Integral equations of first kind

Integral Equations of First Kind

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A V Bitsadze Steklov Institute for Mathematics Moscow, Russia

Series on Soviet and East European Mathematics

Vol.7

Integral Equations of First Kind V f e World Scientific wb

Singapore • New Jersey • London • Hong Kong

Published by World Scientific Publishing Co. Pte. Ltd. P O Box 128, Farrer Road, Singapore 9128 USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

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INTEGRAL EQUATIONS OF THE FIRST KIND Copyright © 1995 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

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V

PREFACE

The book is devoted to the study of classes of linear integral equations of the first kind most often met in applications. Since the general theory of integral equations of the first kind has not been formed yet we shall confine ourselves here to considering the equations whose solutions are either constructed in quadratures or can be reduced to the well-investigated classes of integral equations of the second kind. One-dimensional integral equations are treated on the basis of the theory of onedimensional Cauchy-type integrals. Consideration various multi-dimensional analogs of such integrals makes it possible to study some multi-dimensional integral equations of the first kind as well. Simple models of equations have been used in this book keeping in mind the fact that if necessary the reader himself, while investigating the problems he is interested in, might apply the methods developed here whenever it is possible in principle. The author wishes to express his thanks to Professors E. I. Moiseev and A. P. Soldatov, who after reading the manuscript made a series of valuable suggestions.

A. V. Bitsadze

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VII

CONTENTS

Preface

v

Chapter 1. Brief Review of the General Theory of the Linear Equations in Metric Spaces

1

Chapter 2. General Remarks on Linear Integral Equations of the First Kind

27

Chapter 3. The Picard Theorem of Solvability of a Class of Integral Equations of the First Kind

51

Chapter 4. Integral Equations of the First Kind with Kernels Generated by the Schwarz Kernel

81

Chapter 5. Integral Equations of the First Kind with the Kernels Generated by the Poisson and Neumann Kernels

103

Chapter 6. Some Other Classes of Integral Equation of the First Kind

125

Chapter 7. The Abel Integral Equation and Some of Its Generalizations

181

Chapter 8. A Two-Dimensional Analogue of the Cauchy Type Integral and Some of Its Applications

215

References

259

1 CHAPTER 1

BRIEF REVIEW OF THE GENERAL THEORY OF THE LINEAR EQUATIONS IN METRIC SPACES

Let E

and E be linear metric spaces and let T be a linear x y mapping of E into E x y Tx=y. (1.1)

Elements x e E and y e E related to each other by equality (1.1) are said to be an inverse image and image under mapping respectively. Let us denote the image of space E by L (1.1) = TE . x

y

y y If the mapping of space E

onto L is one-to-one and, x ~ is also linear, together with T, the inverse mapping x y= Ty then E and L are called linearly homeomorphic. When image y is given beforehand and its inverse image is to be found, equality (1.1) is called a linear equation. The study of solvability of this equation and constructing its solutions (exact, if possible, or approximate) constitute one of the main subjects of mathematics. To study equation (1.1), special methods had to be developed. These methods together with results obtained with their aid form a unified branch of mathematical knowledge referred to as the theory of linear equations in metric spaces. Vital interest drawn to this theory is due to its general scientific and applied importance.

1. Linear Equation in Finite-Dimensional Spaces

In this section, a finite-dimensional space is understood as

2 the ^-dimensional Euclidean space E of points x with Cartesian orthogonal coordinates x+ , ..., x . A point x e E is interpreted as a n-dimensional vector with components xv ..., V The linear equation (1.1) is then represented in the form n z Tikxk it=i

l

= yf

= *> •••• »•

(x-2>

where T = W. ,\\ is a quadratic n * n matrix whose elements 7*., are real numbers. A scalar product of two vectors x and y is understood as the number n X xtyt i=\

= xy.

A norm II* II of v e c t o r x i s t h e non-negative number IIJCII

=

(JCJC)1/2.

