UDC 517.948.33

APPLICATION OF A THEOREM OF HART TO THE INVESTIGATION OF NONLINEAR INTEGRAL EQUATIONS

K. T. AKHMEDOV, V. F. VITOV

In the investigation of nonlinear integral equations of Hammerstein type, M. M. Vainberg in [2] (see also [3]) applied Hart’s theorem [1]. In the present article the method proposed by M. M. Vainberg is developed in the following direction:

a) it is extended to equations of Urysohn type

\[ u(x)=\lambda \int_B K[x,s,u(s)]\,ds; \tag{1} \]

b) the case is also studied where \(\lambda\) is a characteristic value of a certain kernel, which for the Hammerstein equation was not done by M. M. Vainberg.

1. We formulate below, without proof, Hart’s theorem.

Let in the space \(l_2\) there be given a set of points \((\lambda,c)=(\lambda,c_1,c_2,\ldots)\), forming a domain \(D\) \(\left(|\lambda|\leq \rho,\ \|c\|=\left(\sum_1^\infty c_i^2\right)^{\frac12}\leq r;\ \rho,r\ \text{fixed}\right)\). A function defined on \(D\) of a countable number of variables \(F(\lambda,c)=F(\lambda;c_1,c_2,\ldots)\) is called continuous at the point \((\lambda^*,c^*)\) if from \(|\lambda^{(n)}-\lambda^*|\to0\) and \(\|c^{(n)}-c^*\|\to0\) as \(n\to\infty\) it follows that
\[ \lim_{n\to\infty} F(\lambda^{(n)},c^{(n)})=F(\lambda^*,c^*). \]

Consider a countable number of functions \(F_i(\lambda,c_1,c_2,\ldots)\), defined on \(D\), and, assuming the existence of all
\[ \frac{\partial F_i}{\partial c_k}\quad (i,k=\overline{1,\infty}), \]
write the infinite determinant

\[ \Delta(\lambda,c)= \left| \begin{array}{cccc} \dfrac{\partial F_1}{\partial c_1} & \dfrac{\partial F_1}{\partial c_2} & \cdots \\ \dfrac{\partial F_2}{\partial c_1} & \dfrac{\partial F_2}{\partial c_2} & \cdots \\ \cdot & \cdot & \cdot \end{array} \right|. \]

We indicate the following hypotheses of Hart’s theorem:

1) \(F_i(\lambda,c)\) are continuous in \(D\) together with their partial derivatives
\[ \frac{\partial F_i}{\partial c_k}\quad (i,k=\overline{1,\infty}). \]

2) The series
\[ \sum_1^\infty [F_i(\lambda,c)]^2,\qquad \sum_{\substack{i,k=1\\(i\ne k)}}^\infty \left[\frac{\partial F_i}{\partial c_k}\right]^2,\qquad \sum_1^\infty \left[1-\frac{\partial F_i}{\partial c_i}\right]^2 \]
converge uniformly in \(D\).

3) \(F_i(\lambda^*, c^*) = 0\ (i=\overline{1,\infty})\), where \((\lambda^*, c^*)\) is an interior point of the domain \(D\).

4) \(\Delta(\lambda^*, c^*) \ne 0\).

Hart’s theorem holds:

From the fulfillment of the first three conditions follows the convergence of the infinite determinant \(\Delta(\lambda,c)\).

If, in addition, the 4th condition is also fulfilled, then there exist numbers \(\alpha>0\) and \(\beta>0\) such that to each \(\lambda\) satisfying the condition \(|\lambda-\lambda^*|\le \alpha\) there corresponds a unique point \(c(\lambda)=(c_1(\lambda),c_2(\lambda),\ldots)\), for which \(\|c(\lambda)-c^*\|\le \beta\) and \(F_i(\lambda;c(\lambda))\equiv 0\ (i=\overline{1,\infty})\). Moreover, all \(c_k(\lambda)\) are continuous functions satisfying the condition \(c_k(\lambda^*)=c_k^*\).

1. Let in the domain \(x,s\in B\times B\), \(-\infty<u<+\infty\), where \(B\) is a bounded closed set of a finite-dimensional space, the following hold:

I. \(|K(x,s,u)|\le P(x,s)(a+b|u|),\ (a>0,\ b>0)\), where

\[ \iint_{BB} P^2(x,s)\,dx\,ds\le P^2<+\infty . \]

II. There exists a partial derivative \(K'_u(x,s,u)\), and

\[ |K'_u(x,s,u_1)-K'_u(x,s,u_2)|\le Q(x)|u_1-u_2|, \]

where

\[ \int_B Q^2(x)\,dx\le Q^2<+\infty . \]

III. In some ball \(\|u(x)\|_{L_2(B)}\le \gamma \|u^*\|_{L_2(B)}\) \((\gamma>1)\), the absolute norm \(N(u)\) of the linear operator

\[ Rh \equiv \int_B K'_u[x,s,u(s)]h(s)\,ds \]

is continuous with respect to \(u\).

Conditions I, II, III may be modified in one way or another. Thus, condition I may be replaced by any other, so long as, in combination with the remaining conditions, it would imply the action of the operator

\[ \int_B K[x,s,u(s)]\,ds \]

in the space \(L_2(B)\). Condition II could be replaced by the following: there exist partial derivatives of first and second order with respect to \(u\) of the function \(K(x,s,u)\).

