Mathematics

I. V. Sukharevsky

On the Stability of Solutions of Integral Equations under Discontinuous Variation of the Kernel

(Presented by Academician V. I. Smirnov on 30 V 1958)

I. Statement of the problem. Consider the Fredholm equation

[
u(x)-\mu\int_0^1 K(x,s)u(s)\,ds=f(x),
\tag{1}
]

for which (\mu=1) is an eigenvalue.

Let (\varphi_1(x),\ldots,\varphi_n(x)) be the eigenfunctions corresponding to (\mu=1); let (\psi_1(x),\ldots,\psi_n(x)) be the eigenfunctions of the adjoint kernel (K(s,x)), and, moreover,

[
\int_0^1 f(x)\psi_j(x)\,dx=0
\qquad (j=1,2,\ldots,n).
]

Denote by (M) the set of solutions of equation (1) for (\mu=1).

The principal question studied in this note is the following. Put (\mu=1) in (1) and replace the interval ((0;1)) by the interval ((\lambda;1)), (0<\lambda<1) (or, equivalently, replace (K(x,s)) by the discontinuous kernel (K_\lambda(x,s)), coinciding with (K(x,s)) for (\lambda<s<1) and equal to zero for (0<s<\lambda)). Will the integral equation obtained in this way,

[
u_\lambda(x)-\int_\lambda^1 K(x,s)u_\lambda(s)\,ds=f(x),
\tag{2}
]

be uniquely solvable for every sufficiently small (\lambda)? If this question can be answered in the affirmative, then it is necessary to determine whether (\lim_{\lambda\to 0} u_\lambda(x)) exists; if (\lim_{\lambda\to 0} u_\lambda(x)=u_0(x)) exists, then whether (u_0(x)) belongs to the family of solutions (M), and, finally, what characteristic property singles out (u_0(x)) from (M).* In addition (in Sec. III), we consider discontinuous variation of the kernel in a neighborhood of the straight line (x=s).

II. Let ({\varphi_i}_1^n) and ({\psi_i}_1^n) be orthonormalized:

[
\int_0^1 \varphi_i(x)\varphi_j(x)\,dx
=
\int_0^1 \psi_i(x)\psi_j(x)\,dx
=
\delta_{ij},
]

[
K(x,s)-\sum_{i=1}^n \psi_i(x)\varphi_i(s)=P(x,s),
]

[
(I-P)^{-1}=I+R,
]

[
\text{* A similar question in the case of operator equations with operators analytically dependent on a parameter was considered in the author’s note (1).}
]

where

[
Ph=\int_0^1 P(x,s)h(s)\,ds,\qquad
Rh=\int_0^1 R(x,s)h(s)\,ds.
]

Then one can show that the integral equation (2) is equivalent to the system of equations

[
\sum_{i=1}^{n}\gamma_i(\lambda)\int_0^\lambda \varphi_i(s)\Psi_j(s,\lambda)\,ds
=
-\int_0^\lambda F(s)\Psi_j(s,\lambda)\,ds
\tag{3}
]

[
(j=1,2,\ldots,n),
]

where

[
\gamma_i(\lambda)=\int_\lambda^1 \varphi_i(s)u_\lambda(s)\,ds,
]

[
F(s)=f(s)+\int_0^1 R(s,t)f(t)\,dt,
]

[
\Psi_j(s,\lambda)=\psi_j(s)+\int_0^\lambda R(t,s,\lambda)\psi_j(t)\,dt,
]

and (R(t,s,\lambda)) is the resolvent kernel corresponding to the equation

[
v(s)+\int_0^\lambda R(t,s)v(t)\,dt=h(s).
]

The further analysis is based on the following auxiliary propositions.

Lemma 1. Let (p_1(x),p_2(x),\ldots,p_n(x)) be a system of arbitrary functions, continuously differentiable (n-1) times in a neighborhood of the point (x=0). Put

[
P_i(\lambda)=p_i(\lambda)+\int_0^\lambda R(\lambda,t,\lambda)p_i(t)\,dt,
]

[
Q_i(\lambda)=p_i(\lambda)+\int_0^\lambda R(t,\lambda,\lambda)p_i(t)\,dt
]

and denote by (W(x;\omega_1,\ldots,\omega_n)) the Wronskian of the system of functions (\omega_1(x),\ldots,\omega_n(x)) at the point (x). Then

[
W(0;P_1,\ldots,P_n)=W(0;Q_1,\ldots,Q_n)=W(0;p_1,\ldots,p_n).
]

((K(x,s)) is assumed sufficiently smooth.)

Lemma 2. Let (p_1(x),\ldots,p_n(x);\ q_1(x),\ldots,q_n(x)) be two arbitrary systems of functions, continuously differentiable (n-1) times in a neighborhood of the point (x=0). Then

[
\lim_{\lambda\to 0}\frac{1}{\lambda^{n^2}}
\det\left{\int_0^\lambda p_i(x)q_j(x)\,dx\right}
=
]

[

-\frac{[1!2!\ldots(n-1)!]^3}{n!(n+1)!\ldots(2n-1)!}
\,W(0;p_1,\ldots,p_n)\,W(0;q_1,\ldots,q_n).
]

The main results of this note may be formulated as the following theorems.

