E. S. TSITLANADZE

INVESTIGATION OF A FUNCTIONAL ANALOGUE OF ONE NONLINEAR INTEGRAL EQUATION OF LICHTENSTEIN

(Presented by Academician A. N. Kolmogorov, 10 VII 1957)

In Lichtenstein’s monograph ((^{1})) an integral operator (L) was considered that is a broad generalization of the Fredholm operator. It is expressed by an infinite sum of multiple integrals of increasing multiplicity and is generated by variation of an analogously expressed functional (F). Lichtenstein proved the existence of one nontrivial eigen-element and eigenvalue for the operator (L) in Hilbert space.

Generalizing the result of L. A. Lyusternik ((^{2})) to the case of the unit ball of a regular Banach space, in our papers ((^{3,4})), in particular, we showed that the functional (F), under general conditions, is weakly continuous and satisfies the conditions for the existence of a critical element. Thereby, under more general assumptions, the existence of an eigen-element and eigenvalue of the Lichtenstein operator was proved.

Moreover, if the functional (F) generating the integral operator (L) is even (i.e., (F) is represented by integrals of even multiplicity) and (F > 0), then (L) has an infinite set of positive eigenvalues tending to zero.

In the present note we investigate a class of infinite systems of nonlinear functional equations representing an analogue of the Lichtenstein integral equation in the real space (l_2).

Let (S_1) be the unit closed ball of the space (l_2) with elements

[
x=(x_{\alpha_i}), \qquad \sum_{\alpha_i=0}^{\infty} x_{\alpha_i}^{2}<\infty .
]

Consider a functional (F(x)), defined for elements (x \subset S_1), of the form

[
F(x)=\frac{1}{n+1}\sum_{\alpha_0,\ldots,\alpha_n=1}^{\infty}
a_{\alpha_0\ldots\alpha_n}\,
\varphi(x_{\alpha_0})\cdots \varphi(x_{\alpha_n}),
\tag{1}
]

where (a_{\alpha_0\ldots\alpha_n}) are given coefficients symmetric in the aggregate of indices (\alpha_0,\ldots,\alpha_n); (\varphi) is a twice continuously differentiable function on the segment ([-1,+1]); (n) is a natural number.

We shall assume that

[
\sum_{\alpha_0,\ldots,\alpha_n=1}^{\infty}
a_{\alpha_0\ldots\alpha_n}^{2}<\infty,\qquad
\sum_{\alpha_k=1}^{\infty}\varphi^{2}(x_{\alpha_k})<\infty,\qquad
\sup_{x\subset S_1}\sum_{\alpha_k=1}^{\infty}\varphi^{2}(x_{\alpha_k})=M^{2}.
\tag{2}
]

Then the ((n+1))-fold series (1) converges uniformly in (S_1).

Indeed, successively using Hölder’s inequality, from (1) we obtain

[
F(x)\leq \frac{1}{n+1}\prod_{j=0}^{n}
\left{
\sum_{\alpha_j=1}^{\infty}\varphi^{2}(x_{\alpha_j})
\right}^{1/2}
\left{
\sum_{\alpha_0,\ldots,\alpha_n=1}^{\infty}
a_{\alpha_0\ldots\alpha_n}^{2}
\right}^{1/2}.
]

Hence, by virtue of (2), we have

[
F(x)\leq \frac{M^{n+1}}{n+1}
\left{\sum_{\alpha_0,\ldots,\alpha_n=1}^{\infty}
a_{\alpha_0,\ldots,\alpha_n}^{2}\right}^{1/2}<\infty.
]

Lemma 1. The functional (F(x)) is weakly continuous in (S_1).

