Some results for the family KKM\((X,Y)\) and the \(\Phi\)-mapping
Let \(X\) be a convex subset of a topological vector space (t.v.s.) and \(Y\) a topological space. If \(F,T: X\to 2^Y\) are two set-valued mappings such that \(T(\text{co}(A))\subset F(A)\) for any finite nonempty set \(A\) of \(X\), then \(F\) is called a generalized KKM mapping with respect to \(T\) [\textit{T. H. Chang} and \textit{C. L. Yen}, J. Math. Anal. Appl. 203, 224--235 (1996; Zbl 0883.47067)]. Let \(T: X\to 2^Y\) be a set-valued mapping such that if \(F: X\to 2^Y\) is a generalized KKM mapping with respect to \(T\), the family \(\{\overline{F(x):} x\in X\}\) has the finite intersection property; then we say that \(T\) has the KKM property. Denote by \(\text{KKM}(X, Y)= \{T: X\to 2^Y: T\) has the KKM property\}. Let \(Y\) be a topological space and \(X\) be a convex subset of a topological vector space. A set-valued mapping \(T: X\to 2^Y\) is called a \(\Phi\)-mapping if there exists a set-valued mapping \(F: X\to 2^Y\) such that (i) for each \(y\in Y\), \(\text{co\,}F(y)\subset T(y)\), and (ii) \(F\) has the compactly local intersection property. Let \(E\) be a Hausdorff topological space and \({\mathcal P}= \{P:P\) is a family of seminorms which determine the topology on \(E\}\). If \(p\in P\), \(P\in{\mathcal P}\), the set-measure of noncompactness \(\alpha_p: 2^E\to \mathbb R^+\) is \(\alpha_p(\Omega)= \text{inf}\{\varepsilon> 0\): \(\Omega\) can be covered by a finite number of sets with \(p\)-diameter less than \(\varepsilon\}\). Definition. A mapping \(T: X\to 2^E\) is said to be a \(k\)-set contraction map, if there exists \(P\in{\mathcal P}\) such that for each \(p\in P\), \(\alpha_p(T(\Omega))\leq k\alpha_p(\Omega)\) with \(k\in (0,1)\) for each bounded set \(\Omega\) of \(X\) and \(T(X)\) is bounded. The main result of the paper is Theorem 2. Let \(X\) be a nonempty bounded convex subset of a locally convex space \(E\), and let \(T\in \text{KKM}(X, X)\) be a \(k\)-set contraction, \(0< k< 1\), and closed with \(\overline{T(X)}\subset X\). Then \(T\) has a fixed point in \(X\). As an application of Theorem 2 the authors prove some matching theorems, coincidence theorems and minimax theorems of the family \(\text{KKM}(X, Y)\) and the \(\Phi\)-mapping.
scientific article; zbMATH DE number 5129635