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UDC 517.948.33
INVESTIGATION OF A CLASS OF NONLINEAR INTEGRAL EQUATIONS
K. T. AKHMEDOV, Sh. KUTMATOV, V. F. VITOV
1°. In the present paper we consider a nonlinear integral equation of the form
\[ v(x)=\lambda \sum_{n=0}^{\infty}\int_{G_n} K_n[x,y_1,\ldots,y_n,v(y_1),\ldots,v(y_n)]\,d\tau_n . \tag{1} \]
Here \(G_n\) is an \(n\)-dimensional cube with side equal to unity and with vertex at the point \((0,0,\ldots,0)\).
In the book of N. S. Smirnov [1] the existence of a solution of equation (1) was proved under the following assumptions. For all real \(v(x)\) belonging to a ball of the space \(L_2(0,1)\), i.e.
\[ \|v\|^2=\int_0^1 v^2(x)\,dx \leq C=\mathrm{const}, \tag{2} \]
the following conditions are satisfied:
(I) the functions \(K_n[x,y_1,\ldots,y_n,v(y_1),\ldots,v(y_n)]\) are continuous with respect to the variables \(v(y_1),\ldots,v(y_n)\),
(II) and are quadratically summable with respect to \(x,y_1,\ldots,y_n\);
(III)
\[ \sum_{n=0}^{\infty} \left\{ \int_0^1 \int_{G_n} K_n^2\,d\tau_n\,dx \right\}^{1/2} \leq A=\mathrm{const}, \]
(IV)
\[ \rho\bigl(K_n[x,y_1,\ldots,y_n,v(y_1),\ldots,v(y_n)], \]
\[ K_n[x,y_1,\ldots,y_n,\omega(y_1),\ldots,\omega(y_n)]\bigr) \leq D_n\rho(v,\omega), \]
where
\[ \sum_{n=0}^{\infty}D_n \leq D=\mathrm{const}. \]
From conditions (I), (II), (III) it follows that the operator
\[ N(v)=\sum_{n=0}^{\infty}\int_{G_n} K_n[x,y_1,\ldots,y_n,v(y_1),\ldots,v(y_n)]\,d\tau_n \]
is compact.
These conditions, together with condition (IV), are sufficient for the operator \(N(v)\) to be, moreover, continuous, i.e. \(N(v)\) is completely continuous. Then, according to Schauder’s principle, for every \(\lambda\) satisfying the inequality
\[ \lambda^2 \leq \frac{C}{A^2}, \tag{3} \]
the equation (1) has a solution \(v(x)\) (not necessarily unique) belonging to the ball (2).
Suppose that in some way, for one of the values of the parameter \(\lambda\), \(\lambda=\lambda^*\), a solution \(v^*(x)\) of equation (1) has been found.
The question arises how to obtain a solution of equation (1) in some neighborhood of the value \(\lambda=\lambda^*\), and, in general, whether it is possible to continue the solution \((\lambda^*, v^*(x))\) along the entire axis of values of the parameter \(\lambda\).
In the particular case when each \(\widetilde K_n(x, y_1, \ldots, y_n, v_1, \ldots, v_n)\) is analytic in \(v_1,\ldots,v_n\), the operator \(N(v)\) is an integro-power series
\[ N(v)=\sum_{n=0}^{\infty} W_n \begin{pmatrix} x\\ v \end{pmatrix} = \sum_{n=0}^{\infty}\sum_{m=0}^{\infty} \int_{G_n} K_{n,\alpha_1,\ldots,\alpha_n}(x,y_1,\ldots,y_n) v^{\alpha_1}(y_1)\cdots v^{\alpha_n}(y_n)\,d\tau_n \]
\[ (\alpha_1+\alpha_2+\cdots+\alpha_n=m). \]
The existence and branching of solutions of integro-power series were studied by E. Schmidt in his classical paper [2] (see also [3]).
Under the same requirement of analyticity of \(K_n\) in \(v_1,\ldots,v_n\), the nonlinear equation \(v(x)=\lambda N(v)\) can also be studied by the Nekrasov–Nazarov method (see [4]).
In another particular case, when \(K_n=0\) \((n=0,\overline{2,\infty})\), equation (1) is the Uryson equation
\[ v(x)=\lambda\int_0^1 K(x,y,v(y))\,dy, \]
which has been studied from various points of view by many authors.