Let ({x^{(k)}}\in S_1) be an arbitrary sequence weakly converging to the weak limit (x^), (|x^|\leq 1). Denote (x^{(k)}=(x_\alpha^{(k)})) and (x^=(x_\alpha^)); then (x_\alpha^{(k)}\to x_\alpha^*) as (k\to\infty). Starting from the transformation

[
\begin{aligned}
F(x^{(k)})-F(x^) &=
\frac{1}{n+1}
\sum_{\alpha_0,\ldots,\alpha_n=1}^{\infty}
\left[
\varphi(x_{\alpha_0}^{(k)})\ldots \varphi(x_{\alpha_n}^{(k)})
-\varphi(x_{\alpha_0}^{})\cdots \varphi(x_{\alpha_n}^{})
\right] \
&=
\frac{1}{n+1}
\sum_{\alpha_0,\ldots,\alpha_n=1}^{\infty}
a_{\alpha_0\ldots\alpha_n}
\Big{
[\varphi(x_{\alpha_0}^{(k)})-\varphi(x_{\alpha_0}^{})]
\varphi(x_{\alpha_1}^{(k)})\cdots \varphi(x_{\alpha_n}^{(k)}) \
&\qquad
+[\varphi(x_{\alpha_1}^{(k)})-\varphi(x_{\alpha_1}^{})]
\varphi(x_{\alpha_0}^{})\varphi(x_{\alpha_2}^{(k)})\cdots \varphi(x_{\alpha_n}^{(k)})
+\cdots \
&\qquad
+[\varphi(x_{\alpha_n}^{(k)})-\varphi(x_{\alpha_n}^{})]
\varphi(x_{\alpha_0}^{})\cdots \varphi(x_{\alpha_{n-1}}^{*})
\Big},
\end{aligned}
]

we obtain

[
\begin{aligned}
|F(x^{(k)})-F(x^)|
&\leq
\frac{1}{n+1}
\sum_{\alpha_0,\ldots,\alpha_n=1}^{\infty}
|a_{\alpha_0\ldots\alpha_n}|
\Big[
|\varphi(x_{\alpha_0}^{(k)})-\varphi(x_{\alpha_0}^{})|
|\varphi(x_{\alpha_1}^{(k)})\cdots \varphi(x_{\alpha_n}^{(k)})| \
&\qquad
+|\varphi(x_{\alpha_1}^{(k)})-\varphi(x_{\alpha_1}^{})|
|\varphi(x_{\alpha_0}^{})\varphi(x_{\alpha_2}^{(k)})\cdots \varphi(x_{\alpha_n}^{(k)})|
+\cdots \
&\qquad
+|\varphi(x_{\alpha_n}^{(k)})-\varphi(x_{\alpha_n}^{})|
|\varphi(x_{\alpha_0}^{})\cdots \varphi(x_{\alpha_{n-1}}^{*})|
\Big].
\end{aligned}
\tag{3}
]

By virtue of the continuity of the function (\varphi) and the equality
(\lim_{k\to\infty}x_{\alpha_j}^{(k)}=x_{\alpha_j}^{}), for arbitrary (\varepsilon>0) and sufficiently large (k) we shall have
(|\varphi(x_{\alpha_j}^{(k)})-\varphi(x_{\alpha_j}^{})|<\varepsilon) for all
(j=0,1,\ldots,n). Therefore, from inequality (3), after some transformations we shall have

[
|F(x^{(k)})-F(x^*)|
\leq
\varepsilon M^{n+1}
\left(
\sum_{\alpha_0,\ldots,\alpha_n=1}^{\infty}
a_{\alpha_0\ldots\alpha_n}^{2}
\right)^{1/2}.
]

Since (\varepsilon) is an arbitrary number, it follows from this and from (2) that
(\lim_{k\to\infty}F(x^{(k)})=F(x^*)). Lemma 1 is proved.

Lemma 2. The strong differential of the functional (F(x)) generates the operator (L_F x) with components

[
L_Fx=
\left(
\varphi'(x_{\alpha_0})
\sum_{\alpha_1,\ldots,\alpha_n=1}^{\infty}
a_{\alpha_0\ldots\alpha_n}\,
\varphi(x_{\alpha_1})\cdots \varphi(x_{\alpha_n})
\right),
\qquad
\alpha_0=1,2,\ldots,
\tag{4}
]

mapping elements (x\in S_1) to elements (l_2).