On defining the distance between x and y as II x - y II, space £ becomes a n-dimensional vector metric space. Matrix T is called degenerate or non-degenerate depending on whether detT = 0 or detT * 0, respectively. A simple but, at the same time, rather important statement is valid: if matrix T is non-degenerate, equation (1.2) has a unique solution x = T~ly

(1.3)

for any its right-hand side y e E Here T~ denotes a matrix inverse to matrix T. Formula (1.3) is known as the Kramer formula. The equation

3 Tx0 = 0

(1.4)

is called the homogeneous equation corresponding to equation (1.2). The equation T*x*Q = 0,

(1.5)

where

r

= ,T\ki,

T\k = Tki

is referred to as the equation adjoint or conjugate to equation (1.4). Matrices T and T* are simultaneously either degenerate or non-degenerate. When matrix T is non-degenerate, both equation (1.4) and (1.5) possess only the trivial solutions

*0 = ° and x*Q = 0. In the theory of equation (1.2), the following statements are of primary importance: 1) Homogeneous equations (1.4) and (1.5) possess equal finite number I = n - r of linearly independent solutions xk ', ..., xk ' and x^ ] ..., x^S ' where r is the rank of matrix T. 2) In order that the non-homogeneous equation (1.2) be solvable, it is necessary and sufficient that vector y in its right-hand side be orthogonal to all the linearly independent solutions of equation (1.5), i.e.,

4 yx*Q^

= 0,

k = l, ..., l.

(1.6)

If conditions (1.6) are satisfied, the general solution of the non-homogeneous equation (1.2) is given by the formula I

x = x + X 7 = A,., all the n roots of polynomial (1.9) are real. Equation (1.2) is of a certain interest, also when T is a rectangular m * n matrix. In this case, equality (1.2) is a system of m equations with n unknown variables. The system (1.2) is called compatible if it has a solution. In order that the system of equations (1.2) be compatible, it is necessary and sufficient that the rank of matrix T coincide with that of the so called extended matrix S. The extended matrix S is obtained by the addition to matrix T of the last column which consists of the components of vector y in the succession y. , . . . , y from the top to the bottom. This statement is known as the Kronecker-Capelli theorem. It is clear that a homogeneous system of the form (1.4) or (1.8) is always compatible. When the rank r of matrix T is equal to n, the system (1.4) has only a trivial solution. It starts to possess non-trivial solutions, only for r < n. If m< n, the number of non-trivial solutions is equal to n - r. The theory of linear equations in finite-dimensional spaces plays an important role in mathematics and its applications. It constitutes a branch of linear algebra ' .

2. Linear Equations in Space Z 2

The main principles of the theory of linear equations stated in the preceding section remain valid also when the

6 coefficients T.? and the right-hand sides v^ in equation (1.2) are complex numbers. One can easily see this by introducing a complex ^-dimensional space instead of the n-dimensional real space. In this complex space, the scalar product xy and the square of the norm 11*11 are defined by the formulae 00

xy = Z ^ i '

-11*"

= xx

-

i=\ This should not however be done, since singling out real and imaginary parts of all the quantities in these equations doubles the number of unknown variables and equations of the system. The left - hand side of the system x + Tx = y

(111)

is a linear form, while the expressions 00

B(X,

y) =

s

T

ij*iyj>

Q(*) = B(x>

x

)

are a bilinear and a quadratic forms of the variables xv

...,

xn,

yv

.... yn.

The theory of equations of the form (1.11) is also developed in terms of the above forms (see, for example, References 31, 39, 98). An attempt to study the linear equation (1.11) with infinite number of unknowns and equations was made as early as in the nineteenth century. Among scientists, who investigated these equations, Hilbert was the most outstanding figure. It was Hilbert, who introduced and studied the space Z 2 in regard to the investigation of algebraic linear systems with infinite number of unknowns. The elements of space Z 2 are points JC = (JC1 , x~, ...)

7 with infinite number of coordinates, the sum of whosesquares :onverges. The scalar prodi converges. product xy and the square of the norm ii 2 2 llxll are given in the form

xy = X *;y£,

\x\2 = xx.