In the present paper, the methodology of [6] is used to study the continuability and branching with respect to the parameter of a known solution \((\lambda^*, v^*(x))\) of equation (1). In addition to the conditions I–IV indicated above, assume the existence of the partial derivatives
\[ \frac{\partial}{\partial v_k} K_n(x,y_1,\ldots,y_n,v_1,\ldots,v_n) \qquad (k=\overline{1,n};\ n=1,2,\ldots), \]
with respect to which let
\[ \text{(V)}\qquad \rho\left( \frac{\partial}{\partial v_k}K_n(v),\, \frac{\partial}{\partial \omega_k}K_n(\omega) \right) \le D_{nk}\rho(v,\omega), \]
where
\[ \sum_{n=1}^{\infty}\sum_{k=1}^{\infty} D_{nk}\le D=\mathrm{const}; \]
\[ \text{(VI)}\qquad \left| \frac{\partial}{\partial v_k} K_n(x,y_1,\ldots,y_n,v_1,\ldots,v_n) \right| \le M_{nk}(y_1,\ldots,y_n, \]
\[ v(y_1),\ldots,v(y_n)) \]
and the function
\[ M(y)=\sum_{n=1}^{\infty}\int_{G_{n-1}} M_{nk}[y_1,\ldots,y_{k-1},y,y_{k+1},\ldots,y_n, \]
\[ v(y_1),\ldots,v(y_{k-1}),v(y),v(y_{k+1}),\ldots,v(y_n)]\,d\tau_{n-1} \]
is quadratically summable for all \(v(x)\) satisfying (2). Here
\[ d\tau_{n-1}=dy_1\ldots dy_{k-1}dy_{k+1}\ldots dy_n . \]
Let us denote by \(K(x,y)\) the function
\[ K(x,y)=\sum_{n=1}^{\infty}\int_{G_{n-1}}\sum_{k=1}^{n}\frac{\partial}{\partial v_k} K_n[x,y_1,\ldots,y_{k-1}, \]
\[ y,y_{k+1},\ldots,y_n,v_1,\ldots,v_n]\,d\tau_{n-1}, \]
where
\[ v_1=v^*(y_1),\ldots,v_{k-1}=v^*(y_{k-1}),\quad v_k=v^*(y),\quad v_{k+1}= \]
\[ =v^*(y_{k+1}),\ldots,v_n=v^*(y_n). \]
It is not difficult to verify that this function is quadratically summable. Then, according to Schmidt, it is representable in the form
\[ K(x,y)=\sum_{i=1}^{\infty}\frac{\varphi_i(x)\psi_i(y)}{\lambda_i}, \]
where \(\{\varphi_i(x)\}\), \(\{\psi_i(y)\}\) are orthonormal systems of eigenfunctions of the kernel \(K(x,y)\) in Schmidt’s sense, and \(\{\lambda_i\}\) is the corresponding system of eigenvalues.
Assume that the system \(\{\varphi_i(x)\}\) is complete.
\(2^\circ\). The continuation of the known solution \((\lambda^*,v^*(x))\) of equation (1) to some neighborhood of the point \(\lambda=\lambda^*\) will be sought in the form
\[ v(x,\lambda)=\sum_{i=1}^{\infty}c_i(\lambda)\varphi_i(x), \tag{4} \]
where it is required that \(v(x,\lambda)\in L_2(0,1)\) and \(v(x,\lambda)\to v^*(x)\) as \(\lambda\to\lambda^*\).
It is not difficult to obtain, for determining \(c_i(\lambda)\), the following infinite system:
\[ F_i(\lambda,c_1,c_2,\ldots)=c_i-\lambda\int_0^1\varphi_i(x) \sum_{n=0}^{\infty}\int_{G_n}\left[ K_n\left[x,y_1,\ldots,y_n, \right.\right. \]
\[ \left.\left. \sum_{j=1}^{\infty}c_j\varphi_j(y_1),\ldots, \sum_{j=1}^{\infty}c_j\varphi_j(y_n) \right]\right]\,d\tau_n\,dx=0,\qquad (i=1,\infty). \tag{5} \]
The following holds.
Equivalence theorem. The integral equation (1) is equivalent to the infinite system of transcendental equations (5) with the additional condition
\[ \left(\sum_{i=1}^{\infty}c_i^2\right)^{\frac12}\le C \tag{6} \]
in the sense that every solution \(v(x)\in L_2(0,1)\) of equation (1) is representable in the form (4), where \(\{c_i\}\) will be a solution of the system (5), and conversely
but every solution of system (5) satisfying requirement (6) determines, in the form (4), a solution of equation (1).
The proof of the equivalence theorem is carried out similarly to how this was done in [6].
