Existence of equilibria of set-valued maps on bounded epi-Lipschitz domains in Hilbert spaces without invariance conditions

Abstract

In this paper, we provide a new result of the existence of equilibria for set-valued maps on bounded closed subsets K of Hilbert spaces. We do not impose either convexity or compactness assumptions on K but we assume that K has epi-Lipschitz sections, i.e. its intersection with suitable finite dimensional spaces is locally the epigraph of Lipschitz functions. In finite dimensional spaces, the famous Brouwer theorem asserts the existence of a fixed point for a continuous function from a compact convex set K to itself. Our result could be viewed as a kind of generalization of this classical result in the context of Hilbert spaces and when the function (or the set-valued map) does not necessarily map K into itself ( K is not invariant under the map). Our approach is based firstly on degree theory for compact and for condensing set-valued maps and secondly on flows generated by trajectories of differential inclusions.