μ χ ) ] ·
R e m a r k 2.1. Suppose in system of equations (2.1), η = 2, a = 1,
the roots of characteristic equation (2.2) μ\ and μ2 are real, distinct and
μι > 0, μ2 < 0. Let, moreover, f j ( x ) G C(r), Kjm(x,t) G C(K) and satisfy
the conditions
fS(x) = φ
Kjm(x,t)
for χ —> a, t
system (2.1).
-
a)M],
- Kjm(a,a) = o[(x - α)β2},
> |μ2| - 1
a. Then the statement similar to Theorem 2.2 holds for
2.2. Let in system (2.1) η = 2, a > 1, the roots of characteristic
equation (2.2) μj ( j — 1,2) be real, distinct and μ^ < 0. Let f j ( x ) G C(r),
the functions Kjm(x,t) and the solutions yj{x), j,m = 1,2 of system (2.1)
be such that Fj (x) G C(F) and
Fj{x) = o[exp [(/χι + μ2)ωα(χ)]{χ - α) 7ε ],
7e
> α - 1, j = 1,2.
(2.14)
for χ —> a. In this case, when α > 1, using the results of Theorem 1.1 it is
easy to prove that the general solution of equation (2.3) is given in Γ by the
formula
j = 1,2,
(2.15)
Vj(x) = K^[C1,C2,F1(X),F2(X)],
where
2
Kf1f2[ci,C2,Fl{x),F2{x)}
= Σ
CmexpiVmUJaix))
m= 1
+ Aq 1[cij(c22iïi(a;) - c2lF2(x))
—
+ c2j(cnF2(x)
- C12F1 (χ))]
^o
/ {βχρ[λι(ω α (χ) - u;Q(i))]ÄiCij(JP1(i)c22 - F2{t)c21)
Ja
+ exp [Χ2(ωα(χ) - ua{t))]\2c2j(cnF2{t)
- c i 2 F i ( í ) ) ] ( í - a)~a dt,
3 = 1,2,
ci,c2 are arbitrary constants.
(2.16)
Ν. Rajabov
120
In representation (2.15) substituting instead of Fj(x), (j = 1,2) their
expressions from formula (2.4), we obtain the following system of Volterra
equations for unknown functions yj{x):
n
rs
/ N3kMyk(t)àt
™
—1 J o.
771=1
= Kf^ta[clìC2,Mx)j2(x)],
3 = 1,2,
(2.17)
where
N
jrn = Δΰl[cij{c22{Kim{x,t)
+ c2j(K2m(x,t)
- K2m(a,a))
- AQ 1(t - a)~a J
-Kim(a,a))
~ Ä"i m (o,a)) - c2i(K2m(x,t)
- ci2(Kim(x,t)
-
- ϋΓ1τη(α,α))}](ί
K2m(a,a))}
-
α)~α
[exp [μι(α>α(α;) - ω α (θ))]οι ; ·μι{ο 2 2 (ίίάηι(Μ)
- C2i(K2m(s,t)
- K2m(a,a))}
+ exp [μ2(ωα{χ) - ω α ( β ) ) ]
xc2jß2{cn(K2m(s,t)-K2m(a,a))-ci2(Kim(s,t)-Kim(a,
a ) ) } ] ( s - a ) ~ Q ds.
Suppose the functions fj{x) satisfy the following conditions for χ —»· o:
fj(x) = o[exp ((μι + μ2)ωα{χ))(χ
μ,<0,
79 > « - 1,
- α) 79 ],
j = 1,2.
(2.18)
Then the functions
üj(x) = κ^[α,02,Μχ),/2(χ)1
3=
1,2
will be continuous, i.e. Qj(x) G C ( r ) and their asymptotic behaviour for
χ —> a is as follows:
üj(x) = o[exp (~μωα(χ))},
μ = min(|/ii|, |μ 2 |).
Then the kernels Njm(x,t) of system (2.17) will have the weak singularity
if the functions Kjm(x,t)
satisfy the conditions
2
Kjm(x,t)
- Kjm(a,a)
= o e x p ( ^ μτηω α (ί))(χ - α) 710 ,
771=1
710 > α - 1 , 3 = 1,2
(2.19)
in the neighborhood of the points t = ο, χ = a.
We see that if conditions (2.18), (2.19) hold and yj(x) G C ( f ) , then the
functions Fj(x), ( j = 1,2) satisfy conditions (2.14).
Systems of linear integral equations
121
Thus, in the case when the roots of characteristic equation (2.2) are real
and distinct, η = 2, a > 1, the following statement holds.
Theorem 2.4. Suppose in system (2.1), η = 2, a > 1, the roots ßj (j =
1,2) of characteristic equation (2.2) are real and distinct, and ßj < 0 . Let,
moreover, (2.19) hold for χ —> a,t —• a. Then, homogeneous system (2.1) has
two linearly independent solutions. Inhomogeneous system (2.1) is solvable
in the class C(T) and its general solution has two arbitrary constants. The
general solution of system (2.1) is represented in the form
yj(x) =
Kf^[cl,c2,f1(x)J2(x)}
- ¿ [\jm(x,t)K^[cuC2,h(t)j2(t)]dt,
m—1J a
j = 1,2,
(2.20)
™ _ i
where j = 1,2; Tjm(x, t) are the resolvent of the system of integral Volterra
equations (2.17) with the weak singularity; c\,c2 are arbitrary constants.
In the case when α > 1, η = 2, ßj ( j = 1,2) are positive and distinct,
the following statement holds.
Theorem 2.5. Suppose in system (2.1), α > 1, η = 2 and the roots ßj
(j = 1,2) of characteristic equation (2.2) are real, distinct and positive.
Let, moreover, fj{x) G C(r), Kjm(x,t) G C(R) (m = 1,2) and satisfy the
condition
Kjm(xmt)
- Kjm(a, a) = o[(t - a)7u],
711 > a - 1
for t —• a. Then, homogeneous system (2.1) has the trivial solution only.
Inhomogeneous system (2.1) has a unique solution which is given by formula (2.20) for ci=c2 = 0.
Remark 2.2. The statement similar to Theorem 2.4 and Remark 2.1
is obtained in the cases when α > 1, η = 2, the roots ßj ( j = 1,2) of the
characteristic equation (2.2) are real, distinct and μι < 0, μ2 > 0.
2.3. Suppose in system (2.1) α > 1, η = 2 and the roots μ;· (j = 1,2) of
characteristic equation (2.2) are real and multiple. Let, moreover, fj{x) G
C(r), Kjm(x,t)
G C ( l ) , yj(x) G C(r) are such that the functions Fj(x)
satisfy the conditions
Fj(x) = o[exp H / * K ( z ) ) ( z - α)711],
7ιι > 2(α - 1),
Ν. Rajabov
122
for χ —>· a and μι = μ2 = μ < 0. Then, if there exists the solution of
system (2.3) when μ < 0, then it is expressed in the form (Corollary 1.3)
yj{x)
= Mj[c1,c2,F1(x),F2{x)],
j
= 1,2.
(2.21)
In representation (2.21), instead of F\(x), F2(x), we substitute their
expressions from formula (2.4). As a result we obtain the following system
of integral Volterra equations
2 Aj
ν*(*) + Σ / Tjm(x,t)ym{t)dt^Mj[c1,c2,fl{x),f2{x)],
m=l
j = 1,2, (2.22)
where Tjm(x,t) are set. If, moreover, the functions Kjm(x,t) satisfy the
following conditions in the neighborhood of the points χ = a, t = a
Kjm(x,t)
- Kjm(a,a)
712 > α - 1 ,
= o[exp (\ωα(χ))(χ
713 > α - 1 ,
- α)7ΐ2(< - α)713],
j , m = l,2,
(2.23)
then Tjm(x, t) has the weak singularity in the point t — a and is continuous
when χ = a.
Thus, we have proved the following theorem.
Theorem 2.6. Suppose in system (2.1), a > 1, η = 2, f j ( x ) G C(T),
Kjm(x, t) 6 C(K) (j, m = 1,2) and the roots of characteristic equation (2.2)
are real and equal: μι = μ2 = μ < 0. Let, moreover, f j ( x ) , Kjm(x,t)
satisfy
conditions (1.5), (2) for χ —> α, t —> α. Then homogeneous system (2.1) has
two linearly independent solutions. Inhomogeneous system (2.1) is solvable
in the class C{Γ) and its general solution has two arbitrary constants. In
this case the general solution of system (2.1) is represented explicitly via the
resolvents of system (2.22).
Remark 2.3. Suppose in system (2.1), α > 1, η = 2, the roots of
characteristic equation (2.2) μ^ ( j = 1,2) are real and distinct and have
distinct signs. Then the statement similar to Theorem 2.4 and Remark 2.1
holds for system (2.1).
Remark 2.4. Let in system (2.1), α = 1, η = 2, the roots of characteristic equation (2.2) be real and multiple: either μι = μ2 = μ < 0 or
μ! = μ2 = μ > 0, οτ μι = μ2 = 0. Then the statement holds similar to
Theorems 2.1, 2.2.
123
Systems of linear integral equations
Remark 2.5. System (2.1) is stated also in the case η = 2, a > 1,
μι = ρ + iq, μι = ρ — iq, ρ > 0, ρ < 0 or ρ = 0. In this case the statement
similar to Theorems 2.1-2.5 for solvability of system (2.1) holds.
Remark 2.6. Depending on the roots of characteristic equation (2.2),
system (2.1) is stated in the cases η > 2, a < 1; η > 2, a = 1; η > 2, a > 1.
In these cases the statement similar to Theorems 1.1, 1.2 hold also.
Remark 2.7. The linear system of Volterra integral equations with
the fixed singular or super-singular point χ — b is investigated. This is the
system
b
Vj(x) + Σ
/
" i)~ßKjm(x,
t)ym(t) dt = fjix),
1
<j<m,
with various order of singularity β {β < \\ β — 1\ β >
Depending on the
order of singularity and the roots of the characteristic equation
Δ(λ) = det II Kjm{b, b) + Áájm|| = 0
the statement similar to theorems above hold.
REFERENCES
1. N. Rajabov, Introduction to Ordinary Differential Equations with Singular and Super-Singular Coefficients. Dushanbe, 1998.
2. N. Rajabov, Higher order ordinary differential equations with supersingular points. In: Partial Differential and Integral Equations. Kluwer
Academic Publisher, 1999, 348-358.
3. N. Rajabov, Explicit solution of certain classes of linear integral
Volterra equations with left and right fixed singular and super-singular
points in the kernel. In: Problems of Mathematics
and
Informatics.
Dushanbe, 2001, 31-34 (in Russian).
4. N. Rajabov, Systems of linear integral equations of Volterra type with
singular and super-singular kernels. In: Ill-Posed and Nonclassical
Problems of Mathematical Physics and Analysis. Samarkand, Uzbek-
istan, September 11-15, 2000, p. 73.
5. Ν. Rajabov, To the theory of one class of systems of integral Volterra
equations with an immovable singular and super-singular point. Proc.
124
Ν. Rajabov
of Intern. Sci. Conf. Methods of Functional Theory and their Appli-
cations. Dushanbe, 2000, 33-34 (in Russian).
6. N. Rajabov, General integral equations of Volterra type with the left
and right immovable singular and super-singular points in the kernel.
Izv. Akad. Nauk. Tadj. Otdel Fiz. Mat. Chemical and Geolog. Sci.
(2001) 1, 30-46 (in Russian)
ΠΙ-Posed and Non-Classical Problems of
Mathematical Physics and Analysis, Samarkand, 2000, pp. 125-141
M. M. Lavrent'ev and S. I. Kabanikhin (Eds)
© VSP 2002
Inverses of a family of bounded linear operators,
generalized pythagorean theorems and reproducing
kernels
S. SAITOH*
Abstract — When we consider the inversion of a linear mapping in the framework
of Hilbert spaces, we need naturally the theory of reproducing kernels. We considered recently generalizations of the Pythagorean theorem with geometric meanings
and from the generalizations we can obtain a general and fundamental concept for
the inversion of a family of bounded linear operators on a Hilbert space into various
Hilbert spaces with any index set. After reviewing the applications of the general
theory of reproducing kernels, we shall state the results for the case of operator
versions. As an announcement of our recent results as typical examples, we shall
state explicit inversion formulas of a family of matrices and explicit generalizations
of the Pythagorean theorem. We shall refer to solutions and generalized solutions of
general bounded linear operator equations with continuous parameters on a Hilbert
space.
1.
REPRODUCING KERNELS
We consider any positive matrix K(p, q) on E; that is, for an abstract set E
and for a complex-valued function K(p,q) on Ex E, it satisfies that for any
finite points {pj} of E and for any complex numbers {Cj},
ΣΣ^^·''^·)^0·
j
ϊ
Then, by the fundamental theorem by Moore—Aronszajn, we have:
'Department of Mathematics, Faculty of Engineering, Gunma University, Kiryu 3768515, Japan. El-mail: ssaitoh@math.sci.gunma-u.ac.jp
S. Saitoh
126
Proposition 1.1 [1]. For any positive matrix K(p,q) on E, there exists a uniquely determined functional Hilbert space Η κ (RKHS Η κ ) comprising complex-valued functions { / } on E and admitting the reproducing
kernel K(p, q) satisfying and characterized by
K(-,q) € Ηκ
for
any
q€ E
(1.1)
and, for any q E E and for any / € Ηκ
f(q) = (f(.),K(-,q))„K.
(1.2)
For some general properties for reproducing kernel Hilbert spaces and
for various constructions of the RKHS Ηκ from a positive matrix K(p,q),
see the recent book [15] and its Chapter 2, Section 5, respectively.
2.
C O N N E C T I O N S W I T H LINEAR M A P P I N G S
Let us connect linear mappings in the framework of Hilbert spaces with
reproducing kernels [8].
For an abstract set E and for any Hilbert (possibly finite-dimensional)
space H, we shall consider an //-valued function h on E
h:
E —• H
(2.1)
and the linear mapping for H
f(p) = (f,h(p))H
for feH
(2.2)
into a linear space comprising complex-valued functions on E. For this linear
mapping (2.2), we form the positive matrix K(p,q) on E defined by
K(p,q) = (h(q),h(p))H
on Ε χ E.
(2.3)
Then, we have the following fundamental results:
(I) For the RKHS Ηκ admitting the reproducing kernel K(p, q) defined
by (2.3), the images {/(p)} by (2.2) for Η are characterized as the members
of the RKHS HK.
(II) In general, we have the inequality in (2.2)
\\f\\HK < Μη,
(2-4)
Inverses of a family of bounded linear operators 127
however, for any / G Ηκ there exists a uniquely determined / * G H satisfying
f(p) = (f*Mp))H
and
\\f\\HK =
on E
uni*.
In (2.4), the isometry holds if and only if
(2.5)
(2.6)
{h(p)\p G E} is complete in H.
(III) We can obtain the inversion formula for (2.2) in the form
/
by using the RKHS
r ,
(2.7)
HK.
However, this inversion formula will depend on, case by case, the realizations of the RKHS Ηχ.
(IV) Conversely, if we have an isometric mapping L from a RKHS Η κ
admitting a reproducing kernel K(p, q) on E onto a Hilbert space H, then
the mapping is linear and its isometric inversion L~l is represented in the
form (2.2). Here, the Hilbert space ii-valued function h satisfying (2.1) and
(2.2) is given by
h(p) = LK(- ,p)
and, then
on
E
(2.8)
{h{p)\p G E} is complete in H.
When (2.2) is isometrical, sometimes we can use the isometric mapping
for a realization of the RKHS Η κ, conversely — that is, if the inverse L~l
of the linear mapping (2.2) is known, then we have ||/||ítk =
We shall state some general applications of the results (I)-(IV) to several
wide subjects and their basic references:
(1) Linear mappings [8, 12]. The fact that the image space of a linear
mapping in the framework of Hilbert spaces is characterized as the reproducing kernel Hilbert space defined by (2.3) is the most important one in
the general theory of reproducing kernels. Therefore, the fact will mean that
the theory of reproducing kernels is fundamental and a general concept in
mathematics, as in the idea of Hilbert spaces. To look for the characterization of the image space is a starting point when we consider the linear
equation (2.2). (II) gives a generalization of the Pythagorean theorem (see
also [7]) and means that in the general linear mapping (2.2) there exists essentially an isometric identity between the input and the output. (Ill) gives
a generalized (natural) inverse (solution) of the linear mapping (equation)
(2.2). (IV) gives a general method determining and constructing the linear
128
S. Saitoh
system from an isometric relation between outputs and inputs by using the
reproducing kernel in the output space.
In particular, when we write (2.2) in an integral transform in the framework of Hilbert spaces, it becomes a Fredholm integral equation of the first
kind and it was thought that to solve the equation is a typical ill-posed
problem. At this point, we can see why we meet the ill-posed problem; that
is, we do not consider it in the natural image space Ηκ, but in some artificial spaces. Our method will give a general method solving the Fredholm
integral equations of the first kind in the framework of Hilbert spaces. We
gave new solutions of the many typical integral equations like the Laplace
transform, the Weierstrass transform, the Cauchy integral transform and
the Poisson integral equation. See [12] and [15].
(2) Linear mappings among smooth functions [19]. We considered linear
mappings in the framework of Hilbert spaces, however, we can also consider
linear mappings in the framework of reproducing kernel Hilbert spaces comprising smooth functions, similarly. Conversely, reproducing kernel Hilbert
spaces are considered as the images of some Hilbert spaces by considering
some Hermitian representations (2.3) of the reproducing kernels. Such decompositions of reproducing kernels are, in general, possible. This idea is
important in [19] and also in the following items (6) and (7).
(3) Nonharmonic linear mappings [9]. If the linear system vectors h(p)
move in a small way (perturbation of the linear system) in the Hilbert
space H, then we can not calculate the related positive matrix (2.3), however, we can discuss the inversion formula and an isometric identity of the
linear mapping, by restricting the image functions in a subspace. The prototype result is the Paley—Wiener theorem on nonharmonic Fourier series.
(4) Various norm inequalities [9, 13]. Relations among positive matrices
correspond to those of the associated reproducing kernel Hilbert spaces, by
the minimum principle. So, we can derive various norm inequalities among
reproducing kernel Hilbert spaces. We were able to derive many beautiful
norm inequalities.
(5) Nonlinear mappings [13, 16]. In a very general nonlinear mapping of
a reproducing kernel Hilbert space, we can look for a natural reproducing
kernel Hilbert space containing the image space and furthermore, we can
derive a natural norm inequality in the nonlinear mapping. How to catch
nonlinearity in connection with linearity? It seems that the theory of reproducing kernels gives a fundamental and interesting answer for this basic
question.
Inverses
of a family
of bounded
linear
operators
129
(6) Linear integral equations [20].
(7) Linear differential equations with variable coefficients [20]. In linear
integro-differential equations with general variable coefficients, we can discuss the existence and construction of the solutions, if the solutions exist.