The subsequent reasoning is substantially based on Hart’s theorem [5]; in this connection we give it here in the following formulation:
Hart’s theorem. Denote by \(P\) the set of points \((t,\xi)=(t,\xi_1,\xi_2,\ldots)\) for which
\[
|t|\leq \rho,\qquad \|\xi\|=\left(\sum_{i=1}^{\infty}\xi_i^2\right)^{\frac12}\leq r
\]
(\(r,\rho\) are constants).
Suppose that a countable number of functions \(g_i(t,\xi)\) is given, continuous in the domain \(P\) and having continuous partial derivatives
\[
\frac{\partial}{\partial \xi_k}\,g_i(t,\xi)\qquad (i,k=1,2,\ldots).
\]
Suppose further that the following conditions are satisfied: the series
\[
\sum_{i=1}^{\infty}[g_i(t,\xi)]^2,\qquad
\sum_{i,k=1}^{\infty}\left[\frac{\partial}{\partial \xi_k}\,g_i(t,\xi)\right]^2\quad (i\ne k),
\]
\[
\sum_{i=1}^{\infty}\left[1-\frac{\partial}{\partial \xi_i}\,g_i(t,\xi)\right]^2
\]
converge uniformly in \(P\), and
\[
g_i(t^{(0)},\xi^{(0)})=0\qquad (i=1,2,\ldots),
\]
where \((t^{(0)},\xi^{(0)})\) is an interior point of the domain \(P\). Then the infinite determinant
\[
\Delta_{t,\xi}=
\left|
\begin{array}{cccccc}
\dfrac{\partial g_1}{\partial \xi_1} & \dfrac{\partial g_1}{\partial \xi_2} & \cdots & \dfrac{\partial g_1}{\partial \xi_n} & \cdots \\[6pt]
\dfrac{\partial g_2}{\partial \xi_1} & \dfrac{\partial g_2}{\partial \xi_2} & \cdots & \dfrac{\partial g_2}{\partial \xi_n} & \cdots \\[6pt]
\cdots & \cdots & \cdots & \cdots & \cdots \\[6pt]
\dfrac{\partial g_n}{\partial \xi_1} & \dfrac{\partial g_n}{\partial \xi_2} & \cdots & \dfrac{\partial g_n}{\partial \xi_n} & \cdots \\[6pt]
\cdots & \cdots & \cdots & \cdots & \cdots
\end{array}
\right|
\]
converges; moreover, if the condition
\[
\Delta_{t^{(0)},\xi^{(0)}}\ne 0
\]
is satisfied, then there exist two numbers \(\alpha_1>0,\ \alpha_2>0\) such that to each \(t\) satisfying the condition
\[
|t-t^{(0)}|\leq \alpha_1
\]
there corresponds a unique point \(\xi(t)\), for which
\[
\|\xi(t)\|\leq \alpha_2
\]
and
\[
g_i(t,\xi(t))\equiv 0\qquad (i=1,2,\ldots);
\]
in addition, all coordinates \(\xi_n(t)\) are continuous functions and
\[
\xi(t^{(0)})=\xi_n^{(0)}.
\]
We shall show that, for system (5), the conditions of Hart’s theorem are satisfied in the domain
\[
P^*\bigl(|\lambda|\leq 2|\lambda^*|,\ \|c\|\leq 2\|c^*\|\bigr).
\]
1) The continuity of the functions \(F_i(\lambda,c_1,c_2,\ldots)\) follows from condition (IV). Indeed,
\[ \begin{aligned} &\left|F_i(\lambda,c_1,c_2,\ldots)-F_i(\lambda',c'_1,c'_2,\ldots)\right|\leq |c_i-c'_i|+\\ &\quad+\left|\lambda\int_0^1 \varphi_i(x)N(v)\,dx-\lambda'\int_0^1 \varphi_i(x)N(v')\,dx\right|\leq \left\|c_i-c'_i\right\|+\\ &\quad+|\lambda-\lambda'|\cdot\left|\int_0^1 \varphi_i(x)N(v)\,dx\right| +|\lambda'|\cdot\left|\int_0^1 \varphi_i(x)[N(v)-N(v')]\,dx\right|. \end{aligned} \]
Here \(v\) and \(v'\) denote, respectively,
\[ \sum_{i=1}^{\infty} c_i\varphi_i(x)\quad \text{and}\quad \sum_{i=1}^{\infty} c'_i\varphi_i(x). \]
We have
