Fixed point theorem of half-continuous multivalued mappings

Abstract

Let E be a topological vector space over R and let C be a nonempty subset of E. A mapping F from C into the set of nonempty subsets of E, is said to be half-continuous if for each x ∈ C with x ∉ F(x) there exists a (nonzero) continuous linear functional p ∈ E[superscript *] and a neighborhood W of x in C such that if y ∈ W, such that y ∉ F(y), then for every z ∈ F(y), p(z-y) > 0. In this work, we prove that if E is locally convex Hausdorff and C is a nonempty compact convex subset of E, then every half-continuous mapping F on C into the set of nonempty subsets of C there exists a point x[subscript 0] in C such that x[subscript 0] ∈ F(x[subscript 0], that is x[subscript 0] is a fixed point of F.