We shall assume that quantities x and y in equation (1.11) belong to space Z 2 anc* Tx ^ s understood as an expression

J=I In other words, formula (1.11) is an infinite system of linear equations. The left-hand side of each equation of system (1.11) is a linear form of variables x* , Jt2, ..., and the expressions 00

B(x,

y) =

X i, j=l

T

ijxiyj>

Q(*) = B(

C 1 - 12 )

are respectively a bilinear and a quadratic forms. In order to study infinite systems of the form (1.11), Hilbert introduced the concept of completely continuous form with infinite number of variables. It has later become one of the fundamental concepts of the theory of operators. It expresses their property to convert bounded sets of elements in spaces, where they act, into compact sets ' ' I In space l^, the complete continuity of form (1.12) and, therefore, that of operator T is guaranteed by the requirement that the series oo

y

r? • < *

converges. If this condition is satisfied, then either the non-homogeneous equation (1.11) is uniquely solvable for any right-hand side y e l„ or the homogeneous equations

8

x0 + TXQ = o,

xQ + r* 0 = o,

r£j. = r ^ ,

possess equal finite number of linearly independent solutions. If the form B(x, y) is no longer completely continuous, it is still possible to indicate classes of infinite systems (1.11) in which the theory of their solvability in Z 2 c a n be developed. Since they belong to a wider class of linear equations which will be discussed in Section 5 of this Qiapter, here we restrict ourselves to the formulation of the above statement7*.77,92,93,98)^

3. Fredholm Integral Equations of the Second Kind. The Fredholm Theorems

The theory of linear equations has been essentially generalized in the case of linear integral equations. The linear integral equation is understood as an equality of the form a(*)(x) = /(*) + | R(x,

t)f(t)dt.

(1.20)

G

The function R(x, t) in the right-hand side of (1.20) is called the Fredholm resolvent of kernel K(x, t). In particular, the condition of the second Fredholm theorem is always satisfied for a region G of a sufficiently small measure. The

third

Fredholm

theorem.

The

non-homogeneous

11 equation (1,14) is solvable if and only if the conditions k

j f(x)^ \x)dx

= 0,

are satisfied, where j £

k = 1, ..., Z,

(1.21)

} are all the linearly independent

solutions of the associated homogeneous equation (1.19). Under conditions (1.21), the general solution of equation (1.14) is given by the formula I

- f considered in Chapter 1, which is called equation of the second kind. This Chapter is devoted to the clarification of the role played by integral equations of the first kind in mathematics and its applications. Let V denote a domain of the Euclidean space E of points x with Cartesian orthogonal coordinates x-^, ..., x , n ^ 1 with (n -1)-dimensional boundary dT) of class C ' for n * 2. The equality Kcp ^ f K(x, s

t)y(t)dst

= f(x),

x e 5,

(2.2)

is called an integral equation of the first kind. As in Sect. 3 Chap. 1, here the integration takes place over domain D or over its boundary S = dV for n > 1, K(x, t) and

28 f(x) are given functions and cp(*) is the function to be found, K(x, t) being referred to as the kernel of integral operator /C9 or integral equation (2.2) and f(x) being referred to as its right-hand side. In many cases, function Ky is of a higher degree of smoothness than cp(jc). For this reason, equation (2.2) has a solution not for every its right-hand side f(x) at all. As we know from Sects. 3 and 4 of Ch.1, an integral equation of the second kind is also not necessarily solvable for every its right-hand side. However, this is due to some simpler reasons.

1. Integral transformations as integral equations of the first kind

The equality (2.2) can be interpreted as an integral transformation of functions from space E to space Er. For example, the transformation 00

J eTx\(t)dt

= f(x)

(2.3)

0 is known as the Laplace integral equation (see [86], pp 60 61). If /(a) is a finite number, the equality (2.3) can be written as J e^-^^cp^t) = (x - a) I e^-^cp^Orft = 0

f(x)

where x{t)

= J e" a T cp(T)^ 0

for x > a. It follows that, for x > a, the integral in the left-hand side of (2.3) converges and for any E > 0 we have

29

-—

=J e x

E

[e

9l (t)J

rft.

0

In the notations

f (X) =

f(a + s + x) 1 8

cp < t ) = e "9.(0

+ JC

the equality (2.3) can be written as 00

J eTtx92(t)dt

=

f2(x).

0 Since function ^ is boundedand for large values of x > 0 behaves as e~£X, we can assume that function cp(je) in (2.3) is square summable or it belongs to the class of boundedcontinuous functions. Under these assumptions, it can be proved that the transformation (2.3) is uniquely invertible. The Stieltjes equation

r 0 and that the integral J jccr_1cp(jc)^A ix, 0 0

a < a
0,

a = const.

(2.5)

g-loo

Equations (2.4) and (2.5) are generalizations of the well known formulae

f jcz-1e"^jc = r ( z ) , i U

— 9ar1

f

x~z Y(z)dz

= e _A: , a > 0,

J . ff-loo

connecting functions T(z) and eTx (see [57]).

