Continuous selection theorems for Fan-Browder mappings in topological spaces and their applications
If \(X\) is a nonempty set and \(Y\) is a topological space, a set-valued mapping \(W:X\multimap Y\) is said to be a generalized \(R\)-\(KKM\)-mapping if for each \(A\in\langle D\rangle\) with the cardinality \(|A|=n+1\) there exists a continuous function \(\phi_A:\Delta_n\rightarrow Y\) such that \(J\in\langle A\rangle\) implies \(\phi_A(\Delta_J)\subset W(J)\), where \(\Delta_J\) is the standard subsimplex of \(\Delta_n\) corresponding to \(J\). Let \(X\) and \(Y\) be topological spaces and \(D\) be a nonempty subset of \(Y\). A set-valued mapping \(T:X\multimap Y\) is called a \(\Phi\)-mapping if there exists a generalized \(R\)-\(KKM\)-mapping \(W:D\multimap Y\) and a set-valued mapping \(S:X\multimap D\) having the local intersection property such that \(W(M)\subset T(x)\), for each \(x\in X\) and \(M\in \langle S(x)\rangle\). The main result of the paper is the following continuous selection theorem: Let \(X\) be a normal space, \(Y\) a topological space, \(D\) a nonempty subset of \(Y\). A \(\Phi\)-mapping \(T:X\multimap Y\) has a continuous selection whenever there exists a compact subset \(K\) of \(X\) and a finite subset \(M\) of \(D\) such that \(X\setminus K\subset \bigcup\limits_{y\in M} \text{int}\, S^-(y)\),where \(S:X\multimap D\) is the companion mapping of \(T\) in the definition above. As applications, fixed point theorems, coincidence theorems and a nonempty intersection theorem are given.
scientific article; zbMATH DE number 5344601
scientific article