\[ \begin{aligned} \left|\int_0^1 \varphi_i(x)N(v)\,dx\right|^2 &\leq \int_0^1 \varphi_i^2(x)\,dx \int_0^1 N^2(v)\,dx=\\ &=\int_0^1\left[\sum_{n=0}^{\infty}\int_{G_n}K_n\,d\tau_n\right]^2 dx \leq \left[\sum_{n=0}^{\infty}\left(\int_0^1\int_{G_n}K_n^2\,d\tau_n\,dx\right)^{1/2}\right]^2 \leq A^2 . \end{aligned} \]
Further,
\[ \begin{aligned} \left|\int_0^1 \varphi_i(x)[N(v)-N(v')]\,dx\right|^2 &\leq\\ &\leq \int_0^1 \varphi_i^2(x)\,dx \int_0^1\left[\sum_{n=0}^{\infty}\int_{G_n}\bigl(K_n(v)-K_n(v')\bigr)\,d\tau_n\right]^2 dx \leq\\ &\leq \left\{\sum_{n=0}^{\infty}\left[\int_0^1\int_{G_n}\bigl(K_n(v)-K_n(v')\bigr)^2\,d\tau_n\,dx\right]^{1/2}\right\}^2=\\ &= \left\{\sum_{n=0}^{\infty}\rho\bigl[K_n(v),K_n(v')\bigr]\right\}^2 \leq \left[\sum_{n=0}^{\infty}D_n\rho(v,v')\right]^2 \leq D^2\rho^2(v,v')\to 0, \end{aligned} \]
since
\[ [\rho(v,v')]^2=\int_0^1 (v-v')^2\,dx = \int_0^1\left[\sum_{i=1}^{\infty}(c_i-c'_i)\varphi_i(x)\right]^2 dx = \sum_{i=1}^{\infty}(c_i-c'_i)^2\to 0, \]
if \(\|c-c'\|\to 0\). Hence the continuity of the functions \(F_i(\lambda,c)\) follows.
The continuity of the functions \(\dfrac{\partial F_i}{\partial c_k}\) follows from condition (V). The proof of this fact is carried out analogously to the preceding one.
2) Let us prove the uniform convergence in \(P^*\) of the series \(\sum_{i=1}^{\infty}F_i^2(\lambda,c)\).
Denote
\[ v(x)=\sum_{i=1}^{\infty}c_i\varphi_i(x). \]
We have
\[ \sum_{i=1}^{\infty} F_i^2(\lambda, c) \leq 2 \left( \sum_{i=1}^{\infty} c_i^2 + \lambda^2 \sum_{i=1}^{\infty} \left[ \int_0^1 \varphi_i(x) N(v)\, dx \right]^2 \right) \leq \]
\[ \leq 8 \|c^*\|^2 + 2\lambda^2 \sum_{i=1}^{\infty} \left[ \int_0^1 \varphi_i(x) N(v)\, dx \right]^2 , \]
but the expression in square brackets is nothing other than the Fourier coefficient of the function \(N(v) \in L_2(0,1)\) with respect to the system \(\{\varphi_i(x)\}\). By virtue of the completeness of this system,
\[ \sum_{i=1}^{\infty} \left[ \int_0^1 \varphi_i(x) N(v)\, dx \right]^2 = \|N(v)\|^2 = \int_0^1 N^2(v)\, dx = \]
\[ = \int_0^1 \left[ \sum_{n=0}^{\infty} \int_{G_n} K_n(v)\, d\tau_n \right]^2 dx \leq \left[ \sum_{n=0}^{\infty} \left( \int_0^1 \int_{G_n} K_n^2(v)\, d\tau_n\, dx \right)^{1/2} \right]^2 \leq A^2, \]
\[ \sum_{i=1}^{\infty} F_i^2(\lambda, c) \leq 8(\|c^*\| + \lambda^{*2} A^2) = \mathrm{const}. \]
Thus, the uniform boundedness of the series
\(\sum_{i=1}^{\infty} F_i^2(\lambda, c)\) has been proved. Since, moreover, the coefficients of this series are all positive and continuous, and it is easy to see that the sum of the series is continuous, it follows that the series
\[
\sum_{i=1}^{\infty} F_i^2(\lambda, c)
\]
converges uniformly in \(P^*\).
We shall now show that in the domain \(P^*\) the series
\[
\sum_{i,j=1}^{\infty} \left[ \frac{\partial F_i(\lambda, c)}{\partial c_j} \right]^2
\quad (i \neq j).
\]
also converges uniformly.
We obtain
\[
\sum_{\substack{i,j=1\\(i\neq j)}}^{\infty}
\left[ \frac{\partial F_i}{\partial c_j} \right]^2
\leq \lambda^2 \sum_{i,j=1}^{\infty}
\left[
\int_0^1 \varphi_i(x) \sum_{n=1}^{\infty} \int_{G_n}
\sum_{k=1}^{n} \varphi_j(y_k)
\frac{\partial}{\partial v(y_k)}
K_n(x, y_1, \ldots
\right.
