Some fixed-point theorems on an almost \(G\)-convex subset of a locally \(G\)-convex space and its applications (Q2495257)

For a nonempty set \(X\), let \(2^X\) denote the class of all nonempty subsets of \(X\) and \(\langle X\rangle\) denote the class of all nonempty finite subsets of \(X\). A set-valued function \(T: X\to 2^Y\) is said to be compact if the image \(T(X)\) of \(X\) is contained in a compact subset of \(Y\). \(T\) is said to be closed if its graph is a closed subset of \(X\times Y\). In this paper, the author obtains a generalization of almost fixed point theorems on almost \(G\)-convex sets and the Himmelberg fixed point theorem on a locally \(G\)-convex space. By invoking non-convexity of constraint regions in place of convexity, a new fixed point theorem is obtained for mappings having the \(\Gamma^*\)-KKM property. 1. (Theorem 2): Let \(X\) be an almost \(G\)-convex subset of a \(G\)-convex space \(E\) which has a uniformity \(U\) and \(U\) has an open symmetric base family \({\mathcal N}\). Let \(V\in{\mathcal N}\) be such that \(V[x]\) is \(G\)-convex for all \(x\in X\). If \(T: X\to 2^X\) is closed and \(T(X)\) is relatively compact, then \(T\) has a \(V\)-fixed point. 2. (Theorem 3): Let \(X\) be an almost \(G\)-convex subset of a locally \(G\)-convex space \(E\). If \(T: X\to 2^X\) is compact and closed, then \(T\) has a fixed point in \(X\). 3. (Theorem 4): Let \(X\) be an almost \(G\)-convex subset of a locally \(G\)-convex space \(E\). If \(T\in \Gamma^*\text{-KKM}(X,X)\) is compact and closed, then \(T\) has a fixed point.