This method is called a backward transformation method and by reducing
the equations to Fredholm integral equations of the first kind—(2.2)—and
we can discuss the classical solutions, in general linear equations.
(8) Approximation theory [3, 2]. Reproducing kernel Hilbert spaces are
very nice function spaces, because the point evaluations are continuous.
Then, the reproducing kernels are a fundamental tool in the related approximation theory.
(9) Representations of inverse functions [14]. For an arbitrary mapping
from an abstract set into an abstract set, we discussed the problem of representing its inverse in term of the direct mapping and we derived a unified
method for this problem. As a simple example, we can represent the Taylor
coefficients of the inverse of the Riemann mapping function on the unit disc
on the complex plane in terms of the Riemann mapping function. This fact
was important in the representation of analytic functions in terms of local
data in [23, 24].
(10) Various operators among Hilbert spaces [17]. Among various abstract
Hilbert spaces, we can introduce various operators of sum, product, integral
and derivative by using the linear mapping (2.2) or very general nonlinear
mappings. The prototype operator is convolution and we discussed it from
a wide and general viewpoint with concrete examples. See [25, 26], and
[27] for further developments in connection with applications to stability in
inverse problems.
(11) Sampling theorems [15, Chapter 4, Section 2; 5]. The Whittaker—
Kotel'nikov—Shannon sampling theorem may be interpretated by (I) and
(II), very well and we can discuss the truncation error estimates in the
sampling theory. Higgins [5] established a fully general theory for [15].
(12) Interpolation problems of Pick—Nevanlinna type [9, 10]. A general
and abstract theory of Pick—Nevanlinna interpolation problems may be
discussed by using the general theory of reproducing kernels.
(13) Analytic extension formulas and their applications [21, 11]. We were
able to obtain various analytic extension formulas and their applications
from various isometric identities (II). For their applications to nonlinear
partial differential equations, see the survey article by Hayashi [4].
130
S.
Saitoh
In this survey article, we shall present also new results on
(14) Inversions of a family of bounded linear operators on a Hilbert space
into various Hilbert spaces, which are generalizations of [22] and [7].
As an announcement of our recent results as typical applications we
shall state explicit inversion formulas of a family of matrices and explicit
generalizations of the Pythagorean theorem for right triangles. Furthermore,
in connection with Kaczmarz's Method for a finite number of bounded linear
operator equations on a Hilbert space, we shall give our generalized solutions
for general operator equations with continuous parameters.
3.
OPERATOR
VERSIONS
We shall give operator versions of the fundamental theory (I)-(IV) which
may be expected to have many concrete applications. In particular, for
full generalizations of the Pythagorean theorem with geometric meanings,
see [7]. Some special versions were given in [22]. We will give more direct
and clear results in matrix theory in Section 5.
For an abstract set Λ, we shall consider an operator-valued function L\
on Λ ,
Λ —> L\
(3.1)
where L\ are bounded linear operators from a Hilbert space Η into various
Hilbert spaces Η χ,
(3.2)
L\ : Η —> Ηχ.
In particular, we are interested in the inversion formula
L\x
—> χ,
χ
6 H.
(3.3)
Here, we consider {L\x; λ G A} as informations obtained from χ and we wish
to determine χ from the informations. Indeed, for one object x, we will, in
general, obtain many type informations L\x. However, the informations
L\x belong to various Hilbert spaces Ηχ, and so, in order to unify the
informations in a sense, we shall take fixed elements ¿>λ,ω £ Ηχ and consider
the linear mapping from Η
Χΐ,(Χ,ω)
= (Lxx,bxì0J)Hx
= (x,L*xbx¡u!)H,
χ € Η
into a linear space comprising functions on Λ χ Ω. For the informations
Σχχ, we shall consider Χ(,(λ,ω) as observations (measurements, in fact)
Inverses
of a family of bounded linear
operators
131
for χ depending on λ and ω. For this linear mapping (3.4), we form the
positive matrix Kb(X, ω; λ', ω') on Λ χ Ω defined by
Κ0(Χ,ω\Χ',
ω') = {L\,bx'
,L\b\ìW)H
= {LxL\,b>!tf ì bxp)H x
on Λ χ Ω.
(3.4)
Then, as in (I)—(IV), we have the following fundamental results:
(I') For the RKHS
admitting the reproducing kernel Kb{Χ, ω; λ', ω')
defined by (3.4), the images {Χ{,(λ,ω)} by (3.4) for Η are characterized as
the members of the RKHS
(II') In general, we have the inequality in (3.4)
< Μη,
(3-5)
however, for any Xb G Hxb there exists a uniquely determined χ' Ε H
satisfying
Xb(X,u>) = (x',L*xbx!W)H
on
ΛχΩ
(3.6)
and
M\HKB
= WAH.
(3.7)
In (3.5), the isometry holds if and only if {L*xb\ìUJ | (λ,ω) 6 Λ χ Ω} is
complete in H.
(Ill') We can obtain the inversion formula for (3.4) and so, for the mapping (3.3) as in (III), in the form
Lxx — • ( L x x , bX:U)Hx
=
ω) —> χ',
(3.8)
by using the RKHS HKb.
(IV') Conversely, if we have an isometric mapping L from a RKHS Hxb
admitting a reproducing kernel ΛΓ{,(λ,ω; λ', ω') on Λ χ Ω in the form (3.5)
using bounded linear operators L\ and fixed vectors
onto a Hilbert
space H, then the mapping L is linear and the isometric inversion L - 1 is
represented in the form (3.4) by using
L*xbXtU = LKb{·, •; λ, ω) on Λ χ Ω .
Further, then {Lxbx¡u
(3.9)
\ (λ, ω) G Λ χ Ω} is complete in Η.
The author obtained the above concept for the operator versions from a
generalization of the Pythagorean theorem in the following way:
132
S. Saitoh
Let ï G f and { e j } " = 1 be linearly independent unit vectors. We consider the linear mappings
L:x^{x-(x,ej)ej}^
1
(3.10)
from R n into M™. Then we wish to establish an isometric identity and
inversion formula in the operators. Recall the Pythagorean theorem for
n ~ 2. By our operator versions, we can establish the desired results.
Note that in (3.11), for η > 3 if we consider
{II® - (®> e j) e ill}j=i
(3.11)
as scalar valued mappings, then the mappings are not linear more. So, we
must consider the operator valued mappings in the problems.
We see that some related equations were considered as in the following
way [6, pp. 128-157]:
Let H, Hj, j = 1 , 2 , . . . ,p be Hilbert spaces and let
Rj : Η —> Hj,
j = l,2,...,p
(3.12)
be linear continuous maps from Η onto Hj. Let gj G Hj be given. Then,
consider the problem to compute / e H such that
Rjf = 9j,
¿ = l,2,...,p.
(3.13)
This equations axe very important in the theory of computerized tomography
by the discretization. The typical method is Kaczmarz's Method based on
an iterative method by using the orthogonal projections Pj in H onto the
affine subspaces Rjf = gj.
Our direct solutions for (3.14) seem that the result is stable for the sake
of the use (3.14) as data, because we use (3.9) which is given by the inner
product.
In general, in equations (3.14) we have noises and errors for the data gj
and so, in those cases the equations do, in general, not have solutions. So, we
will consider a more general solution which is called a generalized solution
(inverse) in Section 6.
4.
INVERSES OF A FAMILY OF MATRICES
If a reproducing kernel Hilbert space is finite dimensional, then the theory
of reproducing kernels is reduced to that of matrices [15]. Furthermore, the
results derived from the theory of reproducing kernels will be derived also
Inverses
of a family of bounded linear
operators
133
directly by that of matrices. In those two sections, we shall state the results
in the matrix theory derived from the above general concept explicitly and
in the framework of matrix theory, for its importance. Of course, the results
obtained here axe more sharp and more concrete than those in [3]. Indeed,
from the theory of reproducing kernels, we can not discuss the linearly independence of the matrix Q& in Section 5, which is introduced by using an
auxiliary vector 6.
A concept and a basic interest in this section are as follows:
For any matrix A of type m χ n, we consider the linear mapping
Ax = y
(4.1)
from Rn into W 1 . Then its inversion is of course examined in detail. However, in many cases, we shall consider a general mapping
A{kx = y{k
(4.2)
from R n into Wk for various jk and we consider its inversion, because we
have, in general, many type data yJkk for x. Here, the important fact is that
each data y3kk for χ is one unit and it is not decomposed into components,
otherwise, the problem (4.2) will be reduced to the case (4.1) by considering
the augmented matrix
i
A
i \
A?
(4.3)
\M°J
Furthermore, we shall measure the data yj.k as the scalar value
(ylk'bik)wk
in Wk by taking a vector bjk of Wk. This concept comes from a general
approach for inversion formulas for a family of bounded linear operators on
a Hilbert space using the theory of reproducing kernels in Section 3. We
shall give concrete examples in our full general Pythagorean theorem.
In order to simplify the representation of the results, we shall consider
all on the real field R
We see the reference [6, pp. 128-157] for the equation (4.2) and the
equation was examined by many authors in connection with Computerized
Tomography. One typical method to solve the equation is the Kaczmarz's
134
S. Saitoh
Method by iteration and the method is applied to many tomography problems. Meanwhile, we shall give a direct and explicit representation of the
inverse of (4.2). So, we are interested in applications of our direct solutions
for (4.2) to the related many problems.
By the Gauss fundamental procedure, we can obtain
Lemma 4.1. Let A be a matrix of type πι χ η and with rank p. Then,
there exist vectors
in Rm such that the vectors {b^A}^^ are linearly independent row vectors.
Lemma 4.2. Let A and Β be matrices of types m χ η and l χ η,
respectively. Assume that A has rank ρ and the augmented matrix
(4.4)
has rank q. Then, there are vectors {bh}ph=i in Rm and {dh}qhJ[ in Rl
such that the row vectors bjA, bjA, ..., bjA, djB, djB, ..., d^_pB are
linearly independent.
Lemma 4.3. Let Aj1, A^,· ·., A3/ be given matrices of types j\ x n,
j2 xn,..., jr χ n, respectively. Let the rank of the matrix
f
A
i\
A322
(4.5)
\M'J
be ps, for s = 1,2,..., r. Let pr = η but pr-\ < n. Then there are ρ ι
vectors
lPp\ G W, p2 - Pi vectors bi>p\+v &£ +2> ... f
G
r
Jr
..., (η — Pr—i) vectors
b^ r_1+2,·.., bft € R such that the vectors
T
2
T
(tt\) A{\
(&AÍ ,
...] {btr) AÌT, where fa = 1,2,..., ρ J , (s2 = Pl +
l , p i + 2 , . . . ,p2),. •. ,(sr = p r _i + l , p r _ i + 2 , . . . ,n) are linearly independent.
Then, we obtain the direct inversion formula for (4.2)
Theorem 4.1. Let A3± , A^2,..., A3/ be given matrices with the properties as in Lemma 4.3 and let bï\ G Rjl, bí\ G W2, ...,
e R>, for fa =
1,2,...,pi), (S2 =ρι + 1,ρι + 2,...,ρ2), ...,(sT = pr-i + l,pr-i + 2,... ,n)
satisfying Lemma 4.3. Let for ï E f , A3kkx —
for k = 1,2, ...,r, be
Inverses of a family of bounded linear operators
135
given. Then
-ι
ί(Η\)τΑ{Λ
X
=
\mTAï)
l<Sl<Pl,
V(*fc)V/
Pr-1 + 1
Pl + l < « 2 < P 2 ,
< sr < n.
(4.6)
We can also obtain one more constructive and direct solution of (4.2).
Theorem 4.2. Let Α^,Α^,·..,
AJrT be given matrices of types j\ χ η,
j2 χ η , . . . , jr χ η, respectively such that the rank of the augmented matrix
(
A
i\
4
2
(4.7)
\ AfT J
is n. For ι e l " , let AJkkx = yJkk, for k = 1,2,..., r, be given. Then there
are η vectors ti[kk, bfô,
br£k e Wk for k = 1,2, ...,r, such that
-1
(Erk=M)T4k\
χ
(ΣΙ=ΛΚί)τνϊ\
(4.8)
=
V
e
U
( K
k
k
)
T
4
k
J
In Theorems 4.1 and 4.2, we assumed that the rank of the augmented
matrix is n, a full rank for χ G M", however, in general, we can obtain a
generalized inverse with minimum norm, as in (II) and (III).
5.
GENERALIZED P Y T H A G O R E A N THEOREMS
Let {e¿}" =1 be a basis of K". For i = 1 , 2 , . . . ,n, we define the maps L¿ :
W1
R n by Li{x) = (x,e¿)e¿, and the maps M¿ : R n
R n by M¿(®) =
χ — {x,ei)ei, then clearly L¿ and M¿ are linear maps and for 6 6 P ,
(Li(x),b) = (x,Li(b)) and (Mi(x),b) — (x,Mi(b)). Hence, they are self
adjoint. Now, we define the matrices
Px = [L1(x)
L2{X)
···
Ln{x)]
(5.1)
136
S. Saitoh
and
Qx = [Ml(x)
M2(X)
•••
Mn(x)].
(5.2)
n
Then the maps L : R ->· M n (R) and M : R™ ->• Mn{R) defined respectively by L(x) = Px and M(x) = Qx aire injective linear maps. The
problems are (1) to find χ and its norm from the data Li(x) (i = 1 , 2 , . . . , n)
and (2) to find χ and its norm from the data M¿(x) (i = 1 , 2 , . . . , η). Consider the problems in their geometrical meanings and from the viewpoint of
the Pythagorean theorem.
Then we can see, directly
Lemma 5.1. For x,b E R n , (i) Pjb = P^x and (ii) Qjb =
Qjx.
Lemma 5.2. Let b 6 R n . Then, Pb is nonsingular if and only if (b, ej) φ
a e aD
=
e
0 for j = 1 , 2 , . . . , n. For b =
¿ ¿ d
i)> Qb JS nonsingular
if and only if aj φ 0 for all j and
a¿/a¿ φ 1.
Here, we note that if {e¿}"=1 is an orthonormal basis, then
OLÌ/CU =
a
η, i.e., the assumption ΣΓ=ι i / ° ¿ Φ 1 automatically holds but in general,
{b— {b,ej)ej}j=1 may be linearly dependent, for η > 2 under the condition
(b, ej) φ 0 for j = 1 , 2 , . . . , n. So, in Lemma 5.2, we see that the assumption
ΣΓ=ι ai/ai Φ 1 iS) i n general, needed for the regularity of Qb.
Then, we obtain the generalizations of the Pythagorean theorem
Theorem 5.1. Let b G R" as in Lemma 5.2. Then, the inverse linear
map L~l : L(Rn) ->• R n is obtained by the formula
x = (PP~l)Tb,
for
PeL(
R")
(5.3)
and
||x|| 2 = ( b T P ) K r 1 ( P T b ) i
(5.4)
where ij-th entry of Κ ι is the product (b, e¿)(ó, ej)(e¿, ej).
Theorem 5.2. Let b G R n as in Lemma 5.2. Then, the inverse linear
map M " 1 : M{Rn) -)· R n is obtained by the formula
x = {QQ~l)Tb
for
Q e M(R n )
(5.5)
and
||x|| 2 = (bTQ)K^(QTb),
(5.6)
where ij-th entry of Κ-χ is the inner product (b — {b, e¿)e¿, b — (b, ej)ej).
The above theorems can be generalized as follows:
Inverses of a family of bounded linear operators
137
1
, (i = 1,2,... ,p) are linear
Theorem 5.3. Assume that L·
n
operators and Mj : R -» R,(j = 1,2, ...,q) are linear functionals such
that ρ + q = n. Then by the Riesz representation theorem, there exist
unique vectors Cj such that Mj(x) = ( x , C j ) (j = 1,2,..., q). If there is a
vector b such that the set {L\(b), L^b),..., L*(b), c\, C2,..., cq} is linearly
independent. Then,
'(bMx))\
<b,L2(x))
®=([¿Í(6)
L*2(b) ... L*p(b) cica
(b,Lp(x))
M\(x)
M2(x)
..
{
6.
Mq(x)
(5.7)
)
GENERALIZED SOLUTIONS
In order to represent our generalized solutions explicitly, we shall consider
bounded linear operators on a reproducing kernel Hilbert space. So, we
consider the Hilbert space Ηκ on E stated in Section 1. We consider H as
Hk in Section 3. We assume that the direct integral
Η=ΓΗχάμ(λ)
JA
(6.1)
of the Hilbert spaces H\ on Λ converges with a σ finite positive measure άμ
on Λ. We assume that the bounded linear operators L\ in (3.2) are bounded
on Hk into Η in the sense:
f \\Lxf\\2Hxdß(\)<M\\ffHK
JΛ
(6.2)
for some constant M > 0.
In this setting, we consider the extremal problem:
inf
[
EFFJC JA
\\Lxf-g(X)fHxdß(X),
(6.3)
138
S. Saitoh
which gives a generalized solution for the equations
Lxf = g(X)
on HK
and
Hx.
(6.4)
We shall write the operators {L\} as L from Ηχ into Η in the sense
(6.2). Let L* be the adjoint operator of L from Η into Ηκ- We form the
positive matrix
k(p,q) = (L*LK(;q),L*LK(;p))HK
on
Ε χ E.
(6.5)
Then, we obtain
Theorem 6.1. For a function g G H, there exists a function f in Hk
such that
inf f \\L\f - 5(λ)||^
f£HK JA
άμ(λ) = Í \\LJ - 9(\)\\2Ηχάμ(λ)
JA
(6.6)
if and only i f , for the RKHS Hk
L*g e Hk.
(6.7)
Furthermore, if there exist the best approximations f satisfying (6.6), then
there exists a unique extremal function f with the minimum norm in Ηχ,
and this function is expressible in the form
f(p) = (L*g,L*LK(;p))Hk
on
E.
(6.8)
In this theorem, note that
(L*g)(p) = (L*g,K(-,p))nK
= (g,LK(-,p))H;
(6.9)
that is, the adjoint operator L* is expressible in terms of g, L, Κ(·,ρ) and H.
For some proof of this theorem, we can apply the argument in [3].
As a simple example, we shall consider the space Ηκ on [0, oo) for
K(x,y) = min{x,y}. This space is composed of all absolutely continuous real-valued functions on [0, oo) satisfying /(0) = 0 and equipped with
the norm
roo
2
HK = /
Jo
f'(x)2dx.
(6.10)
As a space Η we consider the space L2HO, 00), e xd\) and a bounded linear
operator L:
Lxf = [ Ι(ξ)άξ
J0
(6.11)
Inverses of a family of bounded linear operators
139
from Hk into H. Then, the adjoint operator L* from H into Ηχ is given
by
{Vg){x)
1 rx
= ±J g(X)X2e~xdX
poo
2
x
+J
g(X) x X - \ - x r d\
2
(6.12)
and we can discuss the problem, for any g G H,
r&fljf'<««-**>
f£HK
/
I [Χ f(()d(
- α(Χ) 2 β~λ dX.