\]
\[ \left. \ldots, y_n, v(y_1), \ldots, v(y_n))\, d\tau_n\, dx \right]^2 \leq 4\lambda^{*2} \sum_{i=1}^{\infty} \left[ \int_0^1 \varphi_i(x)\, dx \right]^2 \times \]
\[ \times \sum_{j=1}^{\infty} \left\{ \sum_{n=1}^{\infty} \int_{G_n} \sum_{k=1}^{n} \varphi_j(y_k) M_{nk}(y_1, \ldots, y_n, v(y_1), \ldots, v(y_n))\, d\tau_n \right\}^2 \leq \]
\[ \leq 4\lambda^{*2} \sum_{j=1}^{\infty} \left[ \int_0^1 \varphi_j(y) M(y)\, dy \right]^2 = 4\lambda^{*2} \int_0^1 M(y)\, dy < +\infty, \]
since, by virtue of (VI), the function \(M(y)\in L_2(0,1)\). Thus, the uniform boundedness of the series
\[
\sum_{i,j=1}^{\infty}\left[\frac{\partial F_i}{\partial c_j}\right]^2 \quad (i\ne j),
\]
as well as of the series
\[
\sum_{i=1}^{\infty}\left[1-\frac{\partial F_i}{\partial c_i}\right]^2,
\]
has been proved, for, as can be verified,
\[
\sum_{i=1}^{\infty}\left[1-\frac{\partial F_i}{\partial c_i}\right]^2
\leq 4\lambda^* \int_0^1 M^2(y)\,dy .
\]
Taking into account that the terms of these series are positive and continuous and that their sums are continuous, we may speak of the uniform convergence of these series.
3) Further, since
\[
v^*(x)=\sum_{i=1}^{\infty}c_i^*\varphi_i(x)\in L_2(B)
\]
is a solution of equation (1) for \(\lambda=\lambda^*\), by the equivalence theorem
\[
F_i(\lambda^*,c^*)=0 \qquad (i=\overline{1,\infty}).
\]
Then, as a consequence of the fulfillment of all the above-stated conditions of Hart’s theorem, according to this theorem the infinite determinant
\[
\Delta(\lambda,c)=
\left|
\begin{array}{cccc}
1+\lambda a_{11}(v) & \lambda a_{12}(v) & \cdot & \cdot \ \cdot \\
\lambda a_{21}(v) & 1+\lambda a_{22}(v) & \cdot & \cdot \ \cdot \\
\cdot & \cdot & \cdot & \cdot \ \cdot \ \cdot
\end{array}
\right|
\]
converges. Here
\[
a_{ij}
=
-\int_0^1
\varphi_i(x)\sum_{n=1}^{\infty}\int_{G_n}
\left\{\sum_{k=1}^{n}\varphi_j(y_k)\frac{\partial}{\partial v(y_k)}
K_n[x,y_1,\ldots,y_n,
\right.
\]
\[
\left.
v(y_1),\ldots,v(y_n)]\right\}\,d\tau_n\,dx,
\qquad
v(y_m)=\sum_{i=1}^{\infty}c_i\varphi_i(y_m).
\]
Moreover, if \(\Delta(\lambda^*,v^*)\ne 0\), then, according to Hart’s theorem, there exist two positive numbers \(\alpha\) and \(\beta\) such that to each \(\lambda\) from the interval \(|\lambda-\lambda^*|\leq \alpha\) there will correspond a unique point
\[
c(\lambda)=(c_1(\lambda),c_2(\lambda),\ldots),
\]
\((\|c(\lambda)-c^*\|\leq \beta)\), such that
\[
F_i(\lambda,c(\lambda))=0 \qquad (i=1,\infty),
\]
where all \(c_i(\lambda)\) are continuous and \(c_i(\lambda^*)=c_i^*\); and this means that equation (1), for any \(\lambda\) from the interval \(|\lambda-\lambda^*|\leq \alpha\), has the solution
\[
v(x,\lambda)=\sum_{i=1}^{\infty}c_i(\lambda)\varphi_i(x)\in L_2(B),
\]
and it is not hard to show that
\[
\|v(x,\lambda)-v^*(x)\|\to 0 \quad \text{as } \lambda\to\lambda^* .
