Approximation Methods in Equilibrium and Fixed Point Theory Proximally nondegenerate sets

The aim of this paper is to show how approximation methods allow to extend the Equilibrium and Fixed Point Theory to the nonsmooth and nonconvex setting. Our approach relies heavily on the use of the distance function d _M to a set M. We show that, for our purpose, Clarke’s subdifferential ∂d _M is too big, and that the appropriate notion is the upper limit $$\partial + {{d}_{M}}{{(\bar{x})}^{{\underline{\underline {def}} }}}\lim {{\sup }_{{x \to \bar{x},x \notin M}}}\partial {{d}_{M}}(x) .$$