(6.13)
We can give a complete solution for the problem. We would like to
discuss those concrete problems in separate papers.
We are interested in some concrete results for typical problems such as
generalized solutions for ordinary differential equations in connection with
reproducing kernels, Green's functions and the related completeness in (6.8).
REFERENCES
1. N. Aronszajn, Theory of reproducing kernels. Trans. Amer.
Soc. (1950) 68, 337-404.
Math.
2. D.-W. Byun and S. Saitoh, Approximation by the solutions of the
heat equation. J. Approximation Theory (1994) 78, 226-238.
3. D.-W. Byun and S. Saitoh, Best approximation in reproducing kernel Hilbert spaces. In: Proc. of the 2th International Colloquium on
Numerical Analysis. VSP-Holland, 1994, 55-61.
4. N. Hayashi, Analytic function spaces and their applications to nonlinear evolution equations. In: Analytic Extension Formulas and their
Applications. Kluwer Academic Publishers, 2001, 59-86.
5. J. R. Higgins, A sampling principle associated with Saitoh's fundamental theory of linear transformations. Analytic Extension Formulas
and their Applications. Kluwer Academic Publishers, 2001, 73-86.
6. F. Natterer, The mathematics of Computerized Tomography. SIAM in
Applied Mathematics, 32. SIAM, Philadelphia, 2001.
7. Th. M. Rassias and S. Saitoh, The Pythagorean theorem and linear
mappings. PanAmerican Math. J. (2002) 12, 1-10.
8. S. Saitoh, Hilbert spaces induced by Hilbert space valued functions.
Proc. Amer. Math. Soc. (1983) 89, 74-78.
140
S. Saitoh
9. S. Saitoh, Theory of reproducing kernels and its applications. Pitman
Research Notes in Mathematics Series (1988) 189.
10. S. Saitoh, Interpolation problems of Pick—Nevanlinna type. Pitman
Research Notes in Mathematics Series (1989) 212, 253-262.
11. S. Saitoh, Representations of the norms in Bergman—Selberg spaces
on strips and half planes. Complex Variables (1992) 19, 231-241.
12. S. Saitoh, One approach to some general integral transforms and its applications. Integral Transforms and Special Functions (1995) 3, 49-84.
13. S. Saitoh, Natural norm inequalities in nonlinear transforms. General
Inequalities (1997) 7, 39-52.
14. S. Saitoh, Representations of inverse functions. Proc. Amer.
Soc. (1997) 125, 3633-3639.
Math.
15. S. Saitoh, Integral transforms, reproducing kernels and their applications. Pitman Research Notes in Mathematics Series (1997) 369.
16. S. Saitoh, Nonlinear transforms and analyticity of functions. In: Nonlinear Mathematical Analysis and Applications. Hadronic Press, Palm
Harbor, 1998, 223-234.
17. S. Saitoh, Various operators in Hilbert space induced by transforms.
Intern. J. Appi. Math. (1999) 1, 111-126.
18. S. Saitoh, Applications of the general theory of reproducing kernels.
In: Reproducing Kernels and their Applications. Kluwer Academic
Publishers, 1999, 165-188.
19. S. Saitoh and M. Yamamoto, Integral transforms involving smooth
functions. In: Reproducing Kernels and their Applications. Kluwer
Academic Publishers, 1999, 149-164.
20. S. Saitoh, Linear integro-differential equations and the theory of reproducing kernels. In: Volterra Equations and Applications. C. Corduneanu and I. W. Sandberg (Eds). Gordon and Breach Science Publishers, Amsterdam, 2000.
21. S. Saitoh, Analytic extension formulas, integral transforms and reproducing kernels. In: Analytic Extension Formulas and their Applications. Kluwer Academic Publishers, , 2001, 207-232.
Inverses of a family of bounded linear operators
141
22. S. Saitoh, Applications of the reproducing kernel theory to inverse
problems. Comm. Korean Math. Soc. (2001) 16, 371-383.
23. S. Saitoh and M. Mori, Representations of analytic functions in terms
of local values by means of the Riemann mapping function. Complex
Variables (2001) 45, 387-393.
24. S. Saitoh, Principle of telethoscope. Proceedings of the Intern. Workshop Functional-Analytic and Complex Methods, their Interaction and
Applications to Partial Differential Equations. Graz, Austria, 12-16
February, 2001. World Scientific, 2001, 101-117.
25. S. Saitoh, Weighted L p -norm inequalities in convolutions. In: Survey
on Classical Inequalities. Kluwer Academic Publishers, 2000, 225-234.
26. S. Saitoh, V. K. Tuan, and M. Yamamoto, Reverse L p -norm inequalities in convolutions and stability in inverse problems. J. of Inequalities
in Pure and Applied Mathematics (2000) 1, No. 7.
27. S. Saitoh, V. K. Tuan, and M. Yamamoto, Reverse convolution inequalities and applications to inverse heat source problems. (In preparation) .
28. M. Asaduzzaman and S. Saitoh, Inverses of a family of matrices and
generalizations of Pythagorean theorem. (In preparation).
Ill-Posed and Non-Classical Problems of
Mathematical Physics and Analysis, Samarkand, 2000, pp. 143-172
M. M. Lavrent'ev and S. I. Kabanikhin (Eds)
© VSP 2002
Cauchy problem for the Helmholtz equation
Sh. YARMUKHAMEDOV and I. YARMUKHAMEDOV
1.
N O T A T I O N S A N D SETTINGS OF T H E P R O B L E M
Let E3 be a real Euclidean space, y = {yi,y2,yz), x = (zi,£2)£3) e E3,
a2 = (yi - χι)2 + (3/2 - Z2)2, r2 = \y- x\2 = a2 + (y3 - xz)2, s = a2, D be a
bounded one-connected domain in E3 with the boundary dA consisted of a
compact connected part Τ of the domain 2/3 = 0 and the smooth Lyapunov
surface S lying in the half-space y3 > 0, dD = S |J T, D = D (J dD·, d/(dn)
be the differentiation operator along the external normal to S (or to T);
λ = Λ (y) be a complex-valued function set in D. Let
λ(ί/) = λ ι ( 0 ) + » λ 2 ( ν ) ,
where \i(y) = X\ = Re\(y), X2(y) = Ä2 = ImX(y)·, Ck(D) be the space
consisted of k times continuously differentiable complex-valued functions
defined in D and Ay be the three-dimensional Laplace operator
The problem (Cauchy). Suppose U(y) G C2(D) and
AU(y)-X2(y)U(y)=0,
U(y) = f{y),
y e D,
ïr(y)
= 0(y), y e S,
dn
(1.1)
(1.2)
where f(y) and g (y) are given functions in S from the classes C1(Sr) and
C(S) respectively. We need to recover U(y) in D.
144
Sh. Yarmukhamedov
and I.
Yarmukhamedov
Problem (1.1)-(1.2) is not solvable for arbitrary X(y), f(y), and g(y).
If S is an analytical surface, the function X(y) is analytical in DUS and the
functions f ( y ) and g(y) are analytical in S and may be extended analytically into D, then the solution exists, is unique but is unstable. Therefore,
problem (1.1)-(1.2) is an ill-posed problem.
We suppose that X(y) G C2(D) and the solution of the problem exists
(then it is unique) and belongs to the class C2(D) Π Cl(D).
Under this
assumption we shall construct the solution if the Cauchy date f ( y ) and g(y)
are given exactly. If, moreover, the solution satisfies the condition in the
plane part of the boundary
\U(y)\+
-¿-U{y) < M,
y G Τ,
(1.3)
where M is a given positive number, then we shall construct the regularized
solution, when, instead of f ( y ) and g(y), their continuous approximations
fs{y) and gs(y) are given in S:
s u p | / ( y ) - fs(y)\ < δ,
s
sup \g(y) — gs{y)\ < δ.
s
(1.4)
The formulas written below axe based on the setting and the method of
Lavrent'ev, M. M. [1,2]. The essence of this method is the explicit construction of the Levi function. This function depends on the positive parameter δ
and vanishes with its derivatives in the plane part Τ of the boundary where
i/3 = 0 when the parameter tends to infinity in the case when the pole χ of
the Levi function lies in the domain D. The Levi function with the above
properties is called the Carleman function for problem (1.1)—(1.2) [3] (the
definition of the Carleman function we had given suited for our problem).
We shall construct the Carleman function explicitly. If X(y) = const,
then the Carleman function will be the fundamental solution of equation (1.1) and the solution of problem (1.1), (1.2) and the regularization
will be written explicitly in the form of the difference of the potentials of
simple and double layers. If the function X(y) is not a constant and belongs
to the class C2(D), then the solution of the integral Fredholm equation of
the second kind with the kernel dependent on a positive parameter. The
series consisted of the iterated kernels converges absolutely and uniformly
for the parameter δ sufficiently large, and the solution is expressed via the
resolvents of the integral equation. The formula obtained below allows us
to construct easily the solution of problem (1.1)-(1.2) in both cases: when
the Cauchy data are given explicitly and the regularized solution, when the
Cauchy data are given approximately.
Cauchy problem for the Helmholtz
equation
145
When the number of variables is equal to two, the Cauchy problem for
the Laplace equation was investigated first by Caxleman in 1926 for the
bounded domain with the boundary consisted from the parts of the rays
with the common origin and the smooth curve lying inside the angle [4].
Carleman had founded the formula of recovery the holomorphic function in
the points of the domain lying in the bisectrix of the angle given the values
of this function in the boundary curve. The generalization of the Carleman
formula (both one-dimensional and multidimensional) can be found in the
monograph [5]. There is also the detailed bibliography on this problem.
To obtain the extension formula Caxleman had adapted the Cauchy formula to this problem by introduction a new idea. He had replaced in the
Cauchy integral the kernel by the generalized kernel dependent on a positive
parameter which is holomorphic inside the angle except for a simple pole.
This kernel vanishes in the sides of the angle when the parameter tends to
infinity. The recovery formula being found by Carleman is unstable. The
further investigations of M. M. Lavrent'ev had shown that the introduction
of new positive parameter here was stipulated by the essence of the problem
and for special choice of the parameter the stable formulas of extension in
the class of bounded holomorphic functions may be constructed.
The problem of construction of multidimensional analog of the Carleman
formula on the basis of the classical Green formula had led to the notion of
the Carleman function [1]. The explicit form of the Carleman function and
the multidimensional analog of the Caxleman formula connected with it was
found in [6-10]. In the case when the number of variables is equal to two
and λ is a positive constant, the Carleman function and the regularization of
the Cauchy problem for the Helmholtz equation was constructed explicitly
in [10]. For many variables and for complex λ this function was constructed
in [8],
2.
CONSTRUCTION OF THE CARLEMAN F U N C T I O N
A N D THE INTEGRAL EQUATION OF
THE SECOND KIND
Suppose that X(y) G C2(D) and σ is a positive number. Denote
K(w)
w = iVu2
= e™\
+ a2 + y3.
The function Φσ(χ, y) for α > 0 we define by the equality
„ η TW , _ .
- 2 π2Κ(χ3)Φσ(χ,υ)
,
=
/·°°τ
J0
r K(w) ι cos Xu du
lm\-±-L
—===.
LW-X3I y/u2 + a2
(2.1)
146
Sh. Yarmukhamedov and I. Yarmukhamedov
If we we separate the imaginary part, then Φσ(χ, y) will be as follows:
reo
2π2Φσ(χ, y) = / Fa(x, y, u) cos Xu du,
Jo
(2.2)
where
u2
r2
<Pa{x,y,u) = COST\/u2 + a2 - (1/3 - X3)
, „
ν u¿ + cr
τ = 2ay 3 .
,
(2.3)
Convergence of the improper integral in the right-hand side of equalities (2.1)
and (2.2) is guaranteed by the multiplier e~au .
The basic lemma. The function defìned by equality (2.1) (or (2.2)) is
represented in the form
e~My)r
Φσ(χ y) =
'
"üñT
+
(2 4)
"
where G„{x,y) is the function from the class C2(D)with
variable y including the point y = χ.
respect to the
Proof. Divide the interval of integration in (2.1) into two parts:
(0,oo) = (0,1) U (l,oo). The second integral we denote as gi(y,x). Since
1 < u < 00, the function gi(y, x) belongs to the class C2(D) with respect
to the variable y including the point y — χ. The first integral we denote by
92(^)2/)· Transform this integral as follows:
»<».χ) ~ K{x3) Jo
/ ' im L
W-X3
J y/u* + a2
iu
r 1 ι cos Xu du
+ / Im
,
J0
Lw -X3J y/u2 +a2
where
1
Im
=
w — X3
y/u2 + a2
o ,
•
u¿ + 2r¿
Since the entire function
W — X3
147
Cauchy problem for the Helmholtz equation
is real for real w, then, from the expansion
it follows that the first term is the function from the class C2(D). The
second integral from g2{y,x) we denote by ip{x, y). Taking into account the
equality
cos U — " ( ' 1 )•λ
= Ç - '(2n)!'
we transform it as follows:
-1 cos Ait ,
nX'y>
^
Jo ω2 + r2
r1
Jo
u2
( _ ΐ ) " λ 2 η f1
L·n=Un
du
+ r2+
,-·-,·
(2n)!
u2n
„Ju0 u2 + r2
du
λ^ r1 u2du _ y , (—ΐ)ηΛ2η r1 u2n du
2 Jo u2 + r2
¿ 9
(2η)!
J0 u2 + r2'
„_
ν--,·
„u
The first integral is equal to
2r
du
u2 + r2'
Λ
and the second integral is equal to
λ2
r2A2 /
π
ίτ__
2 V2r
2
Γ
du \
Γ°°
du
2
Λ u + r2)'
Therefore, the function ψ (χ, y) has the form
.
T ,
Φ(χ,υ)
=
V ,U>
π
λ2
1
2r
2
.
+ V
Since
_
2r
we obtain
nr\2
4
2
¿
rX
\¿2\
f°°
n\2n nA
du
A(-l)(—l)
S5ÑV ~ Σ
(2n)!
+
πτ^ _
4 ~
_
2
r
ιλ
2
π
+
2
y
,
'
ff1
u2ndu
/ο u2 + r 2 '
AV""1
92{y,x) = -ne~Xr/{2r) +g3{y,x),
n\ '
148
Sh. Yarmukhamedov
and I.
Yarmukhamedov
where gs{y,x) is a function from the class C2(.D) with respect to the variable y including the point y = χ. Setting
Ga(x,y)
= ~(gi{x,y)
we obtain the lemma statement.
g2(x,y))/{2K2),
+
•
L e m m a 2.1. For y Φ χ the following equality
1 f°°
2^2 J 0
holds
X2$a(x,y).
cos Xudu =
(2.5)
Proof. Denote
ψ,τΜ = -
1
K(w)
2
2n K(xs)
w — X3'
Then
1
2π K(xs)
2
.
Im
K(w)
Φσ(ω)-Φσ{ιν)
—
w — χζ
2i
_
, /
w = — i\Ju¿ + e r +y3
and
1
. Γ„ ,
J
- Φσ(ιϋ)
1 Α. —-
2
\/u
+ α2
The function under the Laplace operator Ay depends on s = α2 and j/3, i.e.
is the function dependent on the point t — (s, î/3). The simple computation
shows that the Laplace operator in the coordinates of the point t has the
form
Δ
·=44+4έ+!τ
M
therefore,
1
r°°
/
Ay[Fa(x,y,u)]
cos Xudu
=Λ[ f c o s A ^ f ^ L W 2 ilJo
ViiHa^
Γ
Jo
cosXUAJ
4 ^ = ) d u \ .
Vu2 + a2/
J
Now, the formula (2.5) follows from the equalities
f°°
\ A f
I cos XuAt I 2
Jo
\\/u
Γ
\ a (
/ cos Ati At I ,
Jo
V«
ΝJ
x2 p0
Γ°
Φ- ασνΗ- ,
,
.
: )du = Χ I
.
cos Xu du,
+ a2'
Jo vu2 + a2
\ ,
f°° φσ(ΰ>)
. ,
: ) du = λ2A /
.2
: cos Ait du
2
2
+a '
Jo Vu + a
(2.7)
Cauchy problem for the Helmholtz
149
equation
and formula (2.1). We prove now the first equality (2.7). Denote the lefthand side of equality by I. Taking into account (2.6) we obtain
• Φ<τΜ
Jo
W
+
4
Γ
a2^
+
d r
Jo
Jo
Φ »
cos Xu du +
ds IVit2 + α2
ds2
f
• \/u 2 + α 2
cos λΐί du
ri2 Γ Φ „ Μ
dyf W
Jo
+ α2
cos Xu du.
Differentiating this equality we have
1
- j
d
ds
y/u2
+
α2
*ΦσΗ
L2(u2 + s)
2{u2
φ»
. S
<M«Q
4s — r ,
= 4s
2
ds y/u2 + a2
4 (u2
+
<Φ'
2(u 2 + s) 2
s)3/2
¿Φ'
ri2
dy\
ΦσΗ
Wu2
+
ΦσΗ
+
4(ÍX 2
+
=
Φ£
s2l
s)2
Vu2
+
3Φσ
2
4 (Η + s ) 2
+ s2 '
Grouping the coefficients we shall obtain
00
' = J o/
u 2 $ " c o s Xu
(u2+s)3/2
+
(2u2 - s)i
f
v
y0
Jo
+
(U2+5)2
f
,
Φ cos Xu du
s-2u2
Φσ cos λ « du = I\ + h + h-
(•u2 + s) 2
Jo
Since
UU^(T
«ιΦ>
—
Vu2 + S '
then, integrating the first integral by parts, we have
h
s — u*2
= ' Γ0 (
\(u2+s)2
cos Xu —
«λ sin Ait
u2 + s
)φ>.
Hence
00
J1+/2
./o
ω
(u2
2φ/
+
s) 2
cos Au du
_ . f°° Φρ-uAi si
sin Xu
Integrating by parts the both integrals, where
du Φσ =
iu
y/u2
+ s
Φ 'du.
σ
du.
150
Sh. Yarmukhamedov and I. Yarmukhamedov
Then, the first integral will be equal to
f00 \(
'Jo
1
3u2
Î I R T ^ -
and the second
(^
η
+ S
\
)V2j
[-uXsinXu
lo
L(u2
+
XusinXu
s )3/2
^-
C 0 S
+
λ 2 cos Au "i,.
+ — Φ σ
,
du.
Thus,
Ii+I2
o f°°
cosXu
= X / Φσ
duJo
Vu2 + s
f°°
Jo
s — 2u 2
. Ψ σ cos Au du.
{u2 + s)5'2
Since I3 is equal to the second integral, we obtain
=
λ2
roo
/
ψ
Jo
σ
Μ
cos A u
xfäT.
du.
Thus function φ(ΐΰ) is obtained from ip(w) by the change i by —i; therefore, the computations do not change and the second equality from (2.7) is
established also.