Condition III is fulfilled, for example, under the following particular conditions

\[ K'_u(x,s,u)\equiv \begin{cases} \displaystyle \sum_1^n P_i(x)Q_i(s), & \text{(a)}\\[1.2em] \displaystyle \text{or }\sum_1^n P_i(x)(a_i+b_i|u|), & \text{(b)}\\[1.2em] \displaystyle \text{or }\sum_1^n P_i(x)\,[Q_i(s)(a_i+b_i|u|)]^{\frac12}. & \text{(c)} \end{cases} \]

Here \(P_i(x), Q_i(s)\in L_2(B)\); \(a_i>0,\ b_i>0\); the series (a), (b), (c) may also be infinite, but in that case an additional condition of convergence in the mean of these series appears.

The aim of the present work is to study, under the indicated conditions, the continuability with respect to the parameter \(\lambda\) of the known solution \((\lambda^*,u^*(x))\) of equation (1).

Let us denote by \(\{\varphi_i(x)\}\), \(\{\psi_i(s)\}\) two orthonormal systems (ON-systems) of eigenfunctions in Schmidt’s sense ([4], p. 136) of the kernel.

\(N(x,s)\equiv K'_u[x,\ s,\ u^*(s)]\), and by \(\{\lambda_i\}\) the corresponding system of eigenvalues. These systems always exist for every kernel \(N(x,s)\in L_2(B\times B)\), and the series

\[ \sum_1^\infty \frac{\varphi_i(x)\psi_i(s)}{\lambda_i} \]

converges in the mean to \(N(x,s)\).

Assume that \(\{\varphi_i(x)\}\) is a complete system; below it will be shown that this assumption is not essential.

1. If there exists a solution of equation (1), \(u(x)\in L_2(B)\), then it is representable in a unique way in the form of the series

\[ u(x)=\sum_1^\infty c_i\varphi_i(x). \tag{2} \]

Here \(c_i\) are the Fourier coefficients of the function \(u(x)\) with respect to the complete ON-system \(\{\varphi_i(x)\}_1^\infty\),

\[ c_i=\int_B u(x)\varphi_i(x)\,dx. \]

Substituting here, instead of \(u(x)\), the expression represented by the right-hand side of equation (1), to determine \(c_i\) we obtain the infinite system

\[ F_i(\lambda,c_1,c_2,\ldots)\equiv c_i-\lambda\int_B\varphi_i(x)\int_B K\left[x,\ s;\ \sum_{k=1}^\infty c_k\varphi_k(s)\right]\,ds\,dx=0 \]

\[ (i=1,\infty). \tag{3} \]

We also have

\[ \sum_{i=1}^\infty c_i^2<+\infty. \tag{4} \]

Thus, if equation (1) has a solution \(u(x)\in L_2(B)\), then it is representable in the form (2), where the \(c_i\) are a solution of the infinite transcendental system (3), and moreover

\[ \sum_1^\infty c_i^2<+\infty. \]

The converse is also true, namely: if a solution \(\{c_i\}\) of system (3) is found for which condition (4) is fulfilled, then the series (2) converges in the mean to its sum \(u(x)\in L_2(B)\), which will be a solution of equation (1). Indeed, denoting the sum \(\sum_1^\infty c_i\varphi_i(x)\) by \(u(x)\), multiplying both sides of (3) by \(\varphi_i(x)\) and summing, we obtain

\[ \sum_1^\infty c_i\varphi_i(x)= \lambda\sum_1^\infty \varphi_i(x)\int_B\varphi_i(t)\int_B K\left[t;\ s;\ \sum_{k=1}^\infty c_k\varphi_k(s)\right]\,ds\,dt, \]

or

\[ u(x)=\lambda\sum_1^\infty \varphi_i(x)\int_B\varphi_i(t)\int_B K[t;\ s;\ u(s)]\,ds\,dt. \tag{5} \]

By virtue of condition 1, according to the theorem cited in the monograph of M. A. Krasnosel’skii ([5], p. 47), the operator

\[ Ku=\int_B K[t,\ s,\ u(s)]\,ds \]

acts in \(L_2(B)\), and therefore for every function \(u(s)\in L_2(B)\) the function

\[ w(t)=\int_B K[t,\ s,\ u(s)]\,ds \]

also belongs to \(L_2(B)\). Then the series on the right-hand side of the equality

(5) is the Fourier series \(\sum_1^\infty (w,\varphi_i)\varphi_i(x)\) of the function \(w(x)\), which, by virtue of the completeness of the system \(\{\varphi_i(x)\}\), converges in the mean to the function \(w(x)\) itself, i.e., the right-hand side of equality (5) is equal to \(\lambda \int_B K[x,s,u(s)]\,ds\). Thus, the following has been proved.

Equivalence theorem. The integral equation (1) is equivalent to the infinite system of transcendental equations (3) with the supplementary condition (4). Equivalence is understood here in the sense that every solution of equation (1), \(u(x)\in L_2(B)\), is representable uniquely in the form (2), where the \(c_i\) satisfy system (3) and condition (4), and every solution of system (3) subject to condition (4) determines a solution (2) of equation (1) belonging to \(L_2(B)\).

