Abstract
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MATHEMATICS
VAN SHEN-VAN
COMPLETE CONTINUITY AND STRONG CONTINUITY OF URYSOHN INTEGRAL OPERATORS
(Presented by Academician A. N. Kolmogorov on 4 III 1963)
It is well known that Urysohn integral equations and Urysohn integral operators play an important role in nonlinear analysis. In studying their properties it is often necessary to consider such function spaces in which these operators act and are continuous or completely continuous. It is especially often required that they be completely continuous. Therefore the question of conditions for complete continuity of a Urysohn operator is important. In papers \((^{1-3})\) this problem was considered under various conditions in the spaces \(C\), \(L^p\), and, in particular, in the Orlicz spaces \(L_M^*\). L. A. Ladyzhenskii in paper \((^2)\) gave rather broad sufficient conditions for complete continuity of a Urysohn operator in the space \(C\). In the present article the author continues the study of this question.
- Let \(G\) be a bounded closed subset of a finite-dimensional Euclidean space, and let \(C\) be the space of all continuous functions on \(G\). Let \(C = C + C + \cdots + C\), i.e. \(C\) is the space of continuous vector-functions, whose norm is defined by the formula
\[ \|\varphi\|=\max_i \|\varphi_i\|, \]
where \(\varphi(t) \in (\varphi_1(t), \ldots, \varphi_m(t)) \in C\).
It is assumed that the functions \(K_j(s,t,u_1,\ldots,u_m)\) \((j=1,2,\ldots,m)\) are defined for \(s,t \in G\), \(-\infty<u_i<\infty\), and satisfy the conditions:
\(\mathrm{I}_1.\) For each \(s \in G\) the functions \(K_j(s,t,u_1,\ldots,u_m)\) are continuous in \(u_1,\ldots,u_m\) for almost all \(t \in G\) and are measurable in \(t\) for each \(s\) and each \((u_1,\ldots,u_m)\).
The operator
\[ \mathcal{K}\varphi(s)=(\mathcal{K}_1\varphi(s),\ldots,\mathcal{K}_m\varphi(s)), \tag{1} \]
where
\[ \mathcal{K}_j\varphi(s)=\int_G K_j[s,t,\varphi_1(t),\ldots,\varphi_m(t)]\,dt \quad (j=1,\ldots,m), \]
will be called a Urysohn operator.
It is easy to see that, in order to prove complete continuity of the operator \(\mathcal{K}\) in \(C\), it suffices to consider only one \(\mathcal{K}_j\), i.e. an operator acting from \(C\) into \(C\). Below only an operator of the form
\[ \mathcal{K}\varphi(s)=\int_G K[s,t,\varphi_1(t),\ldots,\varphi_m(t)]\,dt \]
will be considered.
- Suppose that
\[ \mathrm{I}_2.\quad K(s,t,-u_1,u_2,\ldots,u_m)=-K(s,t,u_1,u_2,\ldots,u_m) \]
for
\[ -\infty<u_1,\ldots,u_m<\infty . \]
Lemma 1. Suppose that conditions \(I_1\) and \(I_2\) are satisfied. If for every \(\varphi \in C\), \(\|\varphi\| \le \alpha\), where \(\alpha\) is some positive number,
\[ \mathcal K \varphi(s)=\int_G K\{s,t,\varphi_1(t),\ldots,\varphi_m(t)\}\,dt<\infty, \qquad s\in G, \]
then for every \(s\in G\) the inequality
\[ \int_G \sup_{\substack{|u_i|\le \alpha\\ (i=1,\ldots,m)}} |K[s,t,u_1,\ldots,u_m]|\,dt<\infty \]
holds.
With the aid of this lemma one proves
Theorem 1. Suppose that conditions \(I_1, I_2\) are satisfied and the operator \(\mathcal K\) acts from \(C\) into \(C\). If \(\mathcal K\) is bounded, then it is weakly continuous and the function \(K(t,s,u_1,\ldots,u_m)\) satisfies the condition:
\(II_1^*\). For every \(\alpha>0\) there exists a \(\beta>0\) such that
\[ \int_G \sup_{\substack{|u_i|\le \alpha\\ j=1,\ldots,m}} |K(s,t,u_1,\ldots,u_m)|\,dt \le \beta, \qquad s\in G. \]
If \(I_1, I_2\) are satisfied, then from the compactness of the operator \(\mathcal K\), acting from \(C\) into \(C\), there follows the complete continuity and the strengthened continuity of \(\mathcal K\).