2. On the representation of the solution of the Dirichlet problem for harmonic functions as a potential of a simple layer

31 In the theory of the Dirichlet problem for harmonic functions (elliptic equations of the second order) the solution of this problem is usually searched for as a potential (generalized potential) of a double layer. This is considered to be natural because, for determining the density of the potential, the Fredholm equation of the second kind is obtained, which is globally (locally) solvable under rather general assumptions concerning the behaviour of the solution to be found in a closed domain 2) U a2) and the smoothness of its boundary a2). The function u(x) harmonic in domain 2) and even continuous in 2) U dV cannot always be represented as a potential of a simple layer. For example, a function which is harmonic in domain 2), continuous in 2) U dV and does not have normal derivative on a2), can not be a potential of a simple layer, because the simple-layer potential should necessarily have normal derivative on a 2) under rather general assumptions. Nevertheless, under the requirement that the boundary dV of domain 2) be sufficiently smooth, a wide class of functions harmonic in this domain can still be represented as a potential of a simple layer. So we search for the solution of the Dirichlet problem "+(*0)

=

/(*o>'

*0

6 8Z)

'

(2

'6)

for functions u{x) harmonic in domain 2) in the form of a potential of a simple layer u(x)

= 1%

[ E(x,

t)y(t)dst

(2.7)

3D

where E(x, t) is an elementary equation in space E

solution of the Laplace

32 - logl* - t |,

H = 2 2%n/2

I E(x,

t) = 1

? „ |* - t\2'n,

" n>

r(n/2)

2,

n - 2 Since the potential of a simple layer is continuous in the whole space E for determining the density cp(t) we obtain the following integral equation of the first kind ([40], [55], [88]) —

f E(xQ9

t) ( t 2 ) l o g | t 2 - tQ\ - y(t1)log\t1

- tQ\]

= 0.

£-0

On the other hand, according to the equality \t - tQ\

= (t - tQ) exp (- id(t, t0))

we conclude that, for t e S , d

t'r, y

l o g |t - t f l | =

d i

*(t,

tn).

We t h u s have 1

d

i^ = -*■

f

l o g | t - tQ\0

L

L

t

f

£

0

The limit r

d

lim

fl(t,

r

tn) 0 in (2.13), we obtain the formula (2.11). Thus, the above statement is proved ([12], [14], [16], [39], [55], [87]).

4. On the representation of the solution of Neumann's problem in the form of a potential of a double layer

Let us assume the boundary S of domain D to be a sufficiently smooth (n - 1)-dimensional surface. We shall consider the Neumann's problem as follows: it is required to find a harmonic in domain V function belonging to class C ' (2) U S) and satisfying the boundary condition dU dv

x

=

-

"

-

\L(t)dt

(2.22)

!

TTl

«i

z

a * ~

with an unknown real density \i which is Hoelder continuous. According to the Sokhotskii-Plemelj formula, we obtain from (2.22) a

1

l

+ 0 (X.)

= \l(x.)

H

\i(t)dt

r |JL -a

- a
1) and u and F are the m-dimensional vectors, the requirement of uniformly ellipticity of this system can not always guarantee that the

42 Dirichlet problem is correctly posed. For example, the system of equations S22U, U, S dx1 dXy

s\ a\

2

-

l

32U, -T±± - 2 dX2 6X2

s8%\ s—_ = 0, 8x^3x2 dx^dx

e\

s\

- + —fdXlXldx axf B dx22 dx\

J- = 0 dx$ dx*

(2.25)

is uniformly elliptic on the whole plane of complex variable z = xx + i*2- However, the domain can be shown, where the problem of finding a regular solution w = u-[ + iu2 °f this system satisfying the Dirichlet boundary condition ™(t Ht00)

=

= ^O^o)' 70('o>'

0 G *0

t

aZ) dD

''

((222. 2 6 6) )

is not correctly posed. We can easily verify this by considering an example when © is a circle \z\ < 1. In this case, the corresponding to (2.26) homogeneous problem

V V oV> = 0,

\t I'Qo\' = 1, 1,

(2.27) (2.27)

has as non-trivial solution - the function wQ(z)