Thus, the investigation of equation (1) is reduced to the investigation of the infinite, generally speaking transcendental, system (3).

1. Let us write the infinite determinant

\[ \Delta(\lambda,u)= \left| \begin{array}{ccc} 1+\lambda a_{11}(u) & \lambda a_{12}(u) & \cdots\\ \lambda a_{21}(u) & 1+\lambda a_{22}(u) & \cdots\\ \cdot & \cdot & \cdot \end{array} \right|, \tag{6} \]

in which

\[ a_{ik}(u)=-\int_B \varphi_i(x)\int_B \varphi_k(s)K'_u[x,s,u(s)]\,ds\,dx. \]

We shall prove the following fundamental theorem.

Under the assumptions I, II, III made above, if \(\Delta(\lambda^*,u^*)\ne 0\), where \(u^*(x)\) is a solution of equation (1) corresponding to the parameter value \(\lambda=\lambda^*\), then there exists a finite neighborhood of the point \(\lambda^*\) in which there is a unique solution of equation (1), \(u(x,\lambda)\in L_2(B)\), such that \(\|u(x,\lambda)-u^*(x)\|\to 0\) as \(|\lambda-\lambda^*|\to 0\). In this case one says that the solution \((\lambda^*,u^*(x))\) is continued uniquely to some neighborhood of the point \(\lambda=\lambda^*\).

Proof. If \(u^*(x)\in L_2(B)\) is a solution of equation (1) for \(\lambda=\lambda^*\), then by the equivalence theorem we have

\[ u^*(x)=\sum_1^\infty c_i^*\varphi_i(x), \]

where

\[ c_i^*=\lambda^*\int_B \varphi_i(x)\int_B K[x,s,u^*(s)]\,ds\,dx,\qquad \|c^*\|<+\infty. \]

Consider the countable sequence of functions

\[ F_i(\lambda,c)\equiv c_i-\lambda\int_B \varphi_i(t)\int_B K\left[t,s;\sum_{k=1}^{\infty}c_k\varphi_k(s)\right]\,ds\,dt \quad (i=\overline{1,\infty}), \tag{7} \]

defined in the domain \(D\bigl(|\lambda|\le \alpha|\lambda^*|,\ \|c\|\le \gamma\|c^*\|,\ \alpha>1,\ \gamma>1\bigr)\). We shall show that this sequence satisfies the first three of the above-indicated conditions of Hart’s theorem.

To prove the continuity of the function \(F_i(\lambda,c)\), it is obviously sufficient to prove the continuity of the functional

\[ J_i(v)=\int_B \varphi_i(t)\int_B K[t,s,v(s)]\,ds\,dt, \]

where

\[ v(s)=\sum_1^\infty c_k\varphi_k(s)\in L_2(B). \]

We obtain

\[ \begin{aligned} |J_i(v+h)-J_i(v)| &=\left|\int_B \varphi_i(x)\int_B \bigl(K[x,s;v(s)+h(s)]-K[x,s;v(s)]\bigr)\,ds\,dx\right| \\ &=\left|\int_B \varphi_i(x)\int_B K_u'[x,s;v(s)+\tau h(s)]\,h(s)\,ds\,dx\right|\leq \\ &\leq \left\{\int_B\left(\int_B K_u'[x,s;v(s)+\tau h(s)]\,h(s)\,ds\right)^2 dx\right\}^{1/2} \\ &=\left\|\int_B K_u'[x,s;v(s)+\tau h(s)]\,h(s)\,ds\right\| \quad \text{for } \tau\in(0,1). \end{aligned} \]

Somewhat below it will be shown that, on the set of functions \(v(s)\) under consideration, the absolute norm of the linear operator

\[ Rh=\int_B K_u'[x,s,v(s)]\,h(s)\,ds \]

is finite, and hence ([6], p. 439) this operator is completely continuous. From this we obtain
\(|J_i(v+h)-J_i(v)|\to 0\) as \(\|h\|\to 0\). Thus the continuity of \(J_i(v)\), and consequently also of \(F_i(\lambda,c)\), is proved.

To prove the continuity of \(\dfrac{\partial F_i(\lambda,c)}{\partial c_k}\), it is enough to prove the continuity of the functional

\[ J_{ik}(v)=\int_B \varphi_i(x)\int_B \varphi_k(s)K_u'[x,s,v(s)]\,ds\,dx, \]

and this latter follows from condition II.

\[ \begin{aligned} |J_{ik}(v+h)-J_{ik}(v)| &=\left|\int_B \varphi_i(x)\int_B \varphi_k(s) \bigl(K_u'[x,s;v(s)+h(s)]-\right.\\ &\qquad\left.-K_u'[x,s;v(s)]\bigr)\,ds\,dx\right| \\ &\leq \int_B |\varphi_i(x)|\int_B |\varphi_k(s)|\,Q(x)\,|h(s)|\,ds\,dx \leq \\ &\leq \left\{\int_B Q^2(x)\,dx \left[\int_B |\varphi_k(s)|\,|h(s)|\,ds\right]^2\right\}^{1/2} \leq Q\|h\|\to 0 \quad \text{as } \|h\|\to 0. \end{aligned} \]

