A unified fixed point theory in generalized convex spaces (Q2385343)

In his earlier papers, the author has introduced the class of `better' admissible multimaps \(\mathcal{B}\) and proved that any compact closed multimap in \(\mathcal{B}\) from an admissible (in the sense of Klee) convex subset of a Hausdorff topological vector space into itself has a fixed point. In the present paper, the author introduces new concepts of admissibility (in the sense of Klee) and of Klee approximability for subsets of \(G\)-convex uniform spaces and generalizes the result mentioned above showing that any compact closed multimap in \(\mathcal{B}\) from a \(G\)-convex space into itself with Klee approximable range has a fixed point. This new theorem contains many known results on topological vector spaces or on various subclasses of the class of admissible \(G\)-convex spaces. The mutual relations among these subclasses and some related results are also investigated.