Fixed point theorems for the class S-KKM mappings in abstract convex spaces
An abstract convex space \((X,{\mathcal C})\) consists of a nonempty set \(X\) and a family \({\mathcal C}\) of subsets of \(X\) such that \(X\) and \(\varnothing\) belong to \({\mathcal C}\) and \({\mathcal C}\) is closed under arbitrary intersection. An abstract convex space \((X,{\mathcal C})\) is called a \(\Phi\)-space if \(X\) is a uniform space and for each neighbourhood \(V\) there is a \(\Phi\)-mapping \(T:X\multimap X\) such that \(\text{Graph}T\subseteq V\). For a \(\Phi\)-space \((X,{\mathcal C})\), we say that a mapping \(F:X\multimap X\) has the approximate fixed point property if for any \(U\in{\mathcal U}\), where \({\mathcal U}\) is a basis of the uniform structure of \(X\), there exists \(x\in X\) such that \(U[x]\cap F(x)\neq\varnothing\). In this paper, the authors define KKM mappings and S-KKM mappings for abstract convex spaces and establish some approximate fixed point theorems for multifunctions with the S-KKM property on \(\Phi\)-spaces. They also establish a new version of Sadovskij's fixed point theorem, similar to that of \textit{M.\,A.\thinspace Khamsi} [Nonlinear Anal., Theory Methods Appl.\ 53, No.\,6(A), 829--837 (2003; Zbl 1028.47042)] in abstract convex spaces.
scientific article; zbMATH DE number 5083392
scientific article