Indeed, let (x=(x_\alpha)), (h=(h_\alpha)\in S_1), and let (t) be a numerical operator, (x+th\in S_1); then

[
\left.\frac{dF(x+th)}{dt}\right|{t=0}
=
\frac{1}{n+1}
\sum}^{\infty
a_{\alpha_0\ldots\alpha_n}
\big[
h_{\alpha_0}\varphi'(x_{\alpha_0})\varphi(x_{\alpha_1})\cdots \varphi(x_{\alpha_n})
+
]

[
+
h_{\alpha_1}\varphi'(x_{\alpha_1})\varphi(x_{\alpha_0})\varphi(x_{\alpha_2})\cdots \varphi(x_{\alpha_n})
+\cdots+
h_{\alpha_n}\varphi'(x_{\alpha_n})\varphi(x_{\alpha_n})\cdots \varphi(x_{\alpha_{n-1}})
\big].
]

Taking into account the symmetry of the coefficients (a_{\alpha_0\ldots\alpha_n}), from this we obtain

[
\left.\frac{dF(x+th)}{dt}\right|{t=0}
=
\sum)}^{\infty} h_{\alpha_0}\varphi'(x_{\alpha_0
\sum_{\alpha_1,\ldots,\alpha_n=1}^{\infty}
a_{\alpha_0\ldots\alpha_n}\varphi(x_{\alpha_1})\cdots\varphi(x_{\alpha_n}).
\tag{5}
]

The right-hand side of equality (5) is a linear functional with respect to (h) and is represented in the form of the scalar product

[
\left.\frac{dF(x+th)}{dt}\right|_{t=0}
=
(h,L_Fx),
\tag{6}
]

where (L_Fx) is the operator with components (4). Let us note that

[
\varphi'(x_{\alpha_0})
\sum_{\alpha_1,\ldots,\alpha_n=1}^{\infty}
a_{\alpha_0\ldots\alpha_n}\varphi(x_{\alpha_1})\cdots\varphi(x_{\alpha_n})
\le
]

[
\le
\max_{x\subset S_1}\varphi'(x_{\alpha_0})M^n
\left(
\sum_{\alpha_1,\ldots,\alpha_n=1}^{\infty}
a_{\alpha_0\ldots\alpha_n}^{2}
\right)^{1/2}
<\infty.
]

This shows that the components (L_F^x) converge uniformly in the ball (x\in S_1). Moreover, we have

[
\sum_{\alpha_0=1}^{\infty}
\left[
\varphi'(x_{\alpha_0})
\sum_{\alpha_1,\ldots,\alpha_n=1}^{\infty}
a_{\alpha_0\ldots\alpha_n}\varphi(x_{\alpha_1})\cdots\varphi(x_{\alpha_n})
\right]^2
\le
]

[
\le
\max_{x\subset S_1}\varphi'^2(x_{\alpha_0})M^{2n}
\sum_{\alpha_0,\ldots,\alpha_n=1}^{\infty}
a_{\alpha_0\ldots\alpha_n}^{2}
<\infty,
]

which completes the proof of Lemma 2.

Lemma 3. The operator (L_Fx) satisfies a Lipschitz condition in (S_1).

Let (x^{(1)}=(x_{\alpha_j}^{(1)})), (x^{(2)}=(x_{\alpha_j}^{(2)})), (\alpha_j=1,2,\ldots), be an arbitrary pair of elements of (S_1). We represent the norm of the difference (L_Fx^{(1)}-L_Fx^{(2)}) in the form

[
|L_Fx^{(1)}-L_Fx^{(2)}|
=
\left{
\sum_{\alpha_0=1}^{\infty}
\left[
\varphi'(x_{\alpha_0}^{(1)})
\sum_{\alpha_1,\ldots,\alpha_n=1}^{\infty}
a_{\alpha_0\ldots\alpha_n}
\bigl(
\varphi(x_{\alpha_1}^{(1)})\cdots\varphi(x_{\alpha_n}^{(1)})
-
\right.\right.\right.
]

