Geometric properties and coincidence theorems with applications to generalized vector equilibrium problems
The paper derives Fan-Knaster-Kuratowski-Mazurkiewicz type theorems and geometric properties of convex spaces, applying these to obtain coincidence theorems and fixed point theorems in convex spaces, with which they obtain existence results for a solution of generalized vector equilibrium problems. Some details follow. Let \(X\) be a convex space, i.e. a non-empty convex set in a vector space with any topology that induces the Euclidean topology on the convex hulls of its finite subsets. Let \(Y\) be a Hausdorff topological space. If \( S,T:X\rightarrow 2^{Y}\)are maps such that \(T(coN)\sqsubseteq S(N)\), for each \(N\in \langle X\rangle \) (where \(coN\) denotes the convex hull of \(N\) and \( \langle X\rangle \) denotes the class of all finite subsets of \(X\) ) then \(S\) is said to be {generalized KKM mapping w.r.t. }\({T}\). The map \(T\) is said to have the {KKM property} if \(S\) is a generalized KKM map w.r.t. \(T\) such that the family \(\{\overline{S(X)}:x\in X\}\) has the finite intersection property, where \(\overline{S(X)}\) denotes the closure of \(S\left( X\right) \). Denote by \(KKM(X,Y)\) the family of all maps having the KKM property. A map \(G:X\rightarrow 2^{Y}\) is {transfer closed} if for any \(x\in X\) and \(y\notin P(x)\), there exists \(\widetilde{x}\in X\) such that \(y\notin P(\widetilde{x})\) and it is {transfer open} if for any \(x\in X\) and \(y\in P(x)\), there exists \(\widetilde{x}\in X\) such that \( y\in intP(\widetilde{x}).\) The following theorem is proved: let \(X\) be a convex space, \(Y\) a topological space and \(T\in KKM(X,Y)\) be compact. Assume that \( G:X\rightarrow 2^{Y}\) is transfer closed and, for any \(N\in \langle X\rangle ,T(coN)\sqsubseteq G(N).\) Then, \(\overline{T(X)}\cap [ \bigcap \{G(x):x\in X\}]\neq \varnothing .\) This leads to results concerning geometric properties of convex spaces and to coincidence theorems such as the following. Let \(Y\) be a convex space, \(X\) a topological space and \( T\in KKM(Y,X)\) be compact. Let \(P:X\rightarrow 2^{Y}\) be a map such that for all \(x\in X\) , \(P(x)\) is convex and \(X=\bigcup \{intP^{-1}(y):y\in Y\}.\) Then there exists \(\left( \widetilde{x},\widetilde{y}\right) \in X\times Y\) such that \(\widetilde{x}\in T(\widetilde{y})\) and \(\widetilde{y}\in P( \widetilde{x}).\) Let \(K\) be a non-empty convex subset of a topological vector space \(X\). Let \(Z\) be a a topological vector space. Let \(C:K\rightarrow 2^{Z}\) be a map such that, for each \(x\in K\), \(C(x)\) is a closed and convex cone with \( intC(x)\neq \emptyset \). Let \(F:K\times K\rightarrow 2^{Z}\backslash \{\emptyset \}\), then the {generalized vector equilibrium problem} ( {GVEP}) is to find \(\widetilde{x}\in K\) such that \(F(\widetilde{x} ,y)\nsubseteqq -int C(\widetilde{x})\), for all \(y\in K\). Theorems establishing the existence of a solution to GVEPs are obtained as applications of coincidence theorems.
scientific article; zbMATH DE number 1997751