•
Lemma 2.2. If A (y) G C2(D),
holds
ΑνΦσ(χ,y)
then, for y φ χ the following formula
- Χ2Φσ(χ,y)
= Τσ(χ,y),
(2.8)
where
2m ,
s
f°° Ux^ dX dFa ι
. λ J
2tt2TJx,y)=
2>
-2- iusmXudu
Jo
\-¿{dykdyki
n
roo
. 3
^^ 2v
roo
— / u2Fa (
( ——) i c o s A udu— / FtjU sin Ait ΔΑ du.
j0
j
uyk
J0
Proof. Apply the Laplace operator with respect to the variable y in
formula (2.2). The differentiation under the integral sign is valid since y φ x
and σ > 0. As a result we obtain
roo
2π 2 Δ 2/ Φ σ (ι/,χ) = / Ay(Fa cos Xu)du,
Jo
Cauchy problem for the Helmholtz equation
151
where
» /
3
,
χ
, » π
«
dF(j d cos Xu
..
. .
cos Xu) = cos XuAvFa + 2 >
h FaA(cos Xu),
;
dyk
n^idyk
dcosXu
. . dX
—
— —it sin Ait-—,
dyk
dyk
A(cos Xu) = — u2 cos Xu
(τ~)
k=i yk
Now, equality (2.8) follows from (2.5).
— u s n
*
^ ·
•
Corollary. If X(y) is constant in D, then the function Φσ(χ, y) is the
fundamental solution of equation (1.1) and Ga(x,y) from equality (2.4) is
the regular solution of equation (1.1) in E3 including the point y = χ.
Now, let U(y) be a solution of equation (1.1) from the class
C2(D)Ç\Cl(D).
Then, by the Green formula, for each χ G D we shall
have [2]
U{x) = JDU(y)Ta(x,y)dy
where Τσ(χ,y)
Denote
+J
[φ,(*,ΐ/) ^ ( y ) - U(y)
^(x,y)]dsy,
is defined in (2).
υσ{χ) = J [φΑζ,νΜν)
- f(y) ^(x>y)]dsy
Ra(x) = J ^ ( z . y ) ^ ( y ) - U(y) ^(x,y)]dsy,
x e D
(2·9)
'
χ e D,
(2.10)
where S U Τ = dD. Then the formula for U(x) will become as follows:
U(x) = [ U(y)T0{x,y)dy
JD
+ Uff(x)+Ra(x)
χ Ç. D.
(2.11)
The functions f(y) and g(y) are known in S and belong to the class C(S).
Therefore, the function Ua(x) is defined everywhere except for the points
from the surface S. As the difference of the simple and double layers the
function Ua(x)is extended continuously from D into D. Since the desired
solution α priori belongs to the class C2(D) Π Cl(D), the function Ra(x) is
defined via the values of solution and its normal derivative on the part Τ of
the boundary dD which axe not given α priori. We only assume that they
152
Sh. Yarmukhamedov and I.
Yarmukhamedov
are continuous and, therefore, bounded. However, as it will follow from
further considerations, for large values of the parameter σ, the function
Ra(x), χ G D becomes small. Therefore, in the right-hand side of equality
(2.11), we shall suppose that the term outside the integral Ua(x) + Ra(x) is
known. So, (2.11) can be considered as the Fredholm integral equation of the
second kind with the kernel Ta{x,y) and with the unknown function U(x).
For construction of solution of integral equation (2.11) we introduce the
new functions
V(x) = εσι3 U(x),
Ρσ(χ) = e^R^x),
νσ{χ) =
Ka{x,y)
=
βσχΐυσ(χ),
β-σ^-χ2^Τσ(χ,ν).
Then, relatively the new unknown function V{x), equation (2.11) will become as follows:
V(x)=
[ ν(ν)Κσ(χ,ν)άν
+ νσ(χ) + Ρσ(χ).
(2.12)
JD
From equality (2) it follows that the function Τσ(χ, y)for y φ χ is continuous
with respect to both variables χ and y and is bounded in the neighborhood
of the point y = χ. Therefore, the solution of equation (2.12) can be constructed by the Fredholm method. So, for each χ E D we shall have
V(x) = J Γ \ { x , y ) [vff(y) + Pa{y)]dy + V„(x) + P„(x),
where Τσ{χ,y)
(2.13)
is the resolvent defined by the Newmann series
00
TlT(x,y)
= '£K^(x,y).
(2.14)
71=1
The iterated kernels in this series axe defined recurrently from the relations
KW(z,y)=Ka(x,y),
KIn+lHx,y)=
ί Κσ(χ,ί)Κ^(ί,υ)άί,
k = 1,2,....
(2.15)
JD
We show below that for large values of σ series (2.14) converges absolutely
and uniformly in D relatively the variables χ, y.
3.
C O N S T R U C T I O N OF S O L U T I O N
A N D T H E R E G U L A R I Z A T I O N B Y L A V R E N T ' E V M. M.
Construction of the solution of problem (1.1)-(1.2) is based on the idea and
method of Lavrent'ev M. M. [3, 4].
Cauchy problem for the Helmholtz equation
First we obtain the estimates for Ka(x,y),
153
Γσ(χ,ΐ/), and Φσ(χ, y).
Lemma 3.1. For each σ > 0, x,y G D the following equality
2n2\Kff(x,y)\ < [m + φ(σ)]β~σα2,
(3.1)
holds, where
m = 6αα5(π + 4ψ0) + [(4 + 3π)(α3 + α4) + (2 + π)(α3 + α4 + as)
+ (6 + 7τ)α5]αι + 2[2(α3 + α4) + 3α5]α2,
ψ(σ) = [7ταα5 + (πα5 + 3 π(α3 + α4) + (α3 + θ4 + θδ)( 2 + π))α2
+ 2 πα(α3 + α 4 )]β°2 /(4σ) +
+ [3(ο§ + αΙ + α25 + α2α6)
+ 4α!(2(α3 + ο 4 ) + 3 a 5 ) ] ^ e a ^ 4 < r > / ( 4 ^ )
+ α5[6α(θι + α2) + 96a? + 24 α?α2 + 24αιθ2 + 8α2
+ (4α2(6α?α2 + 3^0% + 2 4 ) + 3 α α 2 2 ) ^ β α ^ { · ^ / ( 2 ^ )
+ 72αβ_σ]/(12σ),
(3.2)
where ψ{σ) < ψ{1), σ > 1 and
αχ = max |λι(ι/)|; α2 = max |A2(y)|; α = max
|λ|,
D
dÀ(y)
Λ = 1,2,3;
α6 = max|AÀ(y)|;
ûfc+2 = max
dxk
(3.3)
the maximum is taken along the closed domain D,
sup / " ^
t> 0 Jo
V
= φ0.
(3.4)
Lemma 3.2. Suppose the surface S is set by the equation
yz = y3(yi,y2),
(yum) e Τ,
where 3/3 (2/1, y2) is the univalent function satisfying the Lyapunov
Denote
maxy 3 (yi,y 2 ) = b.
τ
(3.5)
conditions.
(3.6)
Let σ > 1 and
σ > nj2b\D\/2,
2π2>γ = τη + ψ{1),
(3.7)
154
Sh. Yarmukhamedov and I. Yarmukhamedov
where |Z>| is the volume of D; τη and ψ (I) axe defìned from (3.2). Then series (2.14) converges absolutely and uniformly for x, y e D. For theresolvent
the following equalities hold
2
Lemma 3.3. For each σ > 0, χ, y G D and χ Φ y the following equality
holds
2π 2 β- σ Μ- χ 1) + σ α 2 ΙΦ σ (χ, Ρ )Ι < (7r + 2Vo + Ç ) J + ^ 0 ( a ) .
(3.10)
If 2/3 = 0, then
2πν( Ι 3+* 2 )|φ σ ( Χ ) 2 / )| <
π/27. + Α ο ( σ ) )
(3.11)
where
Μ σ ) = (2a? + α\ +
8^/σ),
(3.12)
<Ε{σ)+Ε1(σ,χ,ν),
dn
Ε(σ) = [2(5 + ττ)α? + (20 + π )α\ + (2 + π)α^ α '/( 4 σ )]/16
(3.13)
+(5π + 12 φ0 + 3β -<7 /σ)σ + (3>/π/16) [Ι2αχ (α3 + α4 + α5)
+α2[12(α3 + α4 + α5) + α ^ ^ } /
£ ι ( σ , ζ , y) =
+ B(a)y3/r
(3.14)
2
+ C (σ)α + Ζ?(σ)| cos Θ|/γ ,
Α(σ) = (3v^/16)(23a? + 28a¡ +
9a¡ea^4,T))/Vá,
Β(σ) = (3π + 2ψ0 + β~σ/{ 2σ))2σ,
C(a) = [ 3 ^ / ( 8 ^ ) ] (2a? + α| +
ϋ(σ) = 3π/2 +
+ <Γσ ¡σ,
(3.15)
where θ is the angle between the radius vector f , χ (Ξ D and the normal in
the point y G S; a* and uq are defìned from (3.3) and (3.4) respectively.
If j/3 = 0, then
2τΓ2βσα2+σχ3
άΦσ
dys
(vχ , υ ) < ^ 3 + Ε 2 (σ),
'
- 2r
(3.16)
155
Cauchy problem for the Helmholtz equation
where
Ε2(σ) = (4a¡ + 3 a\ + a22ea^)/
8
+ (x/ïF/16) [αια 3 + a2(a¡ +
+ πσ.
(3.17)
The proof of the Lemmas 3.1-3.3 can be found in Section 4.
Solution (2.13) of equation (2.12) we rewrite in the notations
Ua(x), R<r{x):
U(x),
Ησ{χ) = Ua(x) + e-**2* [ Τσ{χ^σ(ν)€^
U{x) = Ησ{χ) + Ms),
dy,
JD
(3.18)
Ms) = Ra{x)
+ e~ax¡
(3.19)
Í Γ a ( x , y ) R a { y ) e ^ > dy,
JD
where Ua(x), Ra(x) are defined by equalities (2.9), (2.10) respectively.
Further, we shall use the following elementary inequalities
[ e'™2 dyidy2<
JT
Jt
r
Γ
Γ e'™2
7-00 J-oo
χ, f
JT
y/Σ
dyidy2<-,
σ
r3
< 2*.
(3.20)
T h e o r e m 3.1. Suppose that the surface S satisfìes the conditions of
Lemma 2.2 and σ (σ > 1) is chosen following condition (3.7). Suppose U(x)
is the solution of problem (1.1)-(1.2) from the class Cl{D) U C2(D),
where f(y) and g(y) set in S are the functions from the classes C1(S)
and C(S) respectively. Then for each χ E D the following inequality holds
IU{x) - Ησ{χ)I < ΜΕ3(σ)β~σχ>,
lim Ε3{σ) = 1,
σ—+00
(3.21)
where
ρ ,„\ _ f1
E 3 { a )
-\2
+
, V^
W¿
,
Α0{σ)+Ε2{σ)\
2πσ
)
+
ν
V
Μ, Γ, Αο(σ), and Ε2(σ) are defìned as in (1.3), (3.7), (3.12), and (3.17)
respectively.
156
Sh. Yarmukhamedov and I. Yarmukhamedov
Proof. Taking into account (3.18), we have for χ E D
\υ{χ)-Ησ(χ)\
= \Κ{χ)\.
The theorem will follow from the equality
\Κ{χ)\ <ME3(a)e-"xl
(3.23)
We shall establish it. Equality (3.19) yields
IMaOl <
+ ε~σχί
[ \Ta{x,y)My)\e°vl
dy.
JD
(3.24)
For each χ e D, taking into account (1.3), (2.10), (3.20), (3.11), and
(3.16), we obtain
|Λ σ (ζ)|
<MJt [|Φ | +
σ
\2
dyidy 2
4-ν/σ
2πσ
/
where Αο(σ) and ^ ( σ ) are defined from (3.12) and (3.17) respectively. Finally, from (3.9) and (3.24), (3.23) follows.
•
Corollary. For each χ € D the following limit holds:
lim UJx) = U(x).
σ->οο
The limit holds uniformly in compact sets in D.
Now, we shall construct the regularization. The function U„{x)defined
by equality (2.9) we represent as follows:
υσ{χ) = υσδ(χ) + Ισδ{χ),
υσδ{χ) = /ί |säΦσ - fs άΦσ dsy,
ΓΓ
άΦ Ί
U = J [(<? - 9ί)Φ* - (/ - U)-¿\
dsy,
(3.25)
where f¡ = fg(y) and g¿ = gs(y) are given function from the class C(S)
satisfying condition (1.4).
*
Denote
Ησδ(χ) = υσδ + e"™» ί Γ σ (χ, y)UaS{y)e°ya dy,
(3.26)
JD
Οσδ{χ) = β-σχΙ
ί Γσ(χ,ν)Ισδ(υ)ε^άυ.
JD
(3.27)
Cauchy problem for the Helmholtz equation
157
Then (3.18) can be rewritten as follows:
(3.28)
U(x) - Ησ6{χ) = ha{x) + Ισ&(χ) + Gffs(x).
Further, we shall use the following elementary inequalities which follow easily
from (3.20):
(3.29)
Here b is defined in (3.6), k denotes the maximal number of points intersection the surface δ with the vector r, where χ E D and y € S. Denote
Ε4(σ) = (2Α;/π)(3π/2 + 4ψο + β" σ /σ) + [ο/(2π)](5π + 12ψ0 +
3β~σ/σ)
+ [β/(4τΓ)]{2ν/^6(3π + 2φ0 + β~σ/{2 σ))>/σ
+ (V5F/16) [&(23α? + 28θ2 + 9 a l e a V W ) + 16(π + 2φ0 +
e~a/a)/V¿]
+ (1/8) [(5 + 4 π)α? + 2(5 + π)α22 + (1 + 2 π ) α ^ ° ^ 4 σ ) ] / σ
-(- (3V5F/16) [2a? + a| + 4ai(a 3 + a 4 + a 5 )
+ a 2 [4(a 3 + a 4 + a 5 ) + a 2 + a i ] e a ^ ] / ( a ^ ) j ,
(3.30)
where ak and ψο are defined from (3.3) and (3.4) respectively.
Theorem 3.2. Suppose that the surface S satisfies the conditions of
Lemma 3.2 and σ (σ > 1) is chosen following condition (3.7). Suppose also
that U{x) solves problem (1.1)-(1.2), belongs to the class
C2{D)f]Cl(D)
where f(y) € Cl(S), gy € C(S). Let, instead of f(y) and g(y), their continuous approximations fs(y), 9s{y) respectively be given satisfying condition (1.4); besides, condition (1.3) holds, where b and M are given positive
numbers so that
δ<
Me-^oW
(3.31)
Then for each χ € D the inequality
IU(x) - Ησδ{χ)I <
Τ{σ)Μι~χl/b>i/62,
(3.32)
158
holds,
Sh. Yarmukhamedov
and I.
Yarmukhamedov
where
1
M
σ=ρΙη—,
τι„\
Τ ( σ )
- Γ1 j . v ^ x P M ,
-
.2
+
1
+
Ε
'
2πσ
J]
T V 3 S f i +
Ε4(σ)
tively; Τ(σ) = 0{y/o),
Α0{σ)+Ε2(σ)
{ σ ) +
\/2σ
and Αο(σ), Ε2(σ),
(3.33)
V
)}
(3.34)
sfïS-η^b\D\}l
are defined in (3.12), (3.17), and (3.30) respec-
σ —^ oo.
P r o o f . For each χ € D and σ (σ > 1) satisfying condition (3.7), the
following inequalities hold
M x ) \ < Ε 4 (σ)β σ 1 , 2 - σ χ ΐδ,
(3.35)
\Οσδ(χ)\<Ε5(σ)βσ1>2-σχ2>δ,
(3.36)
where
(3.37)
E4 is defined by equality (3.30).
First, we prove (3.35). For each χ G D, (3.25) and (1.4) yields
\ Ι σ δ \ < δ [ Ι \ Φ σ \ ά 8 ν +J
άΦσ
dn
dsy
The moduli |Φ σ | and | άΦσ/dn\ in the right-hand side we replace by estimate
(3.10) and (3.13). The elementary integrals we replace by estimates (3.29).
The obtained expression we group by decreasing degrees of y/σ. As a result
we obtain (3.35). Further, (3.27) and (3.35) yield the estimate
\G0s\ <Ε4(σ)βσό2-σχ2>δ
f
JD
\Ta(x,y)\dy.
Now, (3.36) follows from (3.9).
Formulas (3.36), (3.35), (3.23), and (3.28) yield the following inequality
for arbitrary χ E D.
\U(x) - Ησδ(χ)\
< [ΜΕ3{σ)
+ δ{Ε4{σ)
+
E ^ e ^ e - ^ l
Choose σ which satisfies (3.33). As a result we obtain (3.32).
•
Cauchy problem, for the Helmholtz equation
159
Corollary. For each χ G D the limit equality holds
UmHaS(x) = U{x).
í-fO
The limit attains uniformly in compact sets from D.
The linear functional Hag(x) = H„s(fs,gs,x) defined in the set of pairs
ifs,9s), where fs, gs{y) £ C(S) and 0 < δ < SQ is called the regularization
following Lavrent'ev, Μ. M. of solution of problem (1.1)—(1.2).
4.
T H E P R O O F OF LEMMAS 3.1-3.3
The proof of Lemma 3.1 we divide into four lemmas.
Prom (2) for the function Ka(x,y) we have
2 π 2 β σ α 2 Κ σ { χ , υ ) = 2π2β-σ^-χ^+σα2Τσ(χ,ν)
= Τ1+Γ2+Γ3+Γ4+Τ5, (4.1)
where
m
« v ^ d\
d , , 9
ο.
9λ
í°° u sin Χηφσ
_„2 ,
dyk u¿ + Η
dyk Jo
rw,
Γ
. λ
„u2φ ) yk2(yk-xk>
k)du ,
3 = " η2 v^
Σ ìdX
~ / Γ°°usinÀue"™
σ
^
m
=
dyk JQ
f dX\2 JΓ°°
ο
0
cos
χ
{u¿ + r¿)¿
du
mì
T5 = - Δ λ [ usmXue~™2(pff ¿
9.
Jo
u + r¿
Here φσ = (pa(x,y,u) is defined by formula (2.3).
Lemma 4.1. For each σ > 0, χ, y 6 D the following inequality holds
IT4 + n I < [ 3 ^ / ( 4 ^ ) ] [αια6 + (o§ + α\ + a\ + α 2 α 6 )β 0 ^ 4σ >],
(4.2)
where ak, k — 1,6 are deßned in (3.3).
Proof. Taking into account (2.3), (3.3), inequality |cos/i| < chΛ2 and
equality
Γ chX2ue-™2 du =
Jo
2ν/σ
Sh. Yarmukhamedov and I. Yarmukhamedov
160
we obtain
|T4| < (ai + 4 + a¡)[3v^/(4v^)]e ú Í/(^).
Similarly, talcing into account the inequalities
|sinAt¿| < I sinÀiu| + sh\2u,
|sinAiu| < |λχ|ω
we obtain
\T5\ < [3^/(4^)W(ai
+ a2ea^).