[
\left.\left.\left.
-
\varphi(x_{\alpha_1}^{(2)})\cdots\varphi(x_{\alpha_n}^{(2)})
\bigr)
+
\bigl(\varphi'(x_{\alpha_0}^{(1)})-\varphi'(x_{\alpha_0}^{(2)})\bigr)
\sum_{\alpha_1,\ldots,\alpha_n=1}^{\infty}
a_{\alpha_0\ldots\alpha_n}
\varphi(x_{\alpha_1}^{(2)})\cdots\varphi(x_{\alpha_n}^{(2)})
\right]^2
\right}^{1/2}.
]

Using the equalities
(\varphi(x_{\alpha_j}^{(1)})-\varphi(x_{\alpha_j}^{(2)})
=(x_{\alpha_j}^{(1)}-x_{\alpha_j}^{(2)})\varphi'(\xi_1)), where
(j=1,2,\ldots,n), (-1\le \xi_1\le +1),
(\varphi'(x_{\alpha_0}^{(1)})-\varphi'(x_{\alpha_0}^{(2)})
=(x_{\alpha_0}^{(1)}-x_{\alpha_0}^{(2)})\varphi''(\xi_2)), where
(-1\le \xi_2\le +1), and the inequality
(|x_{\alpha_0}^{(1)}-x_{\alpha_0}^{(2)}|\le |x^{(1)}-x^{(2)}|), after some transformations we obtain

[
|L_Fx^{(1)}-L_Fx^{(2)}|
\le
2M_1
\left(
\sum_{\alpha_0,\ldots,\alpha_n=1}^{\infty}
a_{\alpha_0\ldots\alpha_n}^{2}
\right)^{1/2}
|x^{(1)}-x^{(2)}|,
\tag{7}
]

where

[
M_1=\sup(nNM^{n-1}K;\ K_1M^n),\qquad
K=\max \varphi'(\xi),\qquad
K_1=\max \varphi''(\xi),\qquad
0\le \xi\le 1.
]

Lemma 4. (L_Fx) is weakly continuous in (S_1).

Indeed, let (x^{(k)} \xrightarrow{\mathrm{weak}} x), where
(x^{(k)}=(x_{\alpha_j}^{(k)})), (x=(x_{\alpha_j})), (\alpha_j=1,2,\ldots), (j=0,\ldots,n); (x^{(k)},x\in S_1), (k=1,2,\ldots); then (x_{\alpha_j}^{(k)}\to x_{\alpha_j}) as (k\to\infty) for all (j=0,\ldots,n). We shall have

[
\begin{aligned}
|L_F x^{(k)}-L_F x|={}&
\Biggl{\sum_{\alpha_0=1}^{\infty}\Bigl[
\bigl(\varphi'(x_{\alpha_0}^{(k)})-\varphi'(x_{\alpha_0})\bigr)
\sum_{\alpha_1,\ldots,\alpha_n=1}^{\infty}
a_{\alpha_0\ldots\alpha_n}\varphi(x_{\alpha_1}^{(k)})\cdots\varphi(x_{\alpha_n}^{(k)}) \
&\quad+\varphi'(x_{\alpha_0})
\sum_{\alpha_1,\ldots,\alpha_n=1}^{\infty}
a_{\alpha_0\ldots\alpha_n}
\bigl(\varphi(x_{\alpha_1}^{(k)})-\varphi(x_{\alpha_1})\bigr)
\varphi(x_{\alpha_2}^{(k)})\cdots\varphi(x_{\alpha_n}^{(k)}) \
&\quad+\bigl(\varphi(x_{\alpha_2}^{(k)})-\varphi(x_{\alpha_2})\bigr)
\varphi(x_{\alpha_1})\varphi(x_{\alpha_3}^{(k)})\cdots\varphi(x_{\alpha_n}^{(k)})+\cdots \
&\quad+\bigl(\varphi(x_{\alpha_n}^{(k)})-\varphi(x_{\alpha_n})\bigr)
\varphi(x_{\alpha_1})\cdots\varphi(x_{\alpha_{n-1}})\Bigr]^2
\Biggr}^{1/2}.
\end{aligned}
\tag{8}
]