\]
Thus, the solution \((\lambda^*,v^*(x))\) of equation (1) is uniquely continued to a certain neighborhood of the point \(\lambda=\lambda^*\). We do not assert that for every \(\lambda\) of the indicated interval equation (1) has a unique solution; from our reasoning it follows only that through the point \((\lambda^*,v^*(x))\) there passes a unique branch of solutions. The possibility is not excluded that for \(\lambda=\lambda^*\) equation (1) also has another solution \(\omega^*(x)\), \((\omega^*(x)\ne v^*(x))\), and that through the point \((\lambda^*,\omega^*(x))\) there likewise passes a branch of solutions of equation (1).
Let us determine what condition \(\Delta(\lambda^*,v^*)\ne 0\) is equivalent to; \(a_{ij}(v)\) can be written in the form
\[ a_{ij}=-\int_0^1 \varphi_i(x)\int_0^1 \varphi_j(y) \left\{\sum_{n=1}^{\infty}\int_{G_{n-1}}\sum_{k=1}^n \frac{\partial}{\partial v_k}K_n[x,y_1,\ldots,y_{k-1},y,y_{k+1},\ldots \right. \]
\[ \left. \ldots,y_n,v(y_1),\ldots,v(y_{k-1}),v(y),v(y_{k+1}),\ldots,v(y_n)]\,d\tau_{n-1} \right\}\,dx\,dy . \]
Here the expression in braces, when the function \(v^*\) is substituted for \(v\), gives the kernel \(K(x,y)\) defined above. We have
\[ a_{ij}(v^*)=-\int_0^1 \varphi_i(x)\int_0^1 \varphi_j(y)K(x,y)\,dy\,dx =-\frac{1}{\lambda_i}\int_0^1 \psi_i(y)\varphi_j(y)\,dy . \]
Thus the determinant \(\Delta(\lambda,v^*)\) takes the form
\[ \Delta(\lambda,v^*)= \left| \begin{array}{ccc} 1+\dfrac{\lambda}{\lambda_1}\displaystyle\int_0^1 \varphi_1(y)\psi_1(y)\,dy & \dfrac{\lambda}{\lambda_2}\displaystyle\int_0^1 \varphi_2(y)\psi_1(y)\,dy & \cdots \\[2.2ex] \dfrac{\lambda}{\lambda_1}\displaystyle\int_0^1 \varphi_1(y)\psi_2(y)\,dy & 1+\dfrac{\lambda}{\lambda_2}\displaystyle\int_0^1 \varphi_2(y)\psi_2(y)\,dy & \cdots \\[2.2ex] \cdots & \cdots & \cdots \end{array} \right| , \]
but, as Goursat showed, a determinant of this form coincides with the Fredholm function of the kernel \(K(x,y)\).
Consequently, \(\Delta(\lambda,v^*)\) vanishes only when \(\lambda\) is an eigenvalue of the linear integral equation
\[ \varphi(x)=\lambda\int_0^1 K(x,y)\varphi(y)\,dy =\lambda\int_0^1 \left\{\sum_{n=1}^{\infty}\int_{G_{n-1}}\sum_{k=1}^{n} \frac{\partial}{\partial v_k}K_n[x,y_1,\ldots,y_{k-1}, \right. \]
\[ \left. y,y_{k+1},\ldots,y_n, v^*(y_1),\ldots,v^*(y_{k-1}),v^*(y),v^*(y_{k+1}),\ldots,v^*(y_n)]\,d\tau_{n-1} \right\}\varphi(y)\,dy . \]
Hence
Theorem 1. Under the assumptions made above, if \(\lambda^*\) is not an eigenvalue of the kernel \(K(x,y)\), defined in the indicated manner, then the solution \((\lambda^*,v^*(x))\) has a unique continuation from the point \(\lambda=\lambda^*\) to some neighborhood of this point.
3°. Consider the case where \(\lambda\) is an eigenvalue of the kernel \(K(x,y)\). In this case \(\Delta(\lambda^*,v^*)=0\), and we can say nothing about the solvability of system (5).
Let \(\lambda^*\) be an eigenvalue of the kernel \(K(x,y)\) of rank \(q\), with eigenfunctions \(\alpha^{(1)}(x),\ldots,\alpha^{(q)}(x)\) and the corresponding eigenfunctions of the adjoint kernel \(\beta^{(1)}(x),\ldots,\beta^{(q)}(x)\) (the systems \(\{\alpha^{(i)}(x)\}\) and \(\{\beta^{(i)}(x)\}\), \(i=\overline{1,q}\), may be assumed orthonormalized). In this case, instead of equation (1), consider the equation
\[ W(x)=\lambda\sum_{n=0}^{\infty}\int_{G_n} K_n[x,y_1,\ldots,y_n,W(y_1),\ldots,W(y_n)]\,d\tau_n- \]
\[ -\frac{1}{\lambda^*}\sum_{k=1}^{q}\beta^{(k)}(x)\int_{0}^{1}\alpha^{(k)}(y)\,[\lambda^* v^*(y)-\lambda W(y)]\,dy, \tag{7} \]
where
\[ W(x)=v(x)+(\lambda-\lambda^*)\sum_{j=1}^{q}\xi_j\alpha^{(j)}(x) \]
and \(\xi_j\) \((j=\overline{1,q})\) are as yet undetermined constants.