We shall prove the uniform convergence in the domain \(D\) of the series

\[ R_1=\sum_1^\infty F_i^2(\lambda,c) = \sum_1^\infty \left[ c_i-\lambda\int_B \varphi_i(x)\int_B K\left[x,s;\sum_{j=1}^\infty c_j\varphi_j(s)\right]\,ds\,dx \right]^2. \]

Since

\[ R_1\leq 2\sum_1^\infty c_i^2+2|\lambda|^2R_1^*,\qquad R_1^*=\sum_1^\infty \left\{\int_B \varphi_i(x)\int_B K[x,s,v(s)]\,ds\,dx\right\}^2 \]

and \(v(s)\) denotes the sum \(\sum_1^\infty c_j\varphi_j(s)\), it is sufficient to prove the uniform convergence of the series \(R_1^*\). It was indicated above that the operator

\[ Kv=\int_B K[x,s,v(s)]\,ds \]

acts in \(L_2(B)\) and is completely continuous. Hence \(R_1^*=\|Kv\|^2=\)

\[ = \int_B \left\{ \int_B K[x,s,v(s)]\,ds \right\}^2 dx \]
and, in the ball \(\|v\| \le \gamma \|u^*\|\), the series \(R_1^*\) is bounded, converges, and its sum \(\|Kv\|^2\) is a continuous function of \(v\). Since, moreover, all terms of the series \(R_1^*\) are continuous and positive, by Dini’s theorem the series \(R_1^*\) converges uniformly.

The uniform convergence in \(D\) of the series

\[ R_2=\sum_{i=1}^{\infty}\sum_{\substack{k=1\\(i\ne k)}}^{\infty} \left[\frac{\partial F_i(\lambda,c)}{\partial c_k}\right]^2, \]

where

\[ \frac{\partial F_i(\lambda,c)}{\partial c_k} = -\lambda \int_B \varphi_i(x)\int_B \varphi_k(s)K'_u\left[x,s;\sum_{j=1}^{\infty}c_j\varphi_j(s)\right]\,ds\,dx, \]

and

\[ R_3=\sum_{i=1}^{\infty}\left[1-\frac{\partial F_i(\lambda,c)}{\partial c_k}\right]^2 = \lambda^2\sum_{i=1}^{\infty}\int_B\varphi_i(x)\times \]

\[ \times\int_B \varphi_i(s)K'_u\left[x,s;\sum_{j=1}^{\infty}c_j\varphi_j(s)\right]\,ds\,dx \]

follows from the uniform convergence of the series

\[ R_2^*(\lambda,v)= \sum_{i=1}^{\infty}\sum_{k=1}^{\infty} \left\{ \int_B \varphi_i(x)\int_B \varphi_k(s)K'_u[x,s,v(s)]\,ds\,dx \right\}^2, \]

\[ (\lambda^2 R_2^*=R_2+R_3), \]

and this latter is a consequence of conditions II, III.

Indeed, the sum of the series \(R_2^*\) is the absolute norm \(N(v)\) of the linear operator

\[ Rh \equiv \int_B K'_u[x,s,v(s)]h(s)\,ds, \]

which, by condition III, is continuous and is a real continuous functional.

Further, since the linear integral operator with kernel

\[ N(x,s)\equiv K'_u[x,s;u^*(s)] \]

is completely continuous, it may be noted that the system of eigenfunctions (in Schmidt’s sense) \(\{\varphi_i(x)\}\) is equicontinuous in the mean ([6], pp. 291—292). Then the solution of equation (1), determined by formula (2), with the additional condition \(\|v\|\le \gamma\|u^*\|\), will belong to the class of functions equicontinuous in the mean.

Thus the functional \(R_2^*\) should be considered on a set of functions \(v(s)\) compact in itself, and here ([6], p. 297) the absolute norm

\[ N(v)=R_2^* \]

is finite.

Hence the series \(R_2^*\) converges and its sum is continuous; and since the terms of the series \(R_2^*\) are positive and continuous, by Dini’s theorem the series \(R_2^*\) converges uniformly.

Thus, the fulfillment of the second condition of Hart’s theorem has been established. Next, we note directly that, since \(u^*(x)\) is a solution of equation (1) for \(\lambda=\lambda^*\), the third condition of Hart’s theorem is also satisfied:

\[ F_i(\lambda^*,c^*)=0,\quad (i=1,\infty). \]

Therefore, in view of the first conclusion of Hart’s theorem, the infinite determinant (6) converges.

Then, under the assumption that \(\Delta(\lambda^*,u^*)\ne 0\), by virtue of the second conclusion of Hart’s theorem, there exist numbers \(\alpha>0\), \(\beta>0\) such that in the domain \(D^*\) \((|\lambda-\lambda^*|\leqslant \alpha,\ \|c-c^*\|\leqslant \beta)\) there will exist a unique point
\(c(\lambda)=(c_1(\lambda),c_2(\lambda),\ldots)\), for which
\(F_i(\lambda,c(\lambda))\equiv 0\) \((i=1,\infty)\), i.e., in the domain \(D^*\) the system (3) has the unique solution \(\{c_i(\lambda)\}\), and all \(c_i(\lambda)\) are continuous and satisfy the condition \(c_i(\lambda^*)=c_i^*\).