By virtue of the continuity of (\varphi) and (\varphi'), for an arbitrary (\varepsilon>0) there exists a sufficiently large natural number (N) such that, for (k>N), the inequalities
[
|\varphi(x_{\alpha_j}^{(k)})-\varphi(x_{\alpha_j})|<\varepsilon,\quad
|\varphi'(x_{\alpha_j}^{(k)})-\varphi'(x_{\alpha_j})|<\varepsilon,\quad
j=0,1,\ldots,n,
]
will hold; and from (8)
[
|L_F x^{(k)}-L_F x|\leq
\varepsilon c\left(\sum_{\alpha_0,\ldots,\alpha_n=1}^{\infty}
a_{\alpha_0\ldots\alpha_n}^{2}\right)^{1/2},
\qquad
c=2\max(M^n,KnM^{\,n-1}).
]

Since (\varepsilon) is arbitrary, the validity of the lemma follows from this. From the weak continuity of the operator (L_Fx) and the regularity of the space (l_2) follows the complete continuity of (L_Fx).

Now suppose that (a_{\alpha_0\ldots\alpha_n}\geq 0), (\varphi) is an even function, (\varphi>0) and (\varphi(0)=0); then the functional (F(x)) is positive and even in (S_1), and, consequently, (L_F) is odd. By virtue of Theorem 9 of paper ((^3)), there exists an infinite system of geometrically distinct normalized eigen-elements
(x^{(m)}=(x_{\alpha_j}^{(m)})) satisfying the functional equation
[
L_F x^{(m)}=\lambda_m x^{(m)},\qquad
|x^{(m)}|=1,\qquad
\lambda_m=(x^{(m)},L_Fx^{(m)}),\qquad
m=1,2,\ldots .
\tag{9}
]

Equation (9) is equivalent to the following infinite system of equations with an infinite number of unknowns:
[
\varphi'(x_{\alpha_0}^{(m)})
\sum_{\alpha_1,\ldots,\alpha_n=1}^{\infty}
a_{\alpha_0\ldots\alpha_n}\varphi(x_{\alpha_1}^{(m)})\cdots
\varphi(x_{\alpha_n}^{(m)})
=\lambda_m x_{\alpha_0}^{(m)},\qquad
\alpha_0=1,2,\ldots .
\tag{9'}
]

Thus the following theorem has been proved.

Theorem. If a strongly differentiable functional (F(x)) in the unit ball (S_1\subset l_2) is defined by equality (1), where (a_{\alpha_0\ldots\alpha_n}\geq 0) are symmetric real coefficients, (\varphi) is a twice continuously differentiable even function on the segment ([-1,+1]), (\varphi>0), (\varphi(0)=0), and if (L_Fx=\operatorname{grad}F(x)) with components (4), then there exists an infinite system of geometrically distinct normalized eigen-elements (x^{(m)}=(x_{\alpha_j}^{(m)})), (m=1,2,\ldots), satisfying (9) (or ((9'))), where (\lambda_m) are the eigenvalues corresponding to the eigen-elements (x^{(m)}).

Tbilisi State
University

Received
3 I 1957

CITED LITERATURE

(^{1}) L. Lichtenstein, Vorlesungen über einige Klassen nichtlinearer Integralgleichungen und Integro-Differenzialgleichungen nebst Anwendungen, Berlin, 1931, pp. 141–162.
(^{2}) L. A. Lyusternik, Izv. AN SSSR, Ser. Matem., No. 3, 258 (1939).
(^{3}) E. S. Tsitlanadze, Tr. Moskovsk. Matem. Obshch., No. 2, 235 (1953).
(^{4}) E. S. Tsitlanadze, Soobshch. AN GruzSSR, 8, No. 6, 353 (1947).