Equation (7) has the same solution \((\lambda^*, v^*(x))\) as equation (1).
In a certain neighborhood of the point \(\lambda=\lambda^*\), we shall seek the continuation of this solution of equation (7) in the form
\[ v(x)=\sum_{i=1}^{\infty} c_i\varphi_i(x), \tag{8} \]
and we require that
\[ v^*(x)=\sum_{i=1}^{\infty} c_i(\lambda^*)\varphi_i(x), \]
\(\{c_i\}\) will be determined from the infinite system of transcendental equations
\[ \begin{aligned} \Phi_i(\lambda,\xi_1,\ldots,\xi_q,c_1,c_2,\ldots)\equiv{}& c_i +(\lambda-\lambda^*)\sum_{k=1}^{q}\xi_k\alpha_i^{(k)} -\lambda\int_{0}^{1}\varphi_i(x)\sum_{n=0}^{\infty}\int_{\sigma_n} K_n[x,y_1,\ldots,y_n,\\ &\sum_{j=1}^{\infty}c_j\varphi_j(y_1)+(\lambda-\lambda^*)\sum_{k=1}^{q}\xi_k\alpha^{(k)}(y_1),\ldots, \sum_{j=1}^{\infty}c_j\varphi_j(y_n)+\\ &+(\lambda-\lambda^*)\sum_{k=1}^{q}\xi_k\alpha^{(k)}(y_k)]\,d\tau_n\,dx +\sum_{k=1}^{q}\beta_i^{(k)}\times\\ &\times\left[\int_{0}^{1}\alpha^{(k)}(y)v^*(y)\,dy -\frac{\lambda}{\lambda^*}(\lambda-\lambda^*)\xi_k -\frac{\lambda}{\lambda^*}\sum_{j=1}^{\infty}c_j\alpha_j^{(k)}\right]=0. \end{aligned} \tag{9} \]
Here \(\alpha_i^{(k)}\), \(\beta_i^{(k)}\) denote the Fourier coefficients of the functions \(\alpha^{(k)}(x)\), \(\beta^{(k)}(x)\) with respect to the system \(\{\varphi_i(x)\}\).
Analogously to how this was done above, the fulfillment of the conditions of Hart’s theorem is verified in a certain domain
\[ P'\left(|\lambda|\leq 2|\lambda^*|,\ \|c\|\leq 2\|c^*\|,\ \sum_{k=1}^{q}\xi_k^2\leq \rho\right). \]
Equation (7) was constructed by us in a special way, since for it Hart’s determinant \(\Delta^{(1)}(\lambda,v)\) at the point \((\lambda^*,v^*)\) is not equal to zero.
Indeed, at this point we have
\[ \frac{\partial\Phi_i}{\partial c_j}(i\ne j) = -\sum_{k=1}^{q}\beta_i^{(k)}\alpha_j^{(k)} -\lambda^*\int_{0}^{1}\int_{0}^{1}\varphi_i(x)\varphi_j(y)K(x,y)\,dx\,dy = \]
\[ = - \sum_{i=1}^{q} \beta_i^{(k)} a_i^{(k)} - \frac{\lambda^*}{\lambda_i} \int_0^1 \psi_i(y)\varphi_j(y)\,dy = b_{ij}. \]
Since \(b_{ij}\) can also be represented in the form
\[ b_{ij} = -\lambda^* \int_0^1 \varphi_j(y) \left( \sum_{k=1}^{q} \beta_i^{(k)} \frac{\alpha^{(k)}(y)}{\lambda^*} + \frac{\psi_i(y)}{\lambda_i} \right)dy = -\lambda^* \int_0^1 \varphi_j(y)\Omega_i(y)\,dy, \]
we have
\[ \Delta^{(1)}(\lambda^*, v^*) = \begin{vmatrix} 1-\lambda^*\int_0^1 \varphi_1(y)\Omega_1(y)\,dy & -\lambda^*\int_0^1 \varphi_2(y)\Omega_1(y)\,dy & \cdots \\[6pt] -\lambda\int_0^1 \varphi_1(y)\Omega_2(y)\,dy & 1-\lambda^*\int_0^1 \varphi_2(y)\Omega_2(y)\,dy & \cdots \\[6pt] \cdot & \cdot & \cdots \end{vmatrix} \]
coincides with the Fredholm function of the kernel
\[ K^{(1)}(x,y)=\sum_{i=1}^{\infty}\varphi_i(x)\Omega_i(y). \]
Then \(\Delta^{(1)}(\lambda^*,v^*)\) can be equal to zero only in the case when \(\lambda^*\) is an eigenvalue of the kernel \(K^{(1)}(x,y)\). But the kernel \(K^{(1)}(x,y)\) can also be written in the form
\[ K^{(1)}(x,y) = \sum_{i=1}^{\infty}\varphi_i(x)\frac{\psi_i(y)}{\lambda_i} + \sum_{i=1}^{\infty}\varphi_i(x)\sum_{k=1}^{q} \frac{\beta^{(k)}(x)\alpha^{(k)}(x)}{\lambda^*} = \]
\[ = K(x,y)+\sum_{k=1}^{q}\frac{\beta^{(k)}(x)\alpha^{(k)}(x)}{\lambda^*} \]
and, according to E. Schmidt (see [1]), this kernel is composed so that for it \(\lambda^*\) is no longer an eigenvalue, and consequently
\(\Delta^{(1)}(\lambda^*,v^*)\ne 0\).
