A theorem on the minimization of a condensing multifunction and fixed points

Abstract

Let G be a subset of a locally convex separated topological vector space E with int( G ) ≠ Ø , cl( G ) convex and quasi-complete. Let f : cl( G ) → E be a continuous condensing multifunction with compact and convex values and with a bounded range. It is shown that for each w ϵ int( G ), there exists a u = u ( w ) ϵ ∂ (cl( G )) such that p ( f ( u ) − u ) = inf{ p ( x − y ): x ϵ f ( u ), y ϵ cl( G )}, where p is the Minkowski's functional of the set (cl( G ) − w ). Several fixed point results are obtained as a consequence of this result.