On the basis of the equivalence theorem, in a neighborhood of the point \(\lambda^*\) \((|\lambda-\lambda^*|\leqslant \alpha)\) there exists a unique solution of equation (1)

\[ u(x,\lambda)=\sum_{1}^{\infty} c_i(\lambda)\varphi_i(x)\in L_2(B). \]

It can be seen that \(\|u(x,\lambda)-u^*(x)\|\to 0\) as \(|\lambda-\lambda^*|\to 0\). Indeed,

\[ \|u(x,\lambda)-u^*(x)\|^2 = \left\|\sum_{1}^{\infty}(c_i(\lambda)-c_i^*)\varphi_i(x)\right\|^2 = \sum_{1}^{\infty}(c_i(\lambda)-c_i^*)^2 . \]

We have

\[ \sum_{i=n+1}^{n+p}(c_i(\lambda)-c_i^*)^2 \leqslant 2\left(\sum_{n+1}^{n+p} c_i^2(\lambda)+\sum_{n+1}^{n+p} c_i^{*2}\right). \]

Since the numerical series \(\sum_{1}^{\infty}c_i^2(\lambda)\) and \(\sum_{1}^{\infty}c_i^{*2}\) converge, for any \(\varepsilon>0\) there exists a number \(m=m(\varepsilon)\) such that for any natural \(p\)

\[ \sum_{i=m+1}^{m+p}(c_i(\lambda)-c_i^*)^2<\frac{\varepsilon}{2}. \]

Further, owing to the continuity of the functions \(c_i(\lambda)\), for the given \(\varepsilon\) and \(m\) there exists a number \(\delta=\delta(\varepsilon,m)\) such that, for \(|\lambda-\lambda^*|<\delta\),

\[ \sum_{i=1}^{m}(c_i(\lambda)-c_i^*)^2<\frac{\varepsilon}{2}. \]

Then for \(|\lambda-\lambda^*|<\delta\) we have

\[ \sum_{1}^{m}(c_i(\lambda)-c_i^*)^2 + \sum_{m+1}^{m+p}(c_i(\lambda)-c_i^*)^2 <\varepsilon . \]

Since here \(p\) is arbitrary, we obtain that

\[ \sum_{1}^{\infty}(c_i(\lambda)-c_i^*)^2 = \|u(x,\lambda)-u^*(x)\|^2 <\varepsilon, \]

as soon as \(|\lambda-\lambda^*|<\delta\).

Thus, the main theorem is completely proved.

1. Let us show that the condition \(\Delta(\lambda^*,u^*)\ne 0\) is equivalent to the requirement that \(\lambda^*\) is not a characteristic number of the kernel \(N(x,s)\), i.e., the linear homogeneous integral equation

\[ \varphi(x)=\lambda\int_B N(x,s)\varphi(s)\,ds = \lambda\int_B K'_u[x,s;u^*(s)]\varphi(s)\,ds \tag{8} \]

would have no solution in \(L_2(B)\), other than the zero solution, for \(\lambda=\lambda^*\).

Consider the equation

\[ \varphi(x)=\lambda\int_B N_n(x,s)\varphi(s)\,ds, \tag{9} \]

in which

\[ N(x,s)=\sum_1^n \frac{\varphi_i(x)\psi_i(s)}{\lambda_i}. \]

Denote by \(D(\lambda)\) and \(D_n(\lambda)\) the Fredholm functions of equations (8) and (9), respectively.

Equation (9) has the solution \(\varphi(x)=\sum_1^n d_i\varphi_i(x)\), where the \(d_i\) are found from the algebraic system

\[ d_i=\lambda\int_B \varphi_i(x)\int_B N_n(x,s)\sum_{k=1}^n d_k\varphi_k(s)\,ds\,dx \quad (i=1,\overline n) \]

or

\[ d_i+\lambda\sum_{k=1}^n a_{ik}^*d_k=0 \quad (i=1,\overline n), \tag{10} \]

where

\[ a_{ik}^*=-\frac{1}{\lambda_i}\int_B \psi_i(s)\varphi_k(s)\,ds. \]

The determinant of the homogeneous system (10) is

\[ \Delta_n(\lambda)= \left| \begin{array}{cccc} 1+\lambda a_{11}^* & \cdots & \lambda a_{1n}^*\\ \vdots & & \vdots\\ \vdots & & \vdots\\ \lambda a_{n1}^* & \cdots & 1+\lambda a_{nn}^* \end{array} \right|. \]

As Goursat showed ([4], p. 64), the determinant \(\Delta_n(\lambda)\) of system (10) coincides with the Fredholm function \(D_n(\lambda)\) of equation (9), and, moreover, \(\lim_{n\to\infty}D_n(\lambda)=D(\lambda)\). But, on the other hand, \(\lim_{n\to\infty}\Delta_n(\lambda)=\Delta(\lambda,u^*)\) (see expression (6)); here

\[ a_{ik}^* =-\frac{1}{\lambda_i}\int \psi_i(s)\varphi_k(s)\,ds =-\int_B \varphi_i(x)\int_B \varphi_k(s)K_u'[x,s,u^*(s)]\,ds\,dx = \]

\[ =a_{ik}(u^*). \]

Thus, \(D(\lambda)=\Delta(\lambda,u^*)\), i.e. the condition \(\Delta(\lambda^*,u^*)\ne0\) is equivalent to saying that the Fredholm function \(D(\lambda)\) of the kernel \(N(x,s)\) does not vanish for \(\lambda=\lambda^*\), i.e. that \(\lambda^*\) is not a characteristic number of the kernel \(N(x,s)\). Hence

Theorem. Under conditions I, II, III, if \(\lambda^*\) is not a characteristic number of the kernel \(N(x,s)=K_u'[x,s,u^*(s)]\), then the solution \((\lambda^*,u^*(x))\) of equation (1) can be uniquely continued continuously with respect to \(\lambda\) and continuously in the mean with respect to \(x\) into some neighborhood of the point \(\lambda=\lambda^*\).