(z~z - l)cp(z), = (zz

(2.28)

where cp(z) is an arbitrary analytic function in circle 2). In the considered case of an elliptic system, the incorrectness of the Dirichlet problem (2.26) consists in the fact that the homogeneous problem (2.27) has an infinite set of linearly independent solutions, and non-homogeneous problem (2.26) is not necessarily solvable for each right-hand side 7 [4, 14]. To understand the nature of this phenomenon we consider

43 a simply connected domain D of class C1 'h on plane z. If we write system (2.25) in the form W

zz-°

we easily see that all its solutions regular in are represented by the formula w(z) = zcfr^z) + ^ ( z ) ,

domain D

(2.29)

where §i(z) and *h(z) are arbitrary functions analytical in » [3], [4], [14]. We assume 4>j(z) and ty^(z) to be continously extendible on 3D and write the boundary condition (2.26) in the following way w(t)

= ti\>1(t)

+ ^ ( t ) = 70(£),

t

e dD.

(2.30)

The equality (2.30) is the boundary condition for functions §Az) and *h(z) analytical in domain 2). We shall show that this problem is equivalent to the integral equation of the first kind

1

zJU - zTT7

r

ty--

—= 11

IT

H

=

= / ( 0 ,

(2-31)

* - T

l«l = l,

1=^*,

where z = z(£) is an analytic function conformly mapping the domain D on the circle

': {U\ < l} of the plane of complex variable £ and /(£) is a given

44 function defined on dd = cr [14], [42]. In fact, let us assume the existence of functions ^(z) and v|/1(z) of class Clfh (D U dV) analytic in domain D which satisfy the boundary condition (2.30). The functions 4>(0 4>i z ( 0 anc* + ( 0 - + 1 z ( 0 analytic in circle d are Hoelder continuous on d U J. In view of (2.30) these functions satisfy the boundary condition

^(7)4>(«) + +(«) = 7(0.

« e a,

(2.32)

where

7(0 = 70[Z(S)]. In circle d we represent the analytic function (£) in the form of the Cauchy type integral 1

9(T)^T

r

♦ (5) =

•

(2.33)

2iri J T - 5 By virtue of Sokhoskii-Plemelj formula, from yields 1

1

r

(2.33)

(P(T)^T

4> ( O = " 9 ( 0 + , 2 2iri J T - 5

S - cr.

(2.34)

For the expression

+ + (0 = - 7(0 + iTTH + (5)

(2.35)

determined from (2.32) to be the limit values of function i|/(0 analytic in d it is necessary and sufficient that the equality +

1

p ++(T) 0 to the position 2(£n> 0), £Q > £, so that time t is the given function t = t (-n) of coordinate T\ measuring height. The problem is to determine the trajectory of point M(x, y) which is referred to as the tautochrone problem. 2 It is known that the scalar square v of the velocity vector dx

dy ^

dt

dt '

of point M(x, y) satisfies the relation v 2 = 2*(T| - y),

0 < y < ^,

(2.37)

where g is the gravity acceleration. If a(jc, y) is the angle between the velocity vector and the positive direction of axis x in the counterclockwise direction, relation (2.37) yields dy — = vsina = V2g(T\ - y)sina dt Since the trajectory x = x(y)

(2.38)

is unknown the quantity

9 00 =

(2.39) sina[*(y). y]

is unknown either. Eqs. (2.38) and (2.39) imply that

47

}

v{y)dy

J

Mgi-n - y)

0

or = fW 0 v^

-

(2.40)

y

where /(T!) = - •Zgt(T|). Function cp(y) should therefore be a solution of the integral equation of the first kind (2.40) with a variable upper limit referred to as Abel integral equation with exponent 1/2. In view of (2.39) only those solutions cp(y) of the equation (2.40) are physically meaningful which satisfy the condition |cp(y) | > 1. If we manage to find the solution cp(y) of this equation which possesses the above property, the geometrical equality j

= tga = (csc2« - 1)1/2= [ 9 2 ( > ) _ ^ 1 / 2

immediately gives the trajectory of the moving point in the form of the integral x-][,

2

iz)-iy/2dz.