Summing these estimates for T4 and T5 we obtain (4.2).
•
Lemma 4.2. For each σ > 0, χ, y 6 D, the inequality
|T3| < (π + 2)(α3 + α4 + α 5 )(«ι + α 2 β α ^ 4 σ ))
(4.3)
holds, where α^, k = 1,5 are defìned in (3.3).
Proof. Analogously Lemma 4.1 we have
3
\Ά\ <2'S2ak+2\
2ur
2
_
e~g"2 du
(u¿ + r¿)¿
roo
2r 2
2 ι
+ / (αϊu + a2uchα2ω)-—=
^ e cu du\
Jo
(« + H) 2
J
-go
/
L7 o
(aiu + a2ucha2u)
< 2 ¿ afc+2 [ 2 ( β 1 £ + a2 J e ^ )
fc=l
Hence (4.3) follows.
+ αχ + a 2 e < ^ > ] .
•
Lemma 4.3. For each σ > 0; χ, y E D, the inequality
\T2\ < (α3+α4)[(2+3π)αι+2α2]+α 5 (2α(π+4^ο) + 3(αι+θ2))+ν^(σ) (4.4)
holds, where
ψ2(σ) = π(α3 + α4)(2α + 3 α 2 )β α ^ 4σ >
+ [2(α3 + α4) + 3a 5 ][^a¡/(2 N /^)]e a 2/( 4 < T ) + 2α 5 (^ι(σ) + αβ~σ/σ),
(4.5)
φι(σ) = [24 α? + 6αχα2 + 2 α\ + 6 α\α2
+ α2(3αια1 + 2α2 + 6a?a 2 )[^/(2^)]e a i/( 4 < r )]/(24a).
(4.6)
Cauchy problem for the Helmholtz equation
161
Proof. We represent T2 as follows:
2
T2 = 2
+
^ dyk
k=1
(4.7)
dyz
where
i°° d<pk
. ,
.
.
e~J"2 du
and
άψσ
SÌVLTVU2 + a2 .
dyk
= -τ —,
_
2
Vu + α
2
.cos τ Vu2 + a2
(yk - xk) - T{yz - x3)-
V«2 + a 2
y/u2 +
a2
YK~Xk
Vu2 + a 2
« + cr
d^Pc
.
/
/-ö-—2 s i n r y ^ + a:2
—— = —2asinrv^2 + a2 ¿ y/u ¿ + or
= = —
«3/3
V«2 + a 2
— (1/3 — X3)2acosr\/i¿2 + a 2 .
We show that
|Js|
< 2α(π + # ο ) + 3(α 1 + α 2 )
2
+
3£
4 \/σ
+
+
ae^
σ
Taking into account (4.8), (4.9), analogous to the proof of Lemmas 4.1
and 4.2, we deduce the inequality:
f°°
Ι-Γ3Ι < 3 /
Jo
<l(a1
where
_
σ{αχη + a2ucha2u)e~au
+
a2 + ^ a
2
2
du + \B\
e ^ ) + \B\,
(4.11)
f 00 usinÀu sinr\/u2 + a 2 _„2 ,
—τ τ
,„
—e
au.
Β = —
Jo
u2 + r2
y/u2 + a2
The evident representation
sin \u — Xu + fu,
(4.12)
162
Sh.
Yarmukhamedov
and
I.
Yarmukhamedov
fu = Aiit[sin(Aiu)/(Aiu) — 1] — 2sinAii¿ sh 2 (A2t¿/2)
+ ¿{A2u[sh (λ 2 ω)/(λ 2 ΐί) - 1] - 2 sin 2 (λχΐ»/2) sh A 2 u}
and the inequality
|/B| < i¿3[24|Ai|3 + 3|λι|λ2 + (3|Λι|Λ| + 2|λ2|3 + 6λ?|λ2|) eh |À2|u]/12 (4.13)
yield
Β
=
Βγ
+
E i ,
where
Au2 sin τ Vu 2
"J
B2
+ α
y/u
2
Jo
00
u s m r V u
Vu2
+ α2
2
ση
e
du
u + r2 '
2
2
+ a2
f
u
+ a2
e "
U2
σ
+
"
2
du
r2
Jo
Analogously the proofs of Lemmas 4.1 and 4.3, from (4.13) we deduce the
inequality
\B2\ < [24of + 6α 2 α 2 + 6αχθ^ + 2
+ α 2 (6α 2 α 2 + 3oioi + 2a¡) [ν^ Γ /(2 Λ /^)]β α 2/< 4 ^]/(24σ·) = ^ ι ( σ ) .
Further we represent Β ι as the sum
Bi
=
B3
+
B
4
,
where
5
3
=
_λ
Ú
Γ
λ
Estimate
-λ
B
=
Ar
ο
Γ°°
/
J o
sin τ
T
/
\/u2
Jo
Bi
*
Vu2
V i i
2
f
+
+ a2
f
e - ™ ' d u ,
e"™ 2
+ a2
T 7 ~ 2
u2 +
H
+ α2
,
d u
'
r
=
2ay3.
We have
3
=
Jo
v « + er
e
+
du
Γ
Jvi+a2T2
Vu2
+
+ r-* to
a2
Cauchy problem for the Helmholtz equation
163
The second integral we can estimate as follows:
Γ
JV 1+α2τ2
sinrv^T^
, I .
r
\/ω2 + a 2
I
Λ
u e
- ^
d u
The first integral by the Bonne theorem (ψ(υ,) = e~au
decreasing, ψ(0) = 1) is equal to
fi sin τ Vu2 + a2
Jo
_
Vu2 + α2
+
rTV?+<*2
Jar
U
j dt
2
J_e-,
2σ
is monotonically
fW^+a2
s nt
sintdt
Vt2 - α2σ2 ~ Jar
Sin tdt
Ot2T2
L
=
t
^
n
^ - a ^ t W ^ - a M '
.
,
-
-
0 < ξ < ν/ΓΤ^α2.
Here the modulus of the first integral, following (3.4), is not greater than
2ψο and
ι
VS2+<>2
r
I Jar
α
g¿2Tτ¿2
Wt2 -
sin idi
t dt
a2T2
t + Vt2 -
ι
a2T2
Γ°°
ί
dt
I~ Λ
- Llo
t2VW^T
du
{\ + u )y/ï+t?
2
π
< -.
2
Combining these inequalities, we obtain
|#3| < (τ/2 + 2^0 + β~σ/(2σ))|λ|.
(4.14)
Since the function φ{υ) = (u2 + r 2 ) - 1 e _ < 7 " 2 monotonically decreases for
u > 0 and i/>(0) = r - 2 , then, this estimate holds for |i?4| also. Therefore,
|Bi| < (π + 4^ο + β-σ/σ)|λ|.
Combining the estimate for |f?i| and |i?2| and taking into account (4.11),
we obtain (4.10).
We prove now the inequalities
141 < \ [(2 + 3π)αι + 2α2 + π(2α + 3 α 2 ) ε ΰ ^ 4 σ ) + ^
Prom (4.8), (4.9), for J*., we have
/fe = ¿?fci + Bk2 + Bk 3,
α2
β α */( 4σ ) ,
A = 1,2.
(4.15)
164
Sh. Yarmukhamedov and I.
Yarmukhamedov
where
Bki = -T{yk
sin τ Vu 2 + a2 u sin Xue~au2
—, 9 , ,2
u2 + r 2
ν« + α
- xk) /
σω du
sin Xue~cu2
. Z"0 000 cos
c o sτ\Λι
r v u 22 + α
a 22 ti
tisinAue
/
/ 2 ι 2
/ 2 _l_ 2/ 2 _ T ~ 2 ?
./ο
ν« + a
v r +
+ r)
n
,
Bk2 - - r ( y 3 - Ï3j(yfc -
,
w
ükz = — (î/3 -X3)(yk
du
. Ζ"00 ì i s i n r \ / u 2 + a 2 sin Xue~an2 du
-Xk) /
/ 9 , 9, 9
57
2~~i 2
·
Jo VU2 + a 2 (li 2 + a 2 )
" Ήr
Similarly to Lemma 4.1, we obtain the inequality
ι*«'
<π(αι + α2βα^4σ))/4.
As
cos τ \ f u 2 + α 2 / \ / i ¿ 2 + α:2 = du sin τ \ / t t 2 + a 2 ,
—tu
then, integrating by parts, we obtain
„
Bk2
,
.,
= - ( l / 3 - X3){yk
-Xk)\
. r ί00 \β~σν·2 cos \u sin ry/u2 + a2
/
,
J o
L
,
9
,
9 /
Vu¿ + az{wi +
XT
r¿)
,
du
[
Γ00 (-2ωτ)β-<™ 2 sin Xu sin τ y/u2 + α 2 J
+ /
, „
—r
au
y/u2 + a2 (u2 + r2)
Jo
ί
00
sin\ue~ a u 2 sinrVu 2 + α 2
—2udu
y/u 2 + a 2
(ti2 + r 2 ) 2
JO
+
f
00
lo
Jo
u sin τ Vu2 + a2
VrfTrf
e-*™2 sin λιι
(u2 + a2)(u2
+
r2).
The integrals standing in the right-hand side we estimate as in Lemmas 4.1,
4.2. As a result we obtain
_
poo
poo
fOO
—au2
\Bk2¡ < ^|λ|β λ 2 / ( 4 σ ) + /
|λι|aue~ c r u 2 du+
o\X2\uchX2ue
*
Jo
/
2|Λ!|u 2 du
Wo
(« 2 + r 2 ) 2
2\J0
u2 + r¿
Jo
2|Aa|u2 dux
λ|/(4σ)
Λ
J0
(u 2 + r 2 ) 2 /
u¿ +
r¿/
165
Cauchy problem for the Helmholtz equation
For Bkι, we have similarly
l-Sfcil <
+ |
+
f(l
+
0=a2e°
MM)
+ \{ai
Adding the inequalities for |-5jfei|, |-Sfc2|) and
ties (4.15).
•
+a2e«yw).
we obtain inequali-
L e m m a 4.4. For each σ > 0, χ, y G D the following inequality holds
\Ά\ < 2 ( ο ι + α2)(α3 + α4) + α6[4ο(π + 4^)) + (3 + π)αι + 3α2] + ^3(σ), (4.16)
where
+ α22(2α3 + 2 ο 4 +
φ3(σ) = πα5(α +
/
+ [α5/(4σ)]{2α(α? + α2) + 24o? + 6 α\α2 + 6αι<^ + 2α\
+ a2{<òa\a2 + 3aio¡ + 2 α\ + αα|)[>/7?/( 2^)}βα^4σ)
+ 16 ae'*}.
The numbers α, ak, ψο are defined in (3.3) and (3.4).
P r o o f . Represent T\ in the form
. / dX ,.
dX .
dX
dX . \
Τ1=2[ — Α1 + —Α2 + -Γ-Α3),
\>dyi
dm
dm2
du*
)
dy
dy
3
,t „
4.17
where
Ak = -2a(yk-Xk)
f°°
us'mXue
2
" ψσ
σ
du
. Γ ,,
« +
Λ
Γ
. \ -ση2
Α3 = τ
usmXue ou ψσ-ö-,—ö>
u
7o
+r
t
Λ = 1,2
= 2σ2/3·
Since 2u\yk — i/c| <u2 + r 2 , analogously Lemma 2.1, we obtain
\Ak\ < βι + α 2 + [ν/^/(2^)]α^ ΰ 2/( 4 σ ),
fc
= 1,2.
(4.18)
Represent A3 in the form
¿3 =
+ A5,
(4.19)
where
ζ·00
,
_ 2
du
A4 = r I cosryu2 + a2 usinXue cu 2
u + r2 '
Jo
.
. Z"00 sinrV"2 + α2
. . _σω2 du
= r(y3-x3)
/
MSinAue ™ u
2
7o
ν« + α
+ rl
4.20
166
Sh. Yarmukhamedov
and I.
Yarmukhamedov
Since
τ sin τ Vu 2 + a 2
,
,
—r
.
— udu = —du cos r γ vr + er,
2
2
Vu + a
d /sin\ue~au2\
du V it2 + r 2 /
\cos\ue~™2
it2 + r2
2« sin Aite - ™ 2
(u 2 + r 2 ) 2 '
2uasinXue~au2
u2 + r 2
then, integrating by parts A5, we represent it as the sum of three integrals.
Estimating the integrals as in Lemma 4.1, we obtain
roo „u \„7,ρ-σω2
< |A|r /
roo
du + σ I (|Ai|u + |Ä2|uchA2ii)e-^
í°°
+ r /
2U(|À1|U
Jo
+ |À2|UchA2u)e-<7U
2
du
,
2
du
,
+
2,2·
Hence, for \As\ we obtain the final inequality
< (l + ^ax+aa
2
+
π
2
+
+
_ν/π
4y<r
Integrating by parts in (4.20) and using the representation (4.12) and the
formula cos η = 1 — 2 sin 2 (u/2) we represent A4 as the sum A4 = Αβ + Αγ,
where
Λ
Ae=
a - \ f00 í
ί°° Ο
ΠΓ>—2
2uayu+cr
Jo
rr
2
+ β2
A7~ÁJo
+
.'
2r2ß2
(u2 + r2)2
f°° /2u(u2 + a2)
Jo l (u2 + r2)2
u
au2
~ ^ue~.¿ du ,
+ r
u¿
\ _ ff „2sin ry/u2 + α2 ,
e
. _
— du
Γ
y/u2 + a2
\
_σω2 sin τ Vu2 + a2
dU
f°°u
f°° 2u2 + a2 .9
. 2 Xu _ „„22 sin τsmrVu+a
Vu^ a-2
+ 2X /
sin — e au —
„ du
/ Jo u2 + r2
2
Vu 2 + a2
2
,
2 sir 2
™
o u¿ + r¿
and
2 u2(u2 + a2)
(u2 + r2)2
sin
· τ yΓτ~,—2
sin
i r + cr
u2
u2 + r2
u2 + a2
u2 + r 2
2 ß2r2
(it2 + r2)2
r2 + β2
u2 + r2 '
ß = V3~ X3-
Estimate A-j. Similar to Lemma 4.3, the modulus of the sum of the first two
terms in the right-hand side is not greater than
4α(π/2 + 2φ0 + βχρ" σ /2σ) +
&ψι(σ).
Cauchy problem for the Helmholtz
167
equation
The modulus of the third term is not greater than
[α/(8σ)]{2α 2 + 2 a\ + [ 0 F / ( 2 v ^ ) ] ^ ^ } ,
since
I sin2 {\u/2)\
< (u 2 /8)(2of + a\ + a\ cha 2 a 2 ).
Therefore, we obtain
\A7\
<2(π
+ 4ι/Ό +
^ - ) ο + ^
2al + 2
+
+
ù\Ja
3φχ{σ).
Further, since wuVu2 + a2 < u2 + r 2 , then 1-Ael satisfies inequality (4.18).
Therefore, for A\ we obtain the estimate
\Aa\ < 2α(π + 4.ψο +
+
^ 2al1 + 2a122 +
.
2y/ä
a
¿^=ale
+ 3^1 (σ) + αϊ + ο 2 +
yW
λ\/σ
Combining (4), (4.21), (4.19), (4.18), and (4.17) we obtain (4.16).
•
Add the inequalities (4.16), (4.4), (4.3), and (4.2). In the right-hand
side of the result inequality we change the function φι (σ) by its expression
from (4.6). The sum obtained as a result we group by the degrees of q/y/σ.
The statement of Lemma 3.1 then follows.
P r o o f of L e m m a 3.2.. Taking into account (2.15) and (3.1), we obtain
(2)
the estimate for K& (χ,y):
\K^(x,y)\<
ί
< ( JD[Ka(z,y)}2
< 7
2
\Κσ(ζ,ν)\\Κσ(χ,ζ)\άζ
JD
dz)1'2(
( J
jD[KAx,z)Uz)1/2
J
e-2^1
dz — dz\ dz2 dzz. Since
J
2
e-2a[(z1-x1)>+(*2-x2) ildz
< (|£>|&)V2( J
M2
e-Wdyidy2)]
Sh. Yarmukhamedov and I.
168
Yarmukhamedov
then
\Ki2\x,y)\
<
(7/V2W^\D\/(2a).
Continuing the process, from (2.15), for natural η and each (x,y) G D, we
shall have
(j/V2)(^fyy2\D\/(2a))n.
<
As it follows from (4.7), the majorant series is converging and the first part
of Lemma 3.2 follows.
Summing the majorant series, we obtain (3.8) whence (3.9) follows.
Thus, Lemma 3.2 is proved.
•
Proof of Lemma 3.3. The function Φσ(χ, y) we represent as follows:
2π2β-σΜ-χ$+σα2φσ(χ,ν)
=h
+ I2,
where
roo
h
=
e-™
2
2
roo
du
h
h
=
Xu
β-ση
¿u
-2Jo
Taking into account (2.3), we obtain for Ii
f°° cos TVU2
T
7 l =
/
— m —¿
Jo
T.
.
h = -m
+ a2
e
_au2
T,
du + i[,
it + r
, f°° sin rVu2 + a2 e'™2
,
du
- χ3) Jo
/ —y/u
/ 22 -fa
, 2 , „2
·
¿ , „ ¿2
U + r
For J(, similarly Lemma 4.3, we obtain the estimate
|/ί| < (π/2 + 2φ0 + e"7(2a))/r.
The modulus of the first integral is less or equal to 7r/(2r); therefore,
\Ι1\<(π
Β~σ/(2σ))/Γ.
+ 2ψο +
Analogously to Lemma 4.4, we obtain
I/,I < ? j f «»<*?
+
a¡ +
(l
+
Summing the estimate for |Ji| and IJ2I we obtain (3.10).
^
Cauchy
problem
f o r the Helmholtz
equation
169
For j/3=0 the calculation is simplified and (3.11) is deduced easily.
Calculate now άΦ σ /άη and represent it in the form
2π 2 β σα»-σ(„ΐ-*ϊ)
=
an
+ h + h + J4>
h
where
2
h =- 2 π
f
=-/
J0
β ~ ™ ' ( 2 σ α ^ \
an an J
au
2re~
cos Θφσ cos Xu
_
f°° άψσ e -cu*
. J
du,
/
=
/
— —
Xudu,
2
2
2 ? 2 cos
3
2
(u + τ )
y0
dn u
" + ~r
2
2
„—au
Γroo <pae-™
. . d x
h = Jo —
5— du, r = 2ayz.
¿5
u + r « sin Au dn
For h we obtain the estimate from (3.10):
|/i| < [π + 2ψο +
? E 2 ± L + l ( 2 c a + r)A0(r)·,
(4.22)
the integral J 2 we represent in the form I2 = h + h , where
Θφσβ~συ·2
f ° ° 2r cos
h
=
=
-Jo
r°° 2rcos
n
Λ
Jo
dU
(u2 + r 2 ) 2
>
/2sin
. o Au\ e"™2 du
M
Analogous to the proof of Lemmas 4.3 and 4.4, for I5 and Ie, we obtain the
estimates
lr
.