1. Let now \(\lambda^*\) be a characteristic number of the kernel \(N(x,s)\) of rank \(q>1\). In this case we can say nothing about the solvability of the system (3).

Instead of equation (1), consider the equation

\[ u(x)=\lambda\left\{\sum_{k=1}^{q}\beta^{(k)}(x)\left[\frac{\lambda-\lambda^*}{\lambda}\,\xi_k+ \int_B \alpha^{(k)}(s)\left(\frac{u(s)}{\lambda^*}-\frac{u^*(s)}{\lambda}\right)\,ds\right]+ \right. \]
\[ \left. +\int_B K[x,s,u(s)]\,ds\right\}. \tag{11} \]

Here \(\xi_k\) are certain fixed, although undetermined, numbers; \(\{\alpha^{(k)}(x)\}\) and \(\{\beta^{(k)}(x)\}\) are the eigenfunctions corresponding to the characteristic number \(\lambda^*\), respectively, of the kernel \(N(x,s)\) and of the adjoint kernel. (We assume that \(\{\alpha^{(k)}(x)\}\) and \(\{\beta^{(k)}(x)\}\) are ON-systems.)

A solution \((\lambda^*,u^*(x))\) of equation (1) is at the same time also a solution of equation (11). In a neighborhood of the point \(\lambda=\lambda^*\) we shall seek a continuation of this solution of equation (11) in the form

\[ u(x,\lambda)=\sum_{1}^{\infty} c_i(\lambda)\varphi_i(x) \tag{12} \]

\[ \left(\text{for }\lambda=\lambda^*,\quad u(x,\lambda^*)=\sum_{1}^{\infty}c_i^*\varphi_i(x)=u^*(x)\right). \]

To determine the coefficients \(c_i\) we obtain an infinite, generally speaking, transcendental system

\[ \Phi_i(\lambda,c)\equiv c_i-\lambda\left\{\sum_{k=1}^{q}\beta_i^{(k)} \left[\frac{\lambda-\lambda^*}{\lambda}\,\xi_k+\frac{1}{\lambda^*}\sum_{j=1}^{\infty}c_j\alpha_j^{(k)} -\frac{1}{\lambda}\int_B \alpha^{(k)}(s)u^*(s)\,ds\right]\right. \]
\[ \left. +\int_B\varphi_i(x)\int_B K\left[x,s;\sum_{j=1}^{\infty}c_j\varphi_j(s)\right]\,ds\,dx\right\}=0 \]

\[ (i=1,\overline{\infty}). \tag{13} \]

Here \(\alpha_i^{(k)}\), \(\beta_i^{(k)}\) are the Fourier coefficients of the functions \(\alpha^{(k)}(x)\), \(\beta^{(k)}(x)\) with respect to the system \(\{\varphi_i(x)\}\).

It is not difficult to verify that in the domain \(D\), for any fixed finite \(\xi_k\), the system of functions \(\Phi_i(\lambda,c)\) satisfies the first three conditions of Hart’s theorem, analogously to how this was done in the proof of the main theorem.

The fourth condition of Hart’s theorem is also fulfilled, since equation (11) was constructed in such a way that for it the Hart determinant \(\Delta^{(q)}(\lambda,c)\) at the point \((\lambda^*,c^*)\) does not vanish. We shall show this.

We find

\[ \frac{\partial\Phi_i(\lambda,c)}{\partial c_j}\bigg|_{(i\ne j)} =-\lambda\left\{\sum_{k=1}^{q}\beta_i^{(k)}\frac{1}{\lambda^*}\alpha_j^{(k)} +\int_B\varphi_i(x)\times \right. \]

\[ \left. \times\int_B\varphi_j(s)K_u'\left[x,s;\sum_{l=1}^{\infty}c_l\varphi_l(s)\right]\,ds\,dx\right\}. \]

At the point \(\lambda=\lambda^*,\ c=c^*\) we have

\[ \frac{\partial \Phi_i(\lambda^*,c^*)}{\partial c_j}\ _{(i\ne j)} = -\sum_{k=1}^{q}\beta_i^{(k)}\alpha_j^{(k)} -\lambda^*\int_B \varphi_i(x)\int_B \varphi_j(s)N(x,s)\,dx\,ds = \]

\[ = -\sum_{k=1}^{q}\beta_i^{(k)}\alpha_j^{(k)} -\frac{\lambda^*}{\lambda_i}\int_B \psi_i(s)\varphi_j(s)\,ds = b_{ij}, \]

\[ \Delta^{(q)}(\lambda^*,c^*)=\Delta^{(q)}(\lambda^*,u^*)= \left| \begin{array}{ccc} 1+b_{11} & b_{12} & \cdots\\ b_{21} & 1+b_{22} & \cdots\\ \cdot & \cdot & \cdots \end{array} \right|. \]