Theorem 1. If the systems of eigenfunctions ({\varphi_i}_1^n), ({\psi_i}_1^n) are locally biorthogonalizable near (x=0) (i.e., admit biorthogonalization on ([0;\lambda]) for all sufficiently small (\lambda)), then equation (2) is uniquely solvable for all sufficiently small (\lambda). In particular, if the functions ({\varphi_i}_1^n), ({\psi_i}_1^n) are (n-1) times continuously differentiable and if, moreover,
[
W(0;\varphi_1,\ldots,\varphi_n)\ne 0,\qquad
W(0;\psi_1,\ldots,\psi_n)\ne 0,
]
then the systems of eigenfunctions are locally biorthogonalizable near (x=0), and, consequently, for all small (\lambda) equation (2) is uniquely solvable.

Theorem 2. Let (f(x)) be (n-1) times continuously differentiable, and let the kernel (K(x,s)) have derivatives
[
\frac{\partial^j K(x,s)}{\partial x^j},\qquad
\frac{\partial^j K(x,s)}{\partial s^j}\quad (j=1,2,\ldots,n-1),
]
which are continuous or polar with respect to ((x,s)). If
[
W(0;\varphi_1,\ldots,\varphi_n)\ne 0,\qquad
W(0;\psi_1,\ldots,\psi_n)\ne 0,
]
then the solution (u_\lambda(x)) of equation (2) has, as (\lambda\to 0), a limit (u_0(x)\in M), characterized by the conditions
[
u_0(0)=u_0'(0)=\cdots=u_0^{(n-1)}(0)=0.
]

Theorem 2 contains a number of restrictions connected with the smoothness of the kernel and of the function (f(x)). If, however, the interval ([0;1]) is drilled through simultaneously in a neighborhood of a sufficiently large number of points (c_j\in[0;1]) ((j=1,2,\ldots,m;\ m\ge n)), then these restrictions need not be imposed on the equation.

Let (f(x)) be continuous, and let the kernel (K(x,s)) be continuous or polar. Put
[
e_\lambda=\sum_{k=1}^{m}[c_k-\lambda_k',\ c_k+\lambda_k''],\qquad
E_\lambda=[0;1]-e_\lambda,
]
where (\lambda_k'), (\lambda_k'') are sufficiently small positive numbers, and consider the equation
[
u_\lambda(x)-\int_{E_\lambda} K(x,s)u_\lambda(s)\,ds=f(x).
\tag{4}
]

Let us also introduce the matrices
[
\Phi=
\left{
\begin{array}{ccc}
\varphi_1(c_1)&\cdots&\varphi_1(c_m)\
\cdot&\cdot&\cdot\
\varphi_n(c_1)&\cdots&\varphi_n(c_m)
\end{array}
\right},
\qquad
\Psi=
\left{
\begin{array}{ccc}
\psi_1(c_1)&\cdots&\psi_n(c_1)\
\cdot&\cdot&\cdot\
\psi_1(c_m)&\cdots&\psi_n(c_m)
\end{array}
\right}.
]

Theorem 3. If (\det(\Phi\Psi)\ne 0), then equation (4) is uniquely solvable for all sufficiently small
[
\lambda=\sum_{k=1}^{m}(\lambda_k'+\lambda_k'').
]
If (\lambda\to 0) in such a way that all quantities (\lambda_k'+\lambda_k'') are of one and the same order, then there exists
[
\lim_{\lambda\to 0}u_\lambda(x)=u_0(x)\in M.
]
The solution (u_0(x)) satisfies the condition
[
U\Psi=0,
\tag{5}
]
where (U) is the row matrix ({u_0(c_1),u_0(c_2),\ldots,u_0(c_m)}).

(Condition (5), obviously, uniquely selects (u_0(x)) from the (n)-parameter family of solutions (M).)

Theorem 3 admits an obvious generalization to the case of multidimensional integral equations.

III. Let (K(x,s)) be a polar kernel:
[
K(x,s)=\frac{K_0(x,s)}{|x-s|^{1-\alpha}}\qquad (0<x\le 1),
]

(K_0(x,s)) is continuous in the square (0 \leq x, s \leq 1). We shall assume the function (f(x)) to be continuous on ([0;1]).

Put

[
K(x,s,\lambda)=
\begin{cases}
K(x,s), & \text{for } 0 \leq x,\ s \leq 1,\ |x-s|>\lambda;\
0, & \text{for } 0 \leq x,\ s \leq 1,\ |x-s|<\lambda
\end{cases}
]

and construct the integral equation*

[
v_\lambda(x)-\int_0^1 K(x,s,\lambda)v_\lambda(s)\,ds=f(x).
\tag{6}
]

Theorem 4. If

[
\det\left{\int_0^1 K(x,x)\varphi_i(x)\psi_j(x)\,dx\right}\neq 0,
\tag{7}
]

then there exists

[
\lim v_\lambda(x)=v_0(x).
]

Moreover (v_0(x)\in M) and

[
\int_0^1 K_0(x,x)v_0(x)\psi_j(x)\,dx=0
\quad (j=1,2,\ldots,n).
\tag{8}
]

Let us note that in the case of the kernel (K(x,s)=K(x,-s)), (7) is a necessary and sufficient condition for the biorthogonalizability of the systems ({\varphi_i}_1^n), ({\psi_i}_1^n) on the interval ((0;1)).

Kharkov Polytechnic Institute
named after V. I. Lenin

Received
28 V 1958

REFERENCES CITED

1. I. V. Sukharevskii, DAN, 118, No. 3 (1958).

* (K(x,s)) is the kernel of equation (1), (f(x)) is its right-hand side.