πΙΰοβΘΙ
21 cos Θ | / π
| / e | < \ J η2\·ψσ\{2α\
. .
e _<7 \
+ a22 + a22 cha2u)e-™2
(ΐ
/^"2)2
lr.n2
f ° ° i2 f n
r
\ e-"u2tdu
< 7 (2aj
+ a 2l )/
u ( 1 + - = 2= = 2) , 29 , . 2. 2.
4L
7 ο V Vu + a H u + r )
Γ
+
sb)A
02Ue mî
" (Ä5]
Sh. Yarmukhamedov
170
and I.
Yarmukhamedov
Therefore,
I Γ I <Τ π +
Μ2Ι < ^ g ~ ( 2 a l
2 .
+α2
„2a , eΛ22βΑα2/(4σ)χ , π l c o s e l
+ 2
Ί+2
r2
2|cos0| (-π
r2
The following inequality we obtain similarly the proof of Lemma 4.1:
|/4| < (3/4)(a3 + a 4 + a 5 )(ai + a 2 e a ' /(4<T) )v/^/^·
(4.24)
We prove now that for each σ > 0, χ, y € D and y φ χ the following
inequality holds
liai < ¿(3a 2 +
+
+
(3vr +
+
τ^/π
πτ
+ [ll(e? + 2a¡)
3 2ae l e -^-JM ] ^ + 2
2) τ+ ou
6¿\/a
r
(4.25)
Represent the integral I3 in the form
Ιτ3
/Γ e
—— ,du,
Joo « + r ¿ an
/ - _ f°° 28ίη2(ΛΜ/2)6-σ"2 <fy>ff ^
7ο
ω2 + r2
dn
=
Ιτ7 +MT
Is,
Ιτ7
--
Estimate Is. Prom (2.3) we obtain
άφσ
.
r-r-—ó d f ΠΓ,—2λ
sin r Vu2 + a2
—— = — sinry^
+ or — \T\JU¿ + a¿ )
.
—
an
dn
dn
y/u2 + or
rcos Ty/u2 + a2 d(ry/u2 + a2)
~ ( y 3 " X 3 ) Í Λ 2 + α2
Τη
2
sinr-s/u + a 2
a
da'
(4.26)
2
2
2
2
Vu + a
\/í¿ + a dn.
d(rVu2 + a2) _
dn
+ a2 + τ
2σ
dn
a
da
y/u2 + a 2 dn
Hence we deduce the inequality
άφ,
dn
TV
τατ
u2 + a2 + 2t + 2ar + . 02
+ -=¿
2
Vu + a
u + a¿
(4.27)
Cauchy problem for the Helmholtz equation
171
Analogously the proofs of Lemmas 4.2 and 4.3, from (4.27) we shall have
I/,I < ^
£
+ 2r + 2ar
Vu 2 + a 2
U¿ + a ¿ /
/·οο u 2
.
α2
+ J / u 2 + r 2 [ 2 í V " 2 + α 2 + 2r + 2σΓ
+
rr
/ 9
arr \ ,
_,.2 .
9 + ~2~i 2 ) c h ° 2 U e
or
2α2 + α 2 /
S |
8
Γ
«»/(4σ) , ^ ν ^ e a¡/(4„)\
+
4 ^
8 ^
>'
(4.28)
From (4.26) we see that /γ is represented as the sum of six integrals. The
first term we denote by IQ:
|
Il τ^/πΝ
4 ^ i f 4 l 2
+
r
dyz f°° sin rVti 2 + a2 (u2 + c^e"™ 2 ,
n
lg = Ζσ —-— I
.
^
^
au.
dn JQ
y/u2 + a2
u2 + r 2
In the right-hand side the Dirichlet integral stands which we had estimated
before. Therefore,
|J»| <2σ(π + 4ψο +
β~α/σ).
The rest integrals are estimated easily, and for I? we deduce the inequality:
\I7\ < 2πτ/r + πσ + 2σ(π + 4ψ0 + β _<τ /σ).
(4.29)
Estimates (4.29) and (4.28) yield (4.25). Now, from (4.25) we obtain (3.13).
Further, we calculate d^a/dy^ and set yz = 0. In this case the calculations are simplified and we obtain (3.16). Lemma 3.3 is proved.
•
REFERENCES
1. L. A. Aizenberg, The Carleman Formulas in the Complex
Nauka, Novosibirsk, 1990 (in Russian).
Plane.
2. K. Miranda, Equations with Partial Derivatives of Elliptic Type. IL,
Moscow, 1957 (in Russian).
3. M. M. Lavrent'ev, On Certain M-Posed Problems of Mathematical
Physics. Nauka, Novosibirsk, 1962 (in Russian).
172
Sh. Yarmukhamedov and I. Yarmukhamedov
4. M. M. Lavrent'ev, On the Cauchy problem for Laplace equation. Izv.
Akad. Nauk SSSR. Ser. Mat. (1956) 20, 819-842 (in Russian).
5. I. Islomov, On construction of the regularized solution of the Cauchy
problem for the Helmholtz equation. Dokl. Uzb. SSR (1984) 8, 10-11.
6. I. Islomov, On the Cauchy problem for the Helmholtz equation. In:
Uniqueness, Stability and Methods of Solutions of Ill-Posed Problems
of Mathematical Physics and Analysis. Novosibirsk, 1984, 97-105 (in
Russian).
7. Sh. Yarmukhamedov, On the Cauchy problem for the Laplace equation. Dokl. Akad. Nauk SSSR (1977) 235, No. 2, 281-283 (in Russian).
8. Sh. Yarmukhamedov, On the harmonic extension of continuous functions defined on the piece of the boundary. Acad. Sci. Dokl. Math.
(1993) 46, No. 3, 430-433 (in Russian).
9. Sh. Yarmukhamedov, On extension of solution of the Helmholtz equation. Dokl. Ros. Akad. Nauk (1997) 357, No. 5, 320-323 (in Russian).
10. Sh. Yarmukhamedov, On harmonic extension of differentiable functions set in the piece of the boundary. Sib. Mat. J. (2002) 43, No. 1,
228-239 (in Russian).
Πl-Posed and Non-Classical Problems of
Mathematical Physics and Analysis, Samarkand,
M. M. Lavrent'ev and S. I. Kabanikhin (Eds)
© V S P 2002
2000, pp. 173-178
On problem for a third order equation
with multiple characteristics
S. ABDINAZAROV and B. M. KHOLBOEV
Abstract — The goal of this paper is investigation of the equation
in the domain D — {{x,y) | hi(y) < χ < h2(y), 0 < y < Y} with the boundary
conditions
u(x,0) = F(x),
Ux {hi (y ), y) = ψ\ {y
ux(My),y)
hi(0) < χ < h2(0),
),
(2)
Uxx (hi {y ), y) = φ2 (y ),
= <p3(y),
0<y<i,
(3)
where f(x,y), ψί(ν){ί = 1,3), F (χ) are given functions. These functions satisfy the
following concordance conditions
F'(MO)) = ¥>i(0),
F"(hi(0)) = ^ ( 0 ) ,
F'(/i 2 (0)) = ^ 3 (0).
(4)
The analogous research for equation (1) with the other boundary conditions was
done in [1-3].
1.
UNIQUENESS OF SOLUTION
T h e o r e m 1. Ifhi(y)
( l ) - ( 3 ) is unique.
G C'([0,1]), ¿ = 1,2 then the solution of
problem
P r o o f . Suppose there exist two solutions of the problem in question
ui{x,y) and
u2{x,y).
174
S. Abdinazarov and Β. M. Kholboev
We set v(x,y) = u\(x,y) — U2{x,y).
obtain the following problem
Then, for the function v(x,y)
we
L(v) = 0,
υ(ζ,0)=0,
Vx{hi{y),y)
Λι(0)<ζ<Μ0),
= 0,
vx(h2(y),y)
vxx(hi{y),y)
= 0,
= 0,
0 < y < 1,
(5)
Consider the identity
J J
vxx(vxxx
-Vy)dxdy
- 0.
(6)
Integrating by parts and using the homogeneous conditions (5) we obtain
\fl
ö/
vlx{h2{y),y)ày
¿ J0
1 /-Mi)
+ ^ ( χ , 1 ) d e = 0.
¿
Jhi(l)
(7)
Hence, we have
w»x(Mv),y) = 0,
«χ(χ,1)=0.
Therefore, v(x, 1) = const.
Suppose now that
v(x, y)
- ω(χ, y)eMy,
M = const φ 0.
Then we obtain
JJ
ωχχΣω eMy dxd y = y
JJ^
ω2χ eM* dx d y = 0,
where
Lu
- ωχχχ — ων-
Μω.
(8)
Therefore, wx(x, y) — 0 and ω(χ, y) = w{y). Substituting the function w(y)
into (2) and (8), we have
w'{y) - Mw{y) = 0,
ω(0) - 0.
It is known that problem (9) has the trivial solution only.
v(x,y) = 0 in D.
•
(9)
Therefore,
On problem for a third order equation
2.
175
E X I S T E N C E OF S O L U T I O N
Let F(x) G C 3 (/¿i(0), hr¿(0)).
F( x) = 0.
Really, if we set
Without loss of generality we may set
u(x,y) = v(x,y)+F(x),
then, for the function v(x,y),
(10)
we shall obtain the following problem
L(v) = f(x,y),
v(x,0)=0,
vx (h {y), y ) = φ1 {y),
hi(0) < x < h 2 ( 0 ) ,
υχχ (hi {y ), y ) = ψ2 (y ),
vx(h2(y),y)=ìp3(y),
0 < y < 1,
(11)
where
ΨΛν) = φι (y) - F'(hi(y)),
Tp2(y) = ip2(y) -
ΨΛν) = <f%(y) - F'(h2(y)),
f(x,y)
F"(hi(y)),
= f(x,y)
-
F"'(x).
As is shown in [1], the function
W(x, y) = lJJD U{x, y; ξ, η)Πξ, η) άξ
satisfies equation (1) and the condition w(x, 0) = 0 if /(ξ, η) G Cx'^(D),
f(x, 0). Here U (x, y; ξ, η) is the fundamental solution of equation (1) (see [1])
which is as follows:
'
1
f
x
~ £ Λ
y < η,
0,
where
roo
f(t)=
/
Jo
cos(À3 - Xt) dX,
t =
χ-ξ
(y
-77)1/3"
The function f(t) is called the Airy function and satisfies the equation
z"(t) + tz(t)/3 = 0.
(13)
The function f(t) satisfies the following relations (see [4]):
/ » ( í ) ~C+ín/2-1/4sin(2í3/2/3)
n 2 1 4
/ W ( í ) ~ C~t Ζ " /
3 2
sin ( - 2 i / / 3 )
where C~ and C^ are constants.
for
for
t —>· +oo,
t ->· - o o ,
(14)
(15)
176
S. Abdinazarov and Β. M. Kholboev
Taking into account these relations, we shall seek for the solution of
problem (11) in the form
v{x,y) = u(x,y)
+ w(x,y).
(16)
Then, relatively the function ω(χ, y), we obtain the following problem
Lu -- 0,
ω(χ,0)=0,
h^O) < χ < h 2 (0),
<¿x{hi(y),y) = <jp(y),
Ux(h2(y),y)
(17)
w«(/ìi(y),!/) = &2(y),
= <p3(y),
0<y<i,
(18)
where
Φι (y) = <fii(y) -
Wx(hi(y),y),
= <P2{y) - Wxx{hi(y),y),
¿pz(y) = £ 3 (y) -
Wx{h2{y),y).
Solution of problem (17), (18) we seek in the form
ry
u)(x,y)=
ry
U(x,y; 0,η)α>ι(η)άη +
Jo
υ(χ,ν\1,η)α2{η)άη
Jo
+ [ V(x, y; 0,77)03(77)077.
Jo
(19)
Here
(
0,
y < η
roo
_ e
[exp (—λ3 — λί) + sin (λ 3 — λί)] dÀ,
φ(ί)=
Jo
t = {y -
η)1/á
Note that φ(ί) satisfies the relations (13), (14), and (15) also. Satisfying the
boundary conditions (18) we obtain
Φι(ν) = /
Jo
Ux{hi(y)
,ν,^ΐ{η),η)αι{η)
ry
+ /
Jo
άη
υχ(Ηι(ν),ν\Κ2(η),η)οΐ2(η)άη
+
í
Jo
Vx{hi{y),y,^1(77),77)0:3(77)dT7,
(20)
On problem
for
a third
order
equation
...
177
rv
<¿2Ü/)= /
Uxx(hi(y),y,
}ΐι(η),η)αι(η)
άη
Jo
+
/
Jo
Uxx(hi(y),y,h2(v),v)a2{v)fy
+
ry
ry
/ νχχ{1%ι(ν),ν·,1ΐι{η),η)αζ{η)άη,
J0
( 2 1 )
Jo
+
/
Jo
υχ{ίι2(ν),ν;ίι2(η),η)α2{η)άη
+
í
Jo
νχ(Η2(ν),νΜ(η),η)α3{η)άη.
(22)
Applying the Abel transform and after non-tedious computation we obtain
the following system
f
^
*
"
^
οκι(*)
+
^(ΰ)<.3(*)
(24)
178
S. Abdinazarov and Β. M. Kholboev
The system (23)-(25) is equivalent to the system of integral Volterra equations of the second kind
3
pz
at{z) = J2,(z) + V / Nsl{z, η)α3(η) άη,
7Ξί J o
I= Û,
(26)
where ai(z) are the desired functions, Ri(z) are known and are expressed
via the functions <pi(z) (i = 1,3). Nsi(z, η) is the matrix whose elements are
expressed via the fundamental solution of equation (18).
It is easy to show that the kernels Ns¡(z,i7) have the weak singularity:
|iVsi(z,7/)| < C/\z - r?|1/2;
c = const > 0.
(27)
Prom the general theory of Volterra equations [5] it follows that system (26)
is solvable uniquely in the class of continuous functions which admit the
representation of the form
3 pZ
ak{z) = Rl(z) + Y /
Hsl{z^)Rs{iy)
j=o 0
drç;
I = 173.
Here the resolvent Hsi (ζ, η) has a weak singularity of type (27).
REFERENCES
1. L. Cattabriga, Annali della seuola narmole. Superiori di Pisa e Mat.
(1959) 13, No. 2, 163.
2. T. D. Dzhuraev, The Boundary-Value Problems for the Equations of
Mixed Type and of Mixed-Composite Type. FAN, Tashkent, 1979 (in
Russian).
3. S. Abdinazarov, General boundary-value problems for the third order
equation with multiply characteristics. Differential Equations (1981)
17, No. 1, Moscow, 3-12 (in Russian).
4. V. M. Fedoryuk, Saddle-Point Method. Nauka, Moscow, 1977 (in Russian).
5. M. Gervey. J. Math. Pures et Appi. (1913) 9, No. 1-4, 305.
Ill-Posed and Non-Classical Problems of
Mathematical Physics and Analysis, Samarkand, 2000, pp. 179-183
M.M. Lavrent'ev and S.I. Kabanikhin (Eds)
© VSP 2002
On uniqueness and stability of solution of
the Cauchy problem for pseudoparabolie equation
B. K. AMONOV* and S. S. KOBILOV*
Stability and uniqueness play an important role when researching wellposedness of problems of mathematical physics.
Lavrent'ev, M. M. [1] had established stability of determination of a harmonic function in the plane or spatial domain given the Cauchy data in
the piece of the boundary in the class of bounded functions. The stability estimates for solutions of general elliptic equations are obtained in [2].
The contents of these papers and some other results can be found in the
monograph [3]. The uniqueness theorems for the noncharacteristic Cauchy
problem for the general parabolic equation of the second order is established
first in [6].
In our paper the uniqueness theorem is established and the stability estimate is obtained for the Cauchy problem for the pseudoparabolic equation.
Suppose that the function u(x, t) is continuous in the domain
Π = {(χ, t) I 0 < χ < π, 0 < t < oo}.
Inside this domain the function u(x,t)
ut{x,t)
= uxxt(x,t)
satisfies the equation
+
OM)
(1)
and the conditions
u(0,i)=0,
ω(π,ί) = 0
u'x(0,t) = f ( t )
(t > 0)
(t> 0).
(2)
(3)
'Samarkand State University, University prosp., 15, Samarkand 703004, Uzbekistan
Β. Κ. Amonov and S. S. Kobilov
180
Here f(t) is a continuous function and
[ u2(x,0)dx
Jo
<M2,
M>
0.
(4)
We need to recover the function u(x,t), given /(i); to establish the uniqueness theorem and to obtain the stability estimate.
Theorem (Uniqueness theorem). If f(t) — 0 then u(x,t) = 0 in the
domain Π.
Proof. It is easy to verify that the solution of equation (1) satisfying
the conditions (2)-(3) has the form
00
sin kx,
(5)
k=1
where
are the Fourier coefficients. Relations (3) and (5) yield
oo
=/(*).
(6)
k=1
Let F(p) be the Laplace transform of the function /(£), i.e.
roo
F(p)=
f(t)e~*dt.
(7)
Jo
Applying the Laplace transformation with respect to t to both sides of equality (6) we find
00
j
Integrate both sides of this equality along the circles
Γη = {ρ I \P-PnI
= |[(„+t)2)+l ' Ä l l
}
with the centers in the points pn = —n2/(n2 + 1). As a result we obtain
Taking into account the Cauchy formula
«»'hm
/ ( O dí
On uniqueness
and stability
of solution
181
we have
no.n2m=
[
J r„
F{p)dp.
Hence we find the coefficients
Γ„
Thus, the Fourier coefficients are defined uniquely. Therefore, the result of
the uniqueness theorem follows.
Consider now the stability of solution.
Suppose |/(ί)| < ε, 0 < t < oo. Formula (5) yields the expression for
the function f(t):
oo
(fl)
= J2ka*e~k2t/il+k2)·
f(t)=u'x(0,t)
fc=1
Setting in this equality e - t = ζ we obtain
00
φ (ζ) = Y í k a k z k ^ 1 + k 2 \
(10)
k=1
Since e - í = ζ and O < t < oo, the function Φ (ζ) is defined in the segment
0 < z < l ,
|Φ(ζ)| < ε
for
0<Z<1
(11)
and inequality (4) holds. We try now to estimate the solution of problem
(l)-(3). Condition (4) yields
Σ*1<Μ\
fc=l
(12)
Denote by Dp the disk \z\ < ρ < 1. Relations (10) and (12) yield the
inequality
oo
l*MI|«|<P<i < Y , K \ a k \ p
1
k2
2
)
< (ΣΙ«*Ι
< ΣΦκίρ"
1
1 / 2
(E^)
1 / 2
<c(p)M,
182
Β. Κ. Amonov and S. S.