But \(b_{ij}\) can be represented in the form

\[ b_{ij} = -\lambda^*\int_B \varphi_j(s) \left[ \frac{\psi_i(s)}{\lambda_i} + \sum_{k=1}^{q}\beta_i^{(k)}\frac{\alpha^{(k)}(s)}{\lambda^*} \right]ds = -\lambda^*\int_B \varphi_j(s)w_i(s)\,ds. \]

Consider the kernel

\[ N^{(q)}(x,s)=\sum_{1}^{\infty}\varphi_i(x)w_i(s). \]

It can be shown that this series converges in the mean to its sum, which is denoted by \(N^{(q)}(x,s)\). Taking into account the considerations of Sec. 5, we may assert that the condition \(\Delta^{(q)}(\lambda^*,u^*)\ne0\) is equivalent to the condition that the equation

\[ \varphi(x)=\lambda\int_B N^{(q)}(x,s)\varphi(s)\,ds \]

have no solutions in \(L_2(B)\) other than the zero solution, for the value \(\lambda=\lambda^*\), i.e., that \(\lambda^*\) should not be a characteristic number of the kernel \(N^{(q)}(x,s)\). Further,

\[ N^{(q)}(x,s) = \sum_{1}^{\infty}\varphi_i(x)w_i(s) = \sum_{1}^{\infty}\varphi_i(x)\frac{\psi_i(s)}{\lambda_i} + \]

\[ + \sum_{i=1}^{\infty}\varphi_i(x)\sum_{k=1}^{q}\beta_i^{(k)}\frac{\alpha^{(k)}(s)}{\lambda^*} = N(x,s)+\sum_{k=1}^{q}\frac{\beta^{(k)}(x)\alpha^{(k)}(s)}{\lambda^*}. \]

Hence we see that the kernel \(N^{(q)}(x,s)\) is such that, according to Schmidt, \(\lambda^*\) is not a characteristic value for it (in the ordinary sense), i.e., the Fredholm function \(D^{(q)}(\lambda)\) of the kernel \(N^{(q)}(x,s)\) for \(\lambda=\lambda^*\) does not vanish; and since \(D^{(q)}(\lambda^*)\) coincides with \(\Delta^{(q)}(\lambda^*,u^*)\), we obtain \(\Delta^{(q)}(\lambda^*,u^*)\ne0\).

Then, by Hart’s theorem, system (13), for fixed \(\xi_k\), has a unique solution \(\{c_i(\lambda,\xi_1,\ldots,\xi_q)\}\), i.e., equation (11), for fixed \(\xi_k\) \((k=\overline{1,q})\), has a unique solution \(u(x,\lambda,\xi_1,\ldots,\xi_q)\).

1. We now determine those values \(\xi_k\) \((k=\overline{1,q})\) for which the solution \(u(x,\lambda,\xi_1,\ldots,\xi_q)\) of equation (11) would at the same time be a solution of equation (1).

Obviously, for this it is necessary to require that

\[ \sum_{k=1}^{q}\beta^{(k)}(x)\left[\frac{\lambda-\lambda^*}{\lambda}\xi_k+\int_B a^{(k)}(s)\left(\frac{u(s,\lambda,\xi_1,\ldots,\xi_q)}{\lambda^*}-\frac{u^*(s)}{\lambda}\right)\,ds\right]\equiv 0. \]

By virtue of the linear independence of the functions \(\{\beta^{(k)}(x)\}\), this identity decomposes into a system of equations, and as a result one obtains a system of \(q\) branching equations

\[ (\lambda-\lambda^*)\xi_k=\int_B \alpha^{(k)}(s)u^*(s)\,ds-\frac{\lambda}{\lambda^*}\sum_{i=1}^{\infty}\alpha_i^{(k)}c_i(\lambda,\xi_1,\ldots,\xi_q) \]

\[ (k=\overline{1,q}). \tag{14} \]

Thus, one can prove the theorem.

Under the conditions indicated above, if \(\lambda^*\) is a characteristic number of the kernel \(N(x,s)\) of rank \(q\), then equation (1) has, in a neighborhood of the point \(\lambda=\lambda^*\), as many solution branches continuous in \(\lambda\) and continuous on the average in \(x\) as the system of branching equations (14) has solutions \((\xi_1(\lambda),\ldots,\xi_q(\lambda))\).

1. Everywhere above it was assumed that the system of eigenfunctions \(\{\varphi_i(x)\}\) of the kernel \(N(x,s)\) (in the Schmidt sense) is complete. It turns out that this assumption is not essential.

