Fixed Point Theory and Applications 
Volume 2005 (2005), Issue 3, Pages 267-279
doi:10.1155/FPTA.2005.267

Existence of fixed points on compact epilipschitz sets without invariance conditions

Mikhail Kamenskii1 and Marc Quincampoix2

 1Voronezh State University, Universitetskaya pl.1, Voronezh 394063, Russia
2Laboratoire de Mathématiques, Unité CNRS UMR 6205, Université de Bretagne Occidentale, 6 Avenue Victor Le Gorgeu, Brest 29200, France
Received 4 April 2005

Abstract

We provide a new result of existence of equilibria of a single-valued Lipschitz function f on a compact set K of ℝn which is locally the epigraph of a Lipschitz functions (such a set is called epilipschitz set). Equivalently this provides existence of fixed points of the map x↦x−f(x). The main point of our result lies in the fact that we do not impose that f(x) is an “inward vector” for all point x of the boundary of K. Some extensions of the existence of equilibria result are also discussed for continuous functions and set-valued maps.