Coincidence theorems with applications to minimax inequalities, section theorem and best approximation in topological spaces (Q2763760)

Let \(X\) and \(Y\) be two topological spaces. The author introduces the notion of compactly local intersection property: \(G:X\to 2^Y\) is said to have the compactly local intersection property on \(X\) if for each compact subset \(K\) of \(X\) with \(G(x)\cap K\neq \emptyset\), there exists an open neighborhood \(N(x)\) of \(X\) such that \(\bigcap\{G(z): z\in N(x)\cap K\}\neq \emptyset\). In Lemma 2.1 several characterizations of the compactly local intersection property are obtained. For a topological space \(X\), we denote by \(C(X)\) and \(ka(X)\) the family of all nonempty compact subsets of \(X\) and the family of all compact acyclic subsets of \(X\), respectively. For two topological spaces \(X\) and \(Y\) and for a given class \(L\) of set-valued mappings we denote: \(L(X,Y)= \{T\mid T:X\to 2^Y, T\in L\}\); \(L_c= \{T=T_m T_{m-1}\dots T_1\mid T_i \in L\}\). Then: (1) \(T\) is an acyclic mapping, written \(T\in V(X,Y)\), if \(T:X \to ka(Y)\) is upper semicontinuous. (2) \(T\in V^+(X,Y)\) if for any \(\sigma\)-compact subset \(K\) of \(X\) there exists \(T^*\in V(K,Y)\) such that \(T^*(x)\subset T(x)\) for all \(x\in K\). (3) \(T\in V^+_c(X,Y)\) if for any \(\sigma\)-compact subset \(K\) of \(X\) there exists \(T^*\in V_c(K,Y)\) such that \(T^*(x)\subset T(x)\) for all \(x\in K\). The following theorems are proved:NEWLINENEWLINENEWLINETheorem 3.1. Let \(X\) an \(Y\) be two topological spaces and \(K\) a nonempty compact subset of \(X\). Let \(F\in V^+_c (Y,X)\) and \(G:X\to 2^Y\) be such that (i) for each \(x\in K\cap \overline {F(Y)}\), \(G(x)\neq \emptyset\); (ii) \(G\) has the compactly local intersection property; (iii) for each \(N\in F(Y)\), there is a compact contractible subset \(L_N\) of \(Y\) containing \(N\) such that for each compactly open set \(U\) of \(X\), the set \(\bigcap\{G(x) \cap L_N\mid x\in U\}\) is empty or contractible and \(F(L_N)-K \subset\bigcup \{\text{cint} G^{-1}(y) \cap F(L_N): y\in L_N\}\) where \(\text{cint}(A)= \bigcup\{B\subset X\mid B\subset A\) and \(B\) is compactly open in \(X\}\). Then there exists \((x_0,y_0)\in K\times Y\) such that \(x_0\in F(y_0)\) and \(y_0\in G(x_0)\). NEWLINENEWLINENEWLINETheorem 3.2. Let \(X\) be a topological space and \(Y\) a contractible space. Let \(F\in V^+_c(Y,X)\) be a compact mapping and \(G:X\to 2^Y\) be such that (i) for each \(x\in\overline {F(Y)}\), \(G(x)\neq \emptyset\), and \(G\) has the compactly local intersection property, (ii) for each compactly open subset \(U\) of \(X\), \(\bigcap \{G(x)\mid x\in U\}\) is empty or contractible. Then there exists \((x_0,y_0)\in X\times Y\) such that \(x_0\in F(y_0)\) and \(y_0\in G(x_0)\).NEWLINENEWLINENEWLINEThese two theorems generalize some results of \textit{L. Górniewicz} and \textit{A. Granas} [J. Math. Pures Appl., IX. Sér. 60, 361-373 (1981; Zbl 0482.55002)], of \textit{F. E. Browder} [Contemp. Math. 26, 67-80 (1984; Zbl 0542.47046)], of \textit{H. Komiya} [Proc. Am. Math. Soc. 96, No. 4, 599-602 (1986; Zbl 0657.47055)], of \textit{C. D. Horvath} [Lect. Notes Pure Appl. Math. 107, 99-106 (1987; Zbl 0619.55002), of \textit{X.-P. Ding} and \textit{K.-K. Tan} [J. Aust. Math. Soc., Ser. A 49, No. 1, 111-128 (1990; Zbl 0709.47053)], of \textit{S. Park}, \textit{S. P. Singh}, and \textit{B. Watson} [Proc. Am. Math. Soc. 121, No. 4, 1151-1158 (1994; Zbl 0806.47053)], of \textit{E. Tarafdar} and \textit{X.-Z. Yuan} [ibid. 122, No. 3, 957-959 (1994; Zbl 0818.47056)], of \textit{X.-P. Ding} and \textit{E. Tarafdar} [Bull. Aust. Math. Soc. 50, No. 1, 73-80 (1994; Zbl 0814.54028)], of \textit{X.-Z. Yuan} [Nonlinear World 2, No. 2, 131-169 (1995; Zbl 0923.47028)], and of \textit{X.-P. Ding} [Appl. Math. Lett. 10, No. 3, 53-56 (1997; Zbl 0879.54055); Coincidence theorems involving composites of acyclic mappings in contractible spaces, ibid. 11, No. 2, 85-89 (1998)]. As applications, some fixed-point theorems, minimax inequalities, a section theorem, and a best-approximation theorem are obtained.