9. Some general remarks on integral equations the book is concerned with

The reader may ask whether the class of integral equations

48 of the first kind being under consideration in this chapter is natural. The answer is positive and quite clear when we speak of integral transformations or Abel integral equations are meant. But it is also positive in all other cases. We shall now explain why it is so. Let us start with a well known classical electrostatic problem. Let V be the domain of the Euclidean space E^ of points x with Cartesian orthogonal coordinates x-,, x~ ^3 occupied by conducting medium (body). The problem is to determine the density \i(t) of charge distribution along the boundary S of the domain D if the potential u(x) of electrostatic field induced by these charges is known to be constant in D. Since it is the Coulomb potential that is under consideration, function u(x) is a potential of a simple layer with density jx(t)

u(x)

= — 4TT

\i(t)dsi

\ — „ \t s \*

J

-*\

which is constant everywhere in D and in particular on i.e. , 1

4TT

r

S,

\i(t)ds [L

I It

= const,

JC0 e S.

(2.41)

Equality (2.36), which should be used to determine function \L(t), is a special case of integral equation of the first kind (2.8). This problem is also meaningful in the case of a planar electrostatic field. In this case T) is the plane cross section of a long cylindric conductor. The appearance of an integral equation of the first kind in the theory of contact problems (for example, equation (2.18)), in the Wing theory (see equation (2.23)), in the theory of sea rising tides, etc. is also due to the occurrence of the potential of a simple layer in the corresponding boundary problems. The integral equations of the first kind, that will be considered below, are mainly related (except for the Abel

49 integral equation) with the boundary value for equations of elliptic type, which are undoubtelly correctly posed. The only exception is equation (2.31) connected with the Dirichlet boundary value (2.26) for elliptic system (2.25). We remind the reader that, according to Hadamard, a problem is referred to as correctly posed if it has a solution and it is a unique and stable one. The context of Hadamard's studies [69], [70] where this definition was given, shows that he meant classical problems for the main types of partial differential equations of the second kind, in particular, the Dirichlet problem for elliptic equations, the Cauchy problem with a data of a spatial type and the characteristic Cauchy problem for hyperbolic equations, the first boundary problem, and the Dirichlet-Cauchy problem for parabolic equations. The property of correctness is inherent in the very problems considered in physics and other natural sciences modelled in terms of these equations. If in equations there are terms with lower derivatives, these problems are regarded to be investigated locally (not necessarily globally) [14]. When the problem of finding a solution of an equation of the first kind (2.1) is said to be incorrectly posed, it means the following. If operator A is completely continuous, its inverse operator, if any, is not even continuous. (Almost all the integral equations of the first kind considered in this book possess this quality). When it is possible, additional requirements imposed on the right-hand sides of these equations are nevertheless pointed out, under which the existence of a solution is proved. And when it is impossible the reasons for this are analyzed. Though simple methods of differentiation and fractional differentiation, which enable explicit inversion formulae of the considered integral equations to be derived, may be taken for regularization, they play a specific role in the up-to-date general theory of the regularization of incorrectly posed problems [1], [29], [33]. In the past few decades, incorrectly posed problems have been intensively investigated by specialists. It is

50 linear integral equations that a special attention has been paid to. Since in incorrectly posed problems most or some of the Hadamard equirements (existence, uniqueness, stability of solution) imposed on correctly posed problems are not satisfied. Most of the researchers either change the way the problems are posed by regularizing them or generalize the concept of solution by introducing various approximate, asymptotic solutions, quasi-solutions, etc. to make it possible to determine a stable way of developing these solutions. A new rapidly developing field of modern mathematics called "Incorrectly posed problems" has appeared. Though almost all the equations studied below have arisen from applications the development of there approximate solutions goes beyond the scope of the present book. In this sence too the book differs from other known books devoted to incorrectly posed problems [1], [29], [33].

51 CHAPTER 3

THE PICARD THEOREM OF SOLVABILITY EQUATIONS OF THE FIRST KIND

OF

A

CLASS

OF

INTEGRAL

1. Preliminary remarks

In this section we shall assume that in equation (2.2) the integration is performed over a finite interval (a, b) of change of real variables x and t i.e.

b Kip - [ K(x, a

t)q>(t)dt

= /(*),

a ^ x ^ b

.

(3.1)

The problem of determining a solution of Equation (3.1) is incorrectly posed. We can easily see this on simple examples. Example 1. The kernel K(x, t) of Equation (3.1) is a polynomial of power N with respect to variable x

N *(*, t) = X

an{t)xm

with continuous or square summable coefficients a (t). With the requirement that the function cp to be found is continuous or square summable, the given function / must be a polynomial N

/