In the following lemma we consider properties of the function \(K(s,t,u_1,\ldots,u_m)\) when the operator \(\mathcal K\) is compact.
Lemma 2. If conditions \(I_1\) and \(I_2\) are satisfied, the operator \(\mathcal K\) acts from \(C\) into \(C\) and is compact, then it satisfies condition \(II_1\) and the following condition:
\(II_2^{**}\). For every \(\varepsilon>0\) and every \(s\in G\) there exists a \(\delta>0\) such that
\[ \int_G \sup |K(s+h,t,u_1,\ldots,u_m)-K(s,t,u_1,\ldots,u_m)|\,dt<\varepsilon \qquad \text{for } \|h\|<\delta. \]
From Theorem 1 and Lemma 2 there immediately follows
Theorem 2. If \(I_1, I_2\) are satisfied, then the necessary and sufficient condition for compactness of the operator \(\mathcal K\) is that both conditions \(II_1, II_2\) be satisfied.
Remark. It is easy to see that condition \(II_2\) may be replaced by a formally stronger one; that is, if \(\mathcal K\) is compact, then \(K(s,t,u_1,\ldots,u_m)\) satisfies \(II_1\) and
\[ II_2'.\quad \lim_{\|h\|\to 0}\sup_{s,s+h\in G} \int_G \sup |K(s+h,t,u_1,\ldots,u_m)-K(s,t,u_1,\ldots,u_m)|\,dt=0. \]
- Suppose that \(K(s,t,u_1,\ldots,u_m)\) satisfies \(I_1\) and
\[ I_3.\quad K(s,t,0,u_2,\ldots,u_m)\equiv 0. \]
Putting
\[ K^{(1)}(s,t,u_1,\ldots,u_m)= \begin{cases} K(s,t,u_1,\ldots,u_m), & \text{if } u_1\ge 0,\\ -K(s,t,-u_1,\ldots,u_m), & \text{if } u_1\le 0, \end{cases} \]
we easily verify that the function \(K^{(1)}(s,t,u_1,\ldots,u_m)\) also satisfies condition \(I_2\).
\[ \text{* Condition } II_1 \text{ was introduced by L. A. Ladyzhenskii in } (^{2}) \text{ for } m=1. \]
\[ \text{** Condition } II_2 \text{ was also given by L. A. Ladyzhenskii in } (^{2}) \text{ for } m=1. \]
Let \(\mathscr K^{(1)}\) be the operator defined by the function \(K^{(1)}(s,t,u_1,\ldots,u_m)\). For each \(\varphi(t)\in C\) put
\[ \varphi_1^{(+)}(t)=\max\{\varphi_1(t),0\},\qquad \varphi_1^{(-)}(t)=\max\{-\varphi_1(t),0\}, \]
\[ \varphi^{(+)}(t)=\bigl(\varphi_1^{(+)}(t),\varphi_2(t),\ldots,\varphi_m(t)\bigr),\qquad \varphi^{(-)}(t)=\bigl(\varphi_1^{(-)}(t),\varphi_2(t),\ldots,\varphi_m(t)\bigr). \]
Then we have
\[ \begin{aligned} \mathscr K^{(1)}\varphi(s) &=\int_G K^{(1)}[s,t,\varphi_1(t),\ldots,\varphi_m(t)]\,dt \\ &=\int_G K[s,t,\varphi_1^{(+)}(t),\varphi_2(t),\ldots,\varphi_m(t)]\,dt \\ &\quad-\int_G K[s,t,\varphi_1^{(-)}(t),\varphi_2(t),\ldots,\varphi_m(t)]\,dt \\ &=\mathscr K\varphi^{(+)}(s)-\mathscr K\varphi^{(-)}(s). \end{aligned} \]
Therefore, from the boundedness and compactness of \(\mathscr K\) there follow, respectively, the boundedness and compactness of \(\mathscr K^{(1)}\); consequently, if the operator \(\mathscr K\) is bounded, then the function \(K^{(1)}(s,t,u_1,\ldots,u_m)\) satisfies \(\Pi_1\), and therefore the function \(K(s,t,u_1,\ldots,u_m)\) for \(0\le u_1\le \alpha,\ |u_i|\le \alpha\ (i=2,\ldots,m)\) also satisfies condition \(\Pi_1\); if \(\mathscr K\) is compact, then \(K^{(1)}(s,t,u_1,\ldots,u_m)\) satisfies \(\Pi_1,\Pi_2\), and therefore the function \(K(s,t,u_1,\ldots,u_m)\) for \(0\le u_1\le \alpha,\ |u_i|\le \alpha\ (i=2,\ldots,m)\) also satisfies conditions \(\Pi_1,\Pi_2\). In a similar way one can prove that from the boundedness of the operator \(\mathscr K\) there follows condition \(\Pi_1\) for \(-\alpha\le u_1\le 0,\ |u_i|\le \alpha\ (i=2,\ldots,m)\), and from the compactness of \(\mathscr K\) there follow conditions \(\Pi_1,\Pi_2\) for \(-\alpha\le u_1\le 0,\ |u_i|\le \alpha\ (i=2,\ldots,m)\).
