Abstract

Let X be a Hausdorff compact space, E a topological vector space on which E* separates points, F:X2E an upper semicontinuous multifunction with compact acyclic values, and g:XE a continuous function such that g(X) is convex and g1(y) is acyclic for each yg(X). Then either (1) there exists an x0X such that gx0Fx0 or (2) there exist an (x0,z0) on the graph of F and a continuous seminorm p on E such that 0<p(gx0z0)p(yz0)         for all         yg(X). A generalization of this result and its application to coincidence theorems are obtained. Our aim in this paper is to unify and improve almost fifty known theorems of others.