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.