Let \(\{\Omega_i(x)\}_1^\infty\) be some system that completes the system \(\{\varphi_i(x)\}\) (\(\{\Omega_i(x)\}\) may also consist of a finite number of terms), i.e., the system \((\{\varphi_i(x)\},\{\Omega_i(x)\})\) is already complete in \(L_2(B)\). This new system may be assumed orthonormalized, i.e.,

\[ (\varphi_i,\Omega_k)=0\quad (i,k=\overline{1,\infty}); \]

\[ (\varphi_i,\varphi_k)= \begin{cases} 0, & i\ne k,\\ 1, & i=k; \end{cases} \qquad (\Omega_i,\Omega_k)= \begin{cases} 0, & i\ne k,\\ 1, & i=k. \end{cases} \]

For simplicity, let us carry out the argument only for the case when \(\lambda^*\) is not a characteristic number of the kernel

\[ N(x,s)=K'_u[x,s,u^*(s)]. \]

We seek the solution of equation (1) in the form

\[ u(x)=\sum_1^\infty d_i\Omega_i(x)+\sum_1^\infty c_i\varphi_i(x), \tag{15} \]

where for \(u^*(x)\) we have

\[ u^*(x)=\sum_1^\infty d_i^*\Omega_i(x)+\sum_1^\infty c_i^*\varphi_i(x). \]

To determine \(\{d_i\}\), \(\{c_i\}\), we obtain an infinite system of transcendental equations:

\[ \Psi_i(\lambda,d,c)\equiv d_i-\lambda\int_B \Omega_i(x)\int_B K\left[x,s;\sum_1^\infty d_k\Omega_k(s)+ \]

\[ +\sum_1^\infty c_k\varphi_k(s)\right]\,ds\,dx=0, \]

\[ F_i(\lambda,d,c)\equiv c_i-\lambda\int_B\varphi_i(x)\int_B K\left[x,s; \sum_1^\infty d_k\Omega_k(s)+\sum_1^\infty c_k\varphi_k(s)\right]\,dsdx=0 \quad (i=1,\overline{\infty}). \tag{16} \]

As usual, the fulfillment of the first three conditions of Hart’s theorem is proved. Let us show that supplementing the system \(\{\varphi_i(x)\}\) by the system \(\{\Omega_i(x)\}\) does not affect the value of the determinant \(\Delta(\lambda^*,u^*)\). Indeed, at the point \((\lambda,u^*(x))\) we have

\[ \frac{\partial \Psi_i(\lambda,d^*,c^*)}{\partial d_k} = -\lambda\int_B \Omega_i(x)\int_B \Omega_k(s)\times \]

\[ \times K_u\left[x,s;\sum_1^\infty d_i^*\Omega_i(s)+\sum_1^\infty c_i^*\varphi_i(s)\right]\,dsdx = \]

\[ = \lambda\int_B\Omega_i(x)\int_B\Omega_k(s)N(x,s)\,dsdx \quad (i\ne k). \]

Since

\[ N(x,s)=\sum_1^\infty \frac{\varphi_i(x)\psi_i(s)}{\lambda_i}, \]

and \((\varphi_i,\Omega_j)=0\), we obtain

\[ \frac{\partial\Psi_i}{\partial d_k} = \begin{cases} 0, & i\ne k,\\ 1, & i=k \end{cases} \quad (i,k=1,\overline{\infty};\ u=u^*). \]

In exactly the same way

\[ \frac{\partial\Psi_i}{\partial c_k} = -\lambda\int_B\Omega_i(x)\int_B\varphi_k(s)N(x,s)\,dxds=0 \quad (i,k=1,\overline{\infty};\ u=u^*). \]

Then in the case under consideration Hart’s determinant will have the form

\[ \Delta(\lambda,u^*)= \left| \begin{array}{ccccccc} 1&0&\cdot&0&0&0&\cdot\\ 0&1&\cdot&0&0&0&\cdot\\ \cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot\\ 0&0&\cdot&1&0&0&\cdot\\ 0&0&\cdot&0&1+\lambda a_{11}^*&\lambda a_{12}^*&\cdot\\ 0&0&\cdot&0&\lambda a_{21}^*&1+\lambda a_{22}^*&\cdot\\ \cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot\\ \cdot&&&&&&\cdot\\ &&&&&&\cdot \end{array} \right| = \]

\[ = \left| \begin{array}{ccc} 1+\lambda a_{11}^*&\lambda a_{12}^*&\cdot\\ \lambda a_{21}^*&1+\lambda a_{22}^*&\cdot\\ \cdot&\cdot&\cdot\\ &&\cdot \end{array} \right| \]

and does not vanish for \(\lambda=\lambda^*\), if \(\lambda^*\) is not a characteristic number of the kernel \(N(x,s)\).

The same simple arguments are also carried out in the case when \(\lambda^*\) is an eigenvalue of the kernel \(N(x,s)\). Therefore the theorems stated above are valid for both cases, when the system of eigenfunctions \(\{\varphi_i(x)\}\) (in the sense of Schmidt) of the kernel \(N(x,s)\) is complete or incomplete.

The present work was reported in November 1965 in Baku at the All-Union Interuniversity Conference on the application of methods of functional analysis to the solution of nonlinear problems.

References

1. Hart W. L. Transcon. of the American Math. Soc., No. 23, 1922, p. 1—30.
2. Vainberg M. M. Scientific Notes of Moscow State University, 1, issue 100, 1946.
3. Vainberg M. M. Dissertation. Moscow State University, Moscow, 1940.
4. Goursat É. Course of Mathematical Analysis, 3, part 2. ONTI, 1934.
5. Krasnosel’skii M. A. Topological Methods in the Theory of Nonlinear Integral Equations. Gostekhizdat, 1956.
6. Smirnov V. I. A Course of Higher Mathematics, vol. V. Moscow, 1960.

Received by the editors
January 12, 1966

Azerbaijan State University
named after S. M. Kirov