Kobilov
where C(p) is a constant dependent on ρ only. Now, we map the disk
{ζ I \z\ < 1} onto the half-disk {u; | |tu| < 1, Imio > 0 } conformally using
the function w = \fz. In this case Dp passes to the half-disk
dpi/2 = {w I w < p1/2,
Im ω > 0}.
Denote by wp(z) the harmonic measure of diameter relatively the half-disk
dpi/2. It is known [7] that
Μ
1
2
r
ι P1/2
-
z
Since |Φ(ζ)| < ε in [ - p 1 / 2 , p 1 / 2 ] and |Φ(ζ)| < C{p)Mm the half-circle {w |
= ρ 1 / 2 , Imtí) > 0}, then, taking into account the theorem on the two
constants (R. Nevanlinna), we have
\Φ(ζ)\ <
or
e^iMCip)]1-^,
00
J2k2\ak\2\z\2k2
ι
<
e2w^[MC(p)]l-w^z\
Hence, on the bases of the Holder inequality, we obtain
M.,»)| < Σ ψ
s
Σφ" 2
<¿m"{-z)[MC{p))l-w^z)Cl,
where C\ is a constant.
We have obtained the stability estimate for solution of problem ( l ) - ( 3 ) .
•
REFERENCES
1. M. M. Lavrent'ev, On the Cauchy problem for the Laplace equation.
In: Trudy 3 Vsesousn. Math. S'ezda. Moscow, 1956, 2 (in Russian).
2. M. M. Lavrent'ev, On the Cauchy problem for elliptic equations of the
second order. Dokl. Acad. Nauk SSSR (1957) 112, No. 2, 195-197 (in
Russian).
3. M. M. Lavrent'ev, On Certain Ill-Posed Problems of
Mathematical
Physics. Izd. Sib. Otdel. Akad. Nauk SSSR, Novosibirsk, 1962 (in Russian) .
On uniqueness and stability of solution
183
4. S. Mizohata, Unicité du prolongement des solutions pour quelques
operateurs différentiels paraboliquers. Mem. Coll. Sci. Univ. Kyoto.
Sec. A (1958) 31, 219-239.
5. M. H. Protter, Properties of solutions of parabolic equations and inequalities. Canad. J. Math. (1961) 13, 331-345.
6. B. K. Amonov and S. P. Shishatskii, A priori estimate for solution of
the Cauchy problem with the data on a time-like surface for second
order parabolic equation and the uniqueness theorem connected with
it. Dokl. Akad. Nauk SSSR (1972) 206, No. 1, 11-12 (in Russian).
7. M. A. Evgrafov, Analytical Functions. Nauka, Moscow, 1968 (in Russian).
Ill-Posed and Non-Classical Problems of
Mathematical Physics and Analysis, Samarkand, 2000, pp. 185-190
M. M. Lavrent'ev and S. I. Kabanikhin (Eds)
© VSP 2002
Uniqueness theorem for an unbounded domain
Z. R. ASHUROVA* and Yu. J. ZHURAEV*
Abstract — In this paper the uniqueness theorem is established for a class of
harmonic functions defined in an unboundary domain which lies in a layer.
1.
INTRODUCTION
We suppose that V is unbounded domain from R 3 lying inside the layer
0 < 2 / 3 <h,
h — π/ρ;
y = (2/1,2/2,2/3)·
Suppose that dV = Γι U Γ2 are the smooth surfaces given by the equations
2/3 = /¿(y 1,2/2), i = 1,2, so that
|/i(2/i,2/2)| < const,
dfi
7^(2/1,2/2) < const,
¿ = 1,2; i = 1,2.
Let Ap(V) be the space consisted of the functions harmonic in V and continuous together with their first partial derivatives up to the finite points of
the boundary &Ό and
(1.1)
* Samarkand State University, Samarkand 703004, Uzbekistan
186
Ζ. R. Ashurova and Yu. J. Zhuraev
dU
— (y) < exp (c|y|),
y€dV,
(1.2)
(where η is the vector external normal to dV and c is a certain
then U ξ 0 in V.
constant),
In the case when condition (1.2) is replaced by the condition
3U
< c il2/| fc >
c i = const,
yedV,
where A; is a nonnegative integer the theorem was proved by Ashurova [1].
Our proof will use the integral representation of harmonic functions in unbounded domains with a noncompact boundary [3] and the theorem of Landis [2].
2.
THE BASIC INEQUALITY
Let y = (2/1,2/2,2/3), χ = {x\,x2,xz), y ER3, 0 < x3, y3 < h, h = π / ρ , where
ρ > 0. For χ φ y we can define the function Φ^, χ) by the formula
« ( * « ) = - *2
, Γ
'
2n K(x3)J0
Im «<* + *} ÜH,
y3-X3 + iv η'
(2.1)
K
'
where η2 - u2 + α2, α 2 = (y\ - χ\)2 + (?/2 - Χ2)2,
Κ {ω) = (ω + 3/ι-£3) - 1 βχρ(-αάηρι(ω-/ι/ 2 )),
ω = ξ + ίη.
(2.2)
Here ο, ρ\ axe positive and 0 < ρ\ < p. Then, the following estimate holds
for the function <&(y,x):
l1 $ v( yy' , z ) l <
-,
/
i. /n\ \ χ,
\·
- rexp (ocos(pi{y
3 h/2)) chPla)
(2-3)
v
'
Here r 2 = a 2 + (2/3 — £3) 2 and Co = const. In this section we shall establish
inequality (2.3). If ω = x 3 , then, following (2.2), we obtain
K(x3) = (3/i) - 1 exp (—α cos pi (£3 — h/2)).
Since —π/2 < pi(x3 — h/2) < π/2, then cospi(a;3 — h/2) > 0; therefore,
1/K(x3)
= 3/iexp (acospi X3/1/2) < 3/iexpa.
(2.4)
Uniqueness Theorem for an Unbounded
Set ßi = y 3 — xz + 3/i, r 2 = a2 + β2.
Im
K(y
z
+ ΐη)
(ββι
(ys - Xz + ϊη)
187
It can be shown that
- η2) s i n a ( s i n p i ( y 3 - h/2) s h p ι η )
(U2 + r2){u2 + r2) exp ( a c o s p i ( y 3 - h/2)
(•u2
Since α <η,
Domain
άιριη)
η(β + βΐ) cos (a sin pi (j/3 - h/2) sh pi η)
.
+ r2 )(u 2 + r j ) exp (acospi(i/3 - h/2) άιριη)
(2.5)
then
exp ( a c o s p i ( y 3 —
h/2) c h p i a )
< exp (acos pi(2/3 —
h/2)
άιριη).
Taking into account that
lim
7?->o
ffffisin(ashpi7/)
η2 + β2
=
Q
'
we see that there exists such ε > 0, that for \η\ < ε, the function
β β ι sin ( a s h pi 7/)/(rç2 + β 2 ) is bounded. If |τ?| > ε then we have the inequality
β β ι sin (ashpi7/)
εζ
η2 + β2
Besides, we have
u¿ + Η
η2+β2
<
1.
A s a result we obtain
(ββι
(u2
- η2 ) sin a(sin pi (y 3 - h/2) sh pi η)
+ r2)
exp (a cos pi (y 3 —
h/2) ch ρ ι η )
<
Ci
exp (a cos pi (y 3 —
h/2) eh
,
pia)
(2-6)
where Ci is a certain constant. (Further, by C¿ we shall denote the constants). The following estimate can be obtained easily:
η(β + βι) cos(asinpi(y 3 /i/2) s h p i ^ )
exp (a cos pi (y 3 —
h/2) ch ριη)
<
02η
exp (a cos pi(y 3 /i/2) c h p i a ) '
(2.7)
188
Ζ. R. Ashurova and Yu. J. Zhuraev
At first we shall estimate the function Φ(υ,χ).
yield the estimate
|φ(ν,*)Ι<
The inequalities (2.4)-(2.7)
C3
exp (cospi(y 3 - h/2) c h p i a )
X
(Γ
\ Jo
Γ
du
(U 2 + r
) V u
2
+
Jo
)
du
{u2 + r2)(u2
+ r2) ) '
Since
u/\/u2 + a2 < 1,
the integral
f
+
r2)(u2
Jo
is converging and
roo
Joo
we have
du
u2
+ r\
du
(it 2
<
C
+ rf) ~ r '
Co
|Φ(υ,χ)| <
—
'
r exp (a cos pi (2/3/1/2) c h p i a ) '
Thus, inequality (2.3) is established.
3.
PROOF OF THE THEOREM
Under the conditions of the theorem, following [3], we have
0,
Sdvdn
xgV{JdV
j U(x),
® dn)ds
xeV.
Taking into account (1.1), we can rewrite this formula as follows:
U(x)
= J^$(y,x)^ds^(y,x)^ds,
(3.1)
χ e V.
Introduce the notations
f
dU^
I j = - J^ Φ ( ι / , χ ) — ds,
h
αχ = α cos pi —.
i = 1,2,
Since T¿ is given by the equation y3 = /i(yi,2/2), where f i is bounded
together with its partial derivatives, inequalities (1.2) and (2.3) yield
J
Γ ex P ( c v V f + y~2 +
J JR2
r exp (ai eh p i a )
dyi d m
=
C4(Jn +
Ji2 +
Ji3 +
/m)>
Uniqueness
w h e r e
b y lu,
spectively.
i =
Theorem for an Unbounded Domain
1 , 2 , 3 , 4
I u
_
r
v ^ T ^ T T ?
I n .
Γ
Jo
A s
w edenote
F i r s t w ee s t i m a t e
^
J0
ψ
p i a )
Ζ
JO
ψ
+y2
+
)2
e x p ( a i e h pi y/(yi~x2
JO
re-
^
(3.2)
c h p i a )
e x p c ( y i
/o /Jo
the quadrants
f o r m
then w e have
dyi dj/2
poo ρ
-
along
i t i nt h e
< yi + y2 + h f o r yi, y2 G ( 0 , + o o ) ,
r e x p (αχ e h
JO
integrals
re e x p (αϊ
poo pc e x p {cy/yj + y% + h2)
JO
the
Represent
1 8 9
h)
+ (ϊ/2 - X2)2)
dyidy2
V(yi-xi)2
A s y/yl +y% + h2 <yi+y2
poo pe
Jo
Jo
'
+h
e x p c
+ (y2-x2)2
f o r yi,y2
+
(h-xs)2'
6 ( 0 , +00), t h e n w e
have
(yi + y2 + h)
e x p ( a i e h pi sj{yx - x2)2 + (y2 -
x2)2)
dyi dj/2
y/(Vi - χ ι) 2 + fa - χ 2? + (Λ - *3)2
(
= expc{xi+x2
-L.
μ
oo
0 0
<
C5
Γ
+ h)
e x p
Γ
/ -00
0 0 JJ —
—0 0
< C
6
x / í f + i ^ e x p
e x
í
+ 1
2
) d¿i
2
) d í i d í
(ai ch p
x
2
\Ji{ +
t2)
( 3 . 3 )
holds since the
integral
oo pi
r 0
d¿2
Pi y/tf +
P (α1 eh
e x p ( c | z | ) .
This inequality
°
e x p c ( í i + í
/ -00 J—00 \Jt\
is
expc(¿i
/
J-Xl J-X2 \Jt{ + ¿ 2
pe
e x p c ( í i +
/
+
2
) d í i d í
tj e x p ( a i c h pi yjt\
2
+t\)
converging.
A n a l o g o u s l y
w eo b t a i n t h e e s t i m a t e f o r t h e r e s t
\hj\
T a k i n g
into account
< C j
+ 6
e x p ( c | x | ) ,
the inequalities
ι
Γ
|/
Γ
orr
Φ ( ν , ϊ ) — d s
J = 2,3,4.
(3.3),
< C
integrals
( 3 . 4 )
(3.4), w e o b t a i n
u
e x p ( c | x | ) .
( 3 . 5 )
190
Ζ. R. Ashurova and Yu. J. Zhuraev
The estimate of our integral along the component can be obtained analogously. Thus, inequalities (3.1) and (3.5) yield the estimate
\U(x)\ < Ci 2 exp(c|x|).
Suppose that c = const, such that
max| x | =Ä |l/(a;)|
lim
—
~ 0.
R—Kx exp(ci/h)\x\
Now, using Theorem 6.1 [2], we obtain U(x) < 0. Repeating our considerations to the function —U(x), we obtain that —U(x) < 0. Hence it follows
that U(x) = 0, where χ G V. The theorem is proved.
We thank Sh. Ya. Yarmukhamedov for the fruitful discussion.
REFERENCES
1. Z. R. Ashurova, Theorems of Fragmen—Lindelef type for harmonic
functions of many variables. Dokl. Akad. Nauk Uz. SSR (1990) 5, 6-8.
2. Ε. M. Landis, Elliptic and Parabolic Equations of the Second Order.
Nauka, Moscow, 1971 (in Russian).
3. Sh. Ya. Yarmukhamedov, The Green formula for an infinite domain
and its applications. Izv. Akad. Nauk Uzb. SSR, Ser. Fiz. Mat. (1981)
5, 36-42.
Ill-Posed, and Non-Classical Problems of
Mathematical Physics and Analysis, Samarkand,
M. M. Lavrent'ev and S. I. Kabanikhin (Eds)
© VSP 2002
2000, pp. 191-197
Recovery of a function set by integrals
along a curve family in the plane
A. H. BEGMATOV* and Z. H. OCHILOV*
Abstract — This work deals with the problem of integral geometry [1, Chapter 6]
of Volterra type. The integrals are given along the set of parabolas with the special
weight function. The stability estimates for solution are obtained in Sobolev's
spaces which shows the weak ill-posedness of the problem. Also, the inversion
formula is obtained.
Romanov [2] had obtained the uniqueness and stability theorems sufficiently general when the manifolds were paraboloids. The weight functions
and manifolds in those cases were invariant relatively the group of all the
motions along a fixed hyperplane. The weak ill-posed problems of integral
geometry of Volterra type with the weight functions with the singularity were
investigated in [3, 4]. The uniqueness theorems, stability estimates, and inversion formulas for weak ill-posed problems of integral geometry when the
integration goes along the special curves and surfaces with singularities in
vertices are obtained in [5, 6].
Let { V ( x , y)} be a set of curves which fill smoothly the half-plane
R\ ={{x,y)
: χ G R 1 , y > 0}.
Suppose that these curves are parametrized by means of their vertices (x, y)
and an arbitrary curve V(x,y) of this set is defined by the relation
V(xty)
= {{ξ,η) : ν~η
= ( χ - ξ ) \ 0 < η < y}.
'This work was partially supported by Uzbek Foundation for Basic Research (grant
No 2/01).
192
Α. Η. Begmatov and Ζ. Η. Ochilov
P r o b l e m 1. Find the function u(x, y) if for all (χ, y) e
integrals along the curves V{x, y) are known:
[
g(x - ξ)η(ξ, y-(χJV(x,y)
\0,
the following
ξ)2) άξ = f(x, y),
(1)
-^y<x-(<0.
Below we shall denote by / ( λ , y) the Fourier transform of the function f(x, y)
with respect to the variable x.
Denote by U the class of functions u(x,y) whose partial derivatives up
to the second order are continuous and finite with the carrier from R^. :
suppti C D = {(x,y)
oo}.
: —a<x<a,0<a<oo,0<y<l,l<
Define the right-hand side of equation (1) for y < 0.
function
[
0,
Introduce the
" i l
y < 0.
Prom the setting of the problem and the condition imposed on the function u it follows that we can apply the Fourier transformation to the function
f*{x,y) with respect to y.
Consider now the Fourier transform of the function f*(X, y) with respect
to y. Taking into account that f*(x,y) vanishes for y < 0, we obtain
oo
roc
/ -00 e^yr(X,y)dy=
e^f(X,y)dy.
Thus, in the case when we define the function f{x,y) by zero for y < 0,
we can apply the Fourier transformation both sides of equation (1).
Introduce the functions
c
ei(ßT+Xy/r)
í»•oo
I (Χ, μ) = 2
j=
dr,
Jo
Í-+00
h(X,ß)
d
= Γ e-^y
2 f
2
J-oo
(ι + μ )ΐ{Κμ)
h = [
J-oo
"e-^hiX^dX.
T h e o r e m 1. Let the function f{x,y) be known for all y > 0. Then the
solution of Problem 1 in the class U is unique, the following representation
holds
u(x, y) = J
J°°
I2(x-ξ,
y - η) (e -
η) dÇ dr/,
Recovery of a function set
193
and the inequality
IMI¿2 <
c\\f\\wo,HR2+)
holds, where C is a certain constant and E is the unit operator.
Proof. Apply the Fourier transformation to both sides of equation (1).
Denoting h = y / y — η , we can write
r+oc m. fy
dn
/
e
/ u(x-h,V)-+dx
J-00
Jo
n
fy
eiXh
ΰ(Χ,η)—άη
Jo «
=
(2)
= /(Χίν).
We apply to (2) the single Fourier transformation with respect to y:
6 ft
J e™ J ΰ(Χ,η) —άηάν
ißy
ΰ(Χ,η) J e άη dy = jf
r oo
J,\h
ιλ
ry
roc
oo
piXh
roo
y) dy.
/
Making the change τ = y — η , w e obtain
ν(Χ,μ)Ι(Χ,μ)=φ{Χ,μ),
where
ν(Χ,μ)=
roo
7ο
ΰ(Χ,μ)β^άη,
Ι(Χ,μ)=2
ψ{\μ)=
roo
¿(μτ+λ-^/τ)
(3)
roo
JO
e^f(X,y) dy,
dr.
(4)
Jo
ντ
We show now that integral (4) converges uniformly relatively the parameters λ and μ . Without loss of generality we may consider that the
parameter μ is positive.
We see that
1=1
r oo
Jo
+M)
dt=
roo
cos
Jo
(μι2 + Xt)dt + i
roo
sin(/ii 2 +
Jo
Xt)
Taking into account the formulas ([7], p. 411)
· , 2 + 2 b x )λ d, x = -1 /^cos b2
J Γ ° sin(aar
2
s .i nb—\ J y ΠΓ
^-'
• , 2 + 2ox) dx = -1/1 cos fe2 1-sin—1
. b2\ \ ΙΊΓ
If°° sin(aa;
—,
o
2
o / y 2o
J
\ α
a
> 0>
a > 0,
di.
194
A. H. Begmatov and Z.H. Ochilov
we obtain
r
1 ΠΓγ/
λ2
. λ2\
.(
λ2
1 = -»/—
cos - — h sin — I + 1 1 cos
2 y 2μ ΙΛ
4μ
4μ/
V
4μ
. Λ 2 ni
sin— .
4/x/J
Estimate the modulus of the function /(Α, μ)
r*~ Fi
in
1
|/| =
2 V ^ V r %
^
. Λ 2 \2
+
Sin
4^)
/
λ2
.
\ C 0 S 4μ
— Sin
λ 2 \2
4μ/
'
Thus, we have
\I\ = y / φ Ζ 2.
(5)
Taking into account (5), we obtain from equation (3)
υ(Χ,μ)=φ(\,μ)/Ι(Χ,μ).
