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

AbstractLet 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