For system (9) all conditions of Hart’s theorem are satisfied, and on the basis of this theorem, for fixed \(\xi_1,\ldots,\xi_q\) system (9) has a unique solution
\(\{c_i(\lambda,\xi_1,\ldots,\xi_q)\}\), and equation (7) has a unique solution in the form (8) for any fixed \(\xi_1,\ldots,\xi_q\).
Varying \(\xi_1,\ldots,\xi_q\), we obtain a \(q\)-parameter family of solutions for equation (7). From this continuum of solutions let us single out those which would also satisfy equation (1). For this, evidently, it is necessary to require that
\[ \sum_{k=1}^{q}\beta^{(k)}(x)\int_0^1 \alpha^{(k)}(y) \left[ \lambda^* v^*(y)-\lambda W(y,\lambda,\xi_1,\ldots,\xi_q) \right]dy =0. \]
Since the system \(\{\beta^{(k)}(x)\}\) is linearly independent, the indicated identity splits into \(q\) equalities
\[ \int_0^1 \alpha^{(k)}(y) \left[ \lambda^* v^*(y)-\lambda v(y) -(\lambda-\lambda^*)\sum_{j=1}^{q}\xi_j\alpha^{(j)}(y) \right]dy =0 \quad (k=\overline{1,q}). \]
Substituting (8) here, we obtain a transcendental system of branching equations with respect to \(\xi_1,\ldots,\xi_2\)
\[ (\lambda-\lambda^*)\xi_k+\sum_{i=1}^{\infty} c_i(\lambda,\xi_1,\ldots,\xi_2)a_i^{(k)}- \]
\[ -\frac{\lambda^*}{\lambda}\int_0^1 v^*(y)a^{(k)}(y)\,dy=0. \tag{10} \]
Hence
Theorem 2. Under the above conditions, if \(\lambda^*\) is an eigenvalue of the kernel \(K(x,y)\) of rank \(q\), equation (1) has as many branches of solutions passing through the point \((\lambda^* v^*(x))\) as the system of branching equations (10) has solutions \((\xi_1(\lambda),\ldots,\xi_2(\lambda))\).
Everywhere above it was assumed that the system of eigenfunctions \(\{\varphi_i(x)\}\) of the kernel \(K(x,y)\) (in the sense of Schmidt) is complete. It is not difficult to show, similarly to how this was done in [6], that this assumption is not essential, and the theorems 1 and 2 proved above also hold in this case. But these arguments are not presented here.
References
- Smirnov N. S. Introduction to the Theory of Nonlinear Integral Equations. ONTI, 1936.
- Schmidt E. Math. Ann., 65, 370—399, 1908.
- Vainberg M. M., Trenogin V. A. Lyapunov–Schmidt methods in the theory of nonlinear integral equations and their further development. UMN, 17, issue 2, 1962.
- Akhmedov K. T. Doctoral dissertation. Moscow, 1958.
- Hart W. L. Transcon of the Amer. Math. Soc., No. 23, 1—30, 1922.
- Vitov V. F. On one method of investigating nonlinear integral equations. Proceedings of the scientific conference of graduate students of Azerbaijan State University named after S. M. Kirov. Baku, 1965.
Received by the editors
September 23, 1965
Azerbaijan State
University