(6)
Divide and multiply the right-hand side of (6) by q + μ2:
v(X, μ) = (1 + μ2)φ(λ, μ)/[(1 + μ2)Ι(Χ, μ)].
(7)
The conditions imposed on the function u(x, y) and (7) yield that the function φ{ •, μ) belongs to L2 with respect to the argument μ. Equalities (5)
and (6) yield that the function (1 + μ 2 ) - 1 / ( λ , μ ) - 1 is the Fourier transform
of the function / ( λ , μ) with respect to y. Apply now the inverse Fourier
transformation to equation (7) with respect to μ. Taking into account the
inversion theorem, the convolution theorem, and the property of differentiation of Fourier transform, we obtain
y) = J00
h (A, y - η) (e - j ^ ) / ( λ , η) άη,
J-ΟΟ
(8)
(1 + μ 2 ) / ( λ , μ)
From the restrictions imposed on the function u(x, y) and (7) it follows that
the function / ( A , · ) belongs to L2 with respect to A. Applying to equation (8) the inverse Fourier transformation with respect to A, we obtain the
representation of the solution of Problem 1
Φ,y)
=
Γ h{-x ~y η)iE
~
~
'
άξd7?
Iff)·η)
Recovery of a function set
where
195
oo
/ -oo e-iXxh(X,y)dX.
Relation (5) and formula (7) yield
|*>(λ,μ)| < (1 + |μ| 2 ) · \φ{Χ,μ)\·
Hence the estimate follows
oo roo
ι
roo
/
/
\ν\*άλάμ<-$= π
(1 + \μ\)2\φ\2άλάμ.
(9)
•oo J—00
ν J-oo
Use now the property of differentiation of the Fourier transform, the triangle
inequality for norms, and (8), (9). As a result we obtain
\\u\\L2 <
The theorem is proved.
cil/llwf1^)·
•
We investigate now the problem of integral geometry with perturbation.
Denote by S(x, y) the part of R+ bounded by the curve V(x, y) and the
axis y — 0; let i i j be the strip {(x,y) : χ G M', y G [O, ¿]} where δ is
sufficiently small.
Problem 2. Define the function u(x,y) if for all (x, y) G R+ we know
the sums of the integrals along the curves V(x,y) and the domains S(x,y):
ry
/ u(x-h,
./Ο
η)-τΛί+
ry rx+h
/
Ηχ,υ,ξ,η)η{ξ,η)άξάη
Jo Jx-h
= Τ{χ,υ),
(10)
where h is defined above.
Equation (10) corresponds to the problem of integral geometry with the
perturbation. The second term in the left-hand side in the integral with the
weight k along the interiors of the parabolas.
Theorem 2. Suppose that the function
y) is known for all y > 0,
and the weight function k G Cq (íí¿ χ
together with its derivatives up to
the second order inclusively vanishes in the parabolas V{x, y).
Then the solution of Problem 2 is unique in the class of twice continuously differentiable functions and the inequality
IMIl2 < Cill-^II^.^Rp
Α. Η. Begmatov and Ζ. Η. Ochilov
196
holds, where Ci is a certain
constant.
P r o o f . Consider the function
fo(x,y)
= T{x,y)
-
f(x,y),
i.e., the second term from the left hand side of equation (10):
n
x+h
y, É. νΜξ, η) άξ άη = f0(x, y).
(11)
Prom the conditions imposed on the functions u and k it follows that
the function /o will have continuous derivatives up to the second order inclusively.
Taking into account that the function k with its derivatives vanishes in
the curves V(x, y), we have
=
¡o ir:kx{xi
d 2 f o
y
¿,y) = [
'e'ηΜξί
η] d c άη
(i2)
'
ν, ξ, ηΜξ, ν) άξ άη.
(13)
Taking into account the restrictions imposed on the weight function k(-)
and the expressions for the derivatives of the function fo(-), we obtain the
estimate
ll/oO^y)!!^,2 ( R 2j < e\\u{x,y)\\L2,
0 < ε < 1.
(14)
The integral operators standing in the left-hand sides of the equations (1)
and (11) we denote by Aq and Αχ. Equations (1) and (10) in this case will
become as follows:
Λο" = / ,
A0u + Aiu = T.
(15)
1
As it follows from [8], the operator AQ has the left inverse AQ . Acting
by Aq 1 on the both sides of (15) we obtain
u + AqLAIU = AQ1T.
(16)
From the estimates of Theorem 1 it follows that the operator Α^Αχ
from (16) is continuous. Therefore, the operator Aq 1A\ satisfies the estimate
||Α0-1Α1||<ε<1.
(17)
Recovery of a function set
197
From the contraction mapping principle being applied for the operator
from the right-hand side (10) follows the uniqueness of solution of equation
(10). Thus, inequalities (14), (17) and Theorem 1 yield the estimate
IM|l 2 < CillJFII^o,!^,
where C\ is a certain constant.
•
REFERENCES
1. M. M. Lavrent'ev, V. G. Romanov, and S.P. Shishatskii, Ill-Posed
Problems of Mathematical Physics and Analysis. Nauka, Moscow, 1980
(in Russian).
2. V. G. Romanov, Some Inverse Problems for the Equations of Hyperbolic Type. Nauka, Novosibirsk, 1972 (in Russian).
3. A. H. Begmatov, Sib. Math. J. (1995) 36, No. 2, 243-247.
4. A. H. Begmatov, J. Inv. Ill-Posed Problems. (1995) 3, No. 3, 231-235.
5. A. H. Begmatov, Sib. Math. J. (1997) 38, No. 4, 723-737.
6. A. H. Begmatov, Dokl. Ros. Akad. Nauk (1998) 358, No. 2, 151-153
(in Russian).
7. I. S. Gradshtein and I. M. Ryzhik. Tables of Integrals, Summs,
and Product. Fizmatgiz, Moscow, 1962 (in Russian).
Series
8. S. G. Krein (Ed), Functional Analysis. Reference Mathematical Bibliography. Nauka, Moscow (in Russian).
Ill-Posed and Non-Classical Problems of
Mathematical Physics and Analysis, Samarkand,
M. M. Lavrent'ev and S. I. Kabanikhin (Eds)
© VSP 2002
2000, pp. 199-205
Uniqueness of extension of solutions of
differential equations of the second order
A. HAIDAROV and D. SHODIEV
Abstract — For a linear differential equation of the second order the Cauchy
problem with the data in the cylinder surface of the time type is considered.
Some important problems of physics and geophysics are connected with
the uniqueness problem for the Cauchy problem for differential equations [1].
The first uniqueness theorem for extension of solutions of equations with
nonanalytic coefficients belongs to T. Carleman [2]. The method was based
on establishment of the domain weight estimates dependent on the domain and differential operators. This method was improved by F. John [3],
M. M. Lavrant'ev [4, 5], L. Nirenderg [6], L. Hörmander [7] et al. For the
nonanalytic hyperbolic equations with the wave operator in the principal
part and the carrier of the form (—Τ, Τ) x 5, the domain of dependence was
found in [8] if S is the sphere of the radius T. The uniqueness of extension
into a certain neighborhood Γ was established in the case when S is strictly
convex in [9]. When Γ = ( - Τ , Τ ) χ S and 5 is a hyperplane in En of the
class C 2 , for the research of the Cauchy problem for hyperbolic equation
with the wave operator in the principal part in [10] the general theorems
of L. Hörmander [7, Chapter 1] were used. Also, in [10], for the case of
the local problem (with the data in the part of boundary of the cylinder
domain) the case when the wave coefficient c is equal to 1 was considered.
In [11] the special case of the problem considered in [10] with the variable
wave coefficient was developed. The local Cauchy problem for a certain class
of hyperbolic equations was considered in [12], and the estimates and the
uniqueness theorems were established. In our paper, the problem of unique
extension of the solution of one class of differential second order equations
200
A. Haidarov and D. Shodiev
from a part of the boundary was considered. The stability estimate in a
neighborhood of the boundary Γ was obtained for a weaker suppositions
relatively the coefficients of the equation then in [11, 12].
Introduce the notations: χ — (XQ, . . . , xn) is a vector from R n + 1 ; t = XQ,
χ' = ( 0 , χ ι , . . . , χ η _ ι , 0 ) ;
(χ, y) = xoyo + ... xnyn.
For an arbitrary set B, we denote by Βσ — Β Π {φ > σ}, where φ is the
function defined below. Let Ν be an exterior normal to the boundary of
the domain, a = (c*o,... ,a„); Da = Dq° • • · D%n, where Dj = —id/dxj,
II ' ||(fc)(Q) is the norm in the Sobolev space W^iQ); \ · |(Q) is the norm
in the space Ck(Q) of the functions k times continuously differentiate. Let
0,
W2 be the closure of Cq°(Q) with respect to the norm || • ||(fc)(Q)· Suppose
also that Ω is a domain from R n whose boundary consists from a part of the
hyperplane xn = d and a component S from the class C2 which lies in the
layer — d < x n < 0. Denote
Q = (-T,T)xü,
Γ = ( - Τ , Τ ) χ S,
where Τ, d are positive constants and Rq = sup n \x'\.
We consider the problem of uniqueness and stability of solution of the
following Cauchy problem
U
= 9o,
Au = f
dix
qñ=9I
in Qo,
in
Γο,
(1)
UEW$(QO),
(2)
where
j,k=1
j-0
J
The coefficients of A satisfy the following conditions
ajk 6 Cl{Q),
ann > e0,
ajk = ak\
ajn = 0 for j < η - 1,
a^aeLooiQ).
(3)
Define the function φ as follows:
φ(χ) = exp [λ(ζ„ + d(l - xlT2)) + 2~1δ(\χ'\2 - Β2)] - 1,
(4)
Uniqueness of extension of solutions
201
where the parameters λ and δ are chosen below. Denote also the operators V'
and AXk:
^
dx\ ' " ' dxn ) '
dxk ^
Theorem 1. Let
<A£',VW,0>>£liei2,
{ΑΧηξ',ξ')< o,
nn
1
nn
(Αξ',ν(Αχ',ξ')) + 2(a )- (Va ,Ax')(Ax',0
< 2-1ε1|ξ'|2,
(5)
for all ξ G
, χ E Q.
Then there exists such a number M dependent only on la·7^1 (Q), so that
for
\AXox'\ < εχε0M-1,
dT~l < <5ειe§M-1,
δ< ε^Μ^,
the solution of problem (1), (2) satisfies the inequality
Μ ε - 2 < λ,
(6)
NI(d(Q.) < *(e)ll«ll}a,[||/ll(o)(Q) + \\9i\\w(To) + ΙΙ<7ο||(2)(Γο)]",
(7)
where Κ is a certain constant dependent on ε only and μ = ε(2 exp Ad — 2 —
ε)" 1 ·
This theorem is established with the use of the following estimate of the
Karleman type.
Theorem 2. Under the conditions of Theorem 1, there exists a constant K, so that
yy-1*1
[ \Dau\2e2T*dx<K
τ > Κ
for all
[ \Au\2e2^ dx,
o
uewliQo)·
(8)
To prove Theorem 2 we will use the conditional Theorem 8.4.3 of
Hörmander [7]. In order to simplify the calculations we introduce the following change of variables by the formulas:
yo = xo,
...,
yn-i = Xn-i,
yn = xn + d( 1 - T~ xl) + 2~1δ(\χ'\2 - R2).
2
(9)
Denote by Ρ the characteristic polynomial of the principal part of the
operator A in the variables y. The functions a jfc we denote as
and ψ we
202
A. Haidarov and D. Shodiev
denote by φ. As a result we obtain
P(y, 0 = -Co2 + Σ ^CiCfc + Βοζοζη + 2 < W C K n + FCl
j,k=1
φ = βΧνη-1,
,\eXyn),
Αφ = (0,0,...
ζ = (6,£ι, · • -,ξη-ι,ξη
7 = rXeXyn,
+ h),
(10)
where
Bo = AdT~2,
F = bnn - 4d 2 T- 4 y 0 2 + δ2 (By', y'),
n
(11)
k
and Β is a linear operator in R with a matrix (V ). Set
Introduce the following form quadratic with respect to ξ
j,k=0
yj
yK
k=0
where ζ = ξ + irVtp(y).
Let Go be the image of Qo under the mapping (9).
L e m m a 1. If the vector ζ Φ 0 satisfies the characteristic equation
P(y; ζ) = 0 and conditions (5) and (6) of Theorem 1 hold, then the
form H(y; ( ) is positive determine for all y 6 Goand
Proof. Taking into account formulas (10)—(12), the polynomials P ^
, j = 0, η are defined by the formulas
p(0) = -2£o + BoCn,
P(j) = 2 (B?)j + 2S(By')jCn,
l<j<n-l,
p ( n ) = Βοξο + 2 S(By', £') + 2FC„,
P(i) = {By¿',Ú
+ ßoy^oCn + 2δ((Βυ%,ξ')ζη
(14)
+ 2F&
(15)
Separating the real part of the equation P(y] ζ) = 0 and taking into
account that r _ 1 ImP(y; 0 = 0 and (10), we obtain
-Co2 + (Βξ',ξ') + (Βξ o + 2 δ(Βν',ξ'))ξη
+ Ftâ
Boto + 2 S W , ? ) + 2 F £ = 0.
+ τ2) = 0,
(16)
203
Uniqueness of extension of solutions
Taking into account (16), rewrite the last equality as follows:
(Βξ',ξ')=ξ 2 0 + Ρξ2η + Ε Ί 2 .
(17)
Changing now Τ and δ so that
4 d T - 1 < y/ëo,
ô2\(By',y')\ < εο/4
and taking into account (3) and (11), we obtain
F > εο/2.
(18)
This and (16), (17), (18) yield the inequalities
Koi<ciia
i£ni<£ia
where C\ = y/nM0, L = ε 1(y/nM0 |_B0| + 2nMi\y'\S) and the constants M;
(I = 0,1) are defined as follows:
Mi = sup
Q
From (14) and (15) by immediate computation we obtain
τ " 1 Im Ρ(o) · P(°) = Χβ^[-2Β0ν0ξ20
- AS(Byoy',ξ')ξ0 - Β0(Βυοξ',ξ')
-4Ρνοξοξη + ΒοΡνο(ξ2η + Ί 2 ) } ,
71-1
T^Im^Py)
3=1
• PW = Xe^[-2S(By',
ν'(Βξ',ξ')) +4δ(Βξ',
+ 4(VF, Βξ')ξη + 2ö(By'VF)^l
V(By',f))
+
)],
72
r^ImP^
.PW = Xex^[26B0(Byny',ξ')ξ0
2F(ByJ',f)
+ 4 5 2 ( ß y n y ^ e r ( ß y ^ e O + (4(5^ n W,e')+25oF î , n eo)en+2PF î / n (e 2 2 +7 2 )].
Using the definition (13) of the form Η and the equality |p(n)|2 = 4 F 2 7 ,
which follows (16) and inequalities (19), we deduce the estimate
λ
- 1 β - Λ y n H { y . 0 > À£Q72/2 +
similar to those obtained in [12].
Thus, H(y, C) > 0 in the characteristic vector ζ, where 7 = 0, |£'| = 0 if
H(y; ζ) = 0. Therefore, since („ = 0, we have ζ = 0.
•
204
A. Haidarov and D. Shodiev
Lemma 1 yields the conditions of Theorem 8.4.3 from [7] for the operator A with the characteristic symbol P(y; ζ) in (10). Therefore, the
estimate (8) holds with G = GQ. Change the variables by formulas (9) in
integrals (8). Taking into account that Jacobian of this change is equal to
unity and that estimate (8) admits the addition of junior terms to P(Y,C),
we see that Theorem 2 holds.
Theorem 1 can be proved from Theorem 2 with the use of the wellknown scheme [1, p. 96]. In the case when the principal part of A is the
wave operator, the proof of Theorem 1 can be found in [11]. In our case
the proof of Theorem 1 is similar. The estimate (7) of Theorem 1 yields the
uniqueness of solution of the Cauchy problem (1), (2) in Qo· The character
of dependence of the constant Κ on ε in (7) in the case of the operator A
with the wave operator as the principal part is considered in [7]. Prom [7] it
follows that such dependence Κ on ε allows us to prove the uniqueness of the
problem in question. Theorem 1 clarifies the geometry of the domain Ω and
the form of the operator A for which the problem is conditionally well-posed.
Estimate (7) of Theorem 1 is necessary also for research of inverse problems
on determination of coefficients of A in cylinder domains. For hyperbolic
equations such estimate is applied for research of inverse problems in [13].
REFERENCES
1. M. M. Lavrent'ev, V. G. Romanov, and S. P. Shishatskii, El-Posed
Problems of Mathematical Physics and Analysis. Nauka, Moscow, 1980
(in Russian).
2. T. Carleman, Sur un problème de unicité pour des sistemes d'equtions
aux derivees partitielles deux variables independentes. Ark Math. .4stronom. Fys. (1939) 17, 1-9.
3. F. F. John, Continuous dependence on data for solution of partial differential equations with a prescribed bound. Comm. Pure. Appi. Math.
(1960) 13, 551-885.
4. M. M. Lavrent'ev, On the Cauchy problem for the Laplace equation.
Izv. Akad. Nauk SSSR. Ser. Mat. (1956) 20, 819-842 (in Russian).
5. M. M. Lavrent'ev, On the Cauchy problem for linear elliptic equations
of the second order. Izv. Akad. Nauk SSSR. Ser. Mat. (1956) 20, 819842 (in Russian).
Uniqueness of extension of solutions
205
6. L. Nirenberg, Lectures on linear differential equations with partial
derivatives. Uspekhi Mat. Nauk (1975) 30, No. 4, 147-204 (in Russian).
7. L. Hörmander, Linear Differential Operators with Partial
Mir, Moscow, 1965 (in Russian).
Derivatives.
8. A. C. Murray and M. H. Protter, Asymptotic behavior and Cauchy
problem for ultrahyperbolic equations. Indiana Univ. Math. J. (1974)
24, No. 2, 115-130.
9. S. P. Shishatskii, A priori estimates in the problem on extension of the
wave field from the cylinder time-like surface. Dokl. Akad. Nauk SSSR
(1973) 213, No. 1, 49, 50.
10. V. M. Isakov, On uniqueness of extension of solutions of hyperbolic
equations. Mat. Zametki (1982) 32, No. 1, 75-81 (in Russian).
11. A. Khaidarov, Carleman Estimates and Uniqueness Theorems for the
Coefficient c of the Operator cutt — Au. Preprint 475. Calcul. Center, Siberian Branch of the USSR Acad. Sci., Novosibirsk, 1983 (in
Russian).
12. A. Khaidarov, Carleman estimates and inverse problems for second
order hyperbolic equations. Mat. Sbornik (1986) 130, No. 2, 265-274
(in Russian).
13. A. Khaidarov, Stability estimates in inverse problems for hyperbolic
equations. Mat. Zametki (1991) 49, No. 1, 119-144 (in Russian).