A theorem on the minimization of a condensing multifunction and fixed points (Q1081826)

The main result of this paper is the following: Let E be a locally convex separated topological vector space. Let G be a subset of E with int(G)\(\neq \emptyset\), cl(G) convex and quasi-complete. If \(f:cl(G)\to P_{cp,cv}(E)\) is a continuous condensing multifunction with a bounded range, then for each \(w\in int(G)\) there exists a \(u=u(w)\) in cl(G) such that \(p(f(u)-u)=p(f(u)-cl(G))\), where p is the Minkowski' functional of (cl(G)-w). Further, if p(f(u)-w)\(\leq 1\) then \(u\in f(u)\) and if \(p(f(u)-w)>1\), then \(u\in \partial (cl(G))\).