Now suppose that the function \(K(s,t,u_1,\ldots,u_m)\) is arbitrary, but satisfies \(I_1\). Putting
\[ K_1(s,t;u_1,\ldots,u_m) =K(s,t;u_1,\ldots,u_m)-K(s,t;0,u_2,\ldots,u_m), \]
\[ K_2(s,t;u_2,\ldots,u_m) =K(s,t;0,u_2,\ldots,u_m)-K(s,t;0,0,u_3,\ldots,u_m), \]
\[ \ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots \]
\[ K_m(s,t;\ldots;u_m)=K(s,t;0,\ldots,0,u_m)-K(s,t;0,\ldots,0), \]
we obtain that
\[ K(s,t,u_1,\ldots,u_m) =\sum_{j=1}^{m} K_j(s,t,u_j,\ldots,u_m)+K(s,t;0,\ldots,0). \]
It is obvious that each function \(K_j(s,t;u_j,\ldots,u_m)\) satisfies not only \(I_1\), but also \(I_3\); therefore the following holds.
Theorem 3. If the operator \(\mathscr K\) acts from \(C\) into \(C\) and is bounded, then it is weakly continuous and satisfies condition \(\Pi_1\).
Theorem 4. If the operator \(\mathscr K\) acts from \(C\) into \(C\), then a necessary and sufficient condition for the compactness of \(\mathscr K\) is the satisfaction of both conditions \(\Pi_1,\Pi_2\). From the compactness of the operator \(\mathscr K\) there follows its complete and strengthened continuity.
4. Finally, let us consider the question of complete continuity of the Hammerstein integral operator. Suppose that \(K(s,t)\) is defined for \(s,t\in G\), and \(f(t,u_1,\ldots,u_m)\) is defined for \(t\in G,\ -\infty<u_i<\infty\ (i=1,\ldots,m)\) and satisfies the Carathéodory conditions (see \((3)\)). Put
\[ a_\alpha(t)=\sup_{\substack{|u_i|\le \alpha\\ i=1,\ldots,m}} f(t,u_1,\ldots,u_m). \]
Theorem 5. If for every \(\alpha>0\) there exist positive numbers \(a_1<a_2\) such that
\[ a_1 \leqslant a_\alpha(t) \leqslant a_2, \tag{2} \]
then a necessary and sufficient condition for the complete continuity of the operator
\[ \mathfrak{K}f\varphi(s)=\int_G K(s,t)\, f\bigl[t,\varphi_1(t),\ldots,\varphi_m(t)\bigr]\,dt \]
is the fulfillment of the following two conditions:
\[ \overline{\Pi}_1.\quad \text{For every } s\in G \]
\[ \int_G |K(s,t)|\,dt<\infty. \]
\[ \overline{\Pi}_2.\quad \text{For every } \varepsilon>0 \text{ there exists } \delta>0 \text{ such that} \]
\[ \int_G |K(s+h,t)-K(s,t)|\,dt<\varepsilon \]
for every \(s\in G\), whenever \(\|h\|<\delta\).
If \(f(t,u)=u\), then the operator \(Kf\) becomes linear and \(a_\alpha(t)=\alpha\); thus condition (2) is naturally fulfilled. Consequently, a necessary and sufficient condition for the complete continuity of the operator
\[ \mathfrak{K}\varphi(s)=\int_G K(s,t)\varphi(t)\,dt \]
is the fulfillment of the conditions \(\overline{\Pi}_1,\overline{\Pi}_2\).
Mathematics Faculty of Nanking University
Nanking, China
Received
21 VI 1962
CITED LITERATURE
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- L. A. Ladyzhenskii, DAN, 96, No. 6, 1105 (1954).
- M. M. Vainberg, Variational methods for the study of nonlinear operators, 1956.
- I. V. Shragin, Nauchn. dokl. vyssh. shkoly, ser. fiz.-matem. nauk, No. 2 (1958).
- I. V. Shragin, ibid., No. 3 (1958).
- L. A. Lyusternik, UMN, vol. 1, 77 (1936).