On a sharpened form of the Schauder fixed-point theorem

Abstract

If K is a compact convex subset of a locally convex topological vector space X, we consider a continuous mapping f of K into X. A fixed-point theorem is proved for such a map f under the assumption that for a given continuous realvalued function p on K x X with p(x,y) convex in y and for each point x in K not fixed by f, there exists a point y in the inward set I(K)(x) generated by K at x with p(x,y - f(x)) less than p(x,x - f(x)). For X a Banach space, in particular, this yields a sharp extension and a drastic simplification of the fixed point theory of weakly inward (and weakly outward) mappings. The result comes close in the domain of mappings of compact convex sets to the thrust of fixed point conditions of the Leray-Schauder type for compact maps of sets with interior in X.