Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: EXISTENCE OF FIXED POINTS ON COMPACT EPILIPSCHITZ SETS WITHOUT INVARIANCE CONDITIONS | EXISTENCE OF FIXED POINTS ON COMPACT EPILIPSCHITZ SETS WITHOUT INVARIANCE CONDITIONS MIKHAIL KAMENSKII AND MARC QUINCAMPOIX Received 4 April 2005 We provide a new result of existence of equilibria of a single-valued Lipschitz function f on a compact set K of R 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. 1. Introduction This paper is devoted to the following result. Theorem . Let K be an epilipschitz compact subset of R f R R be a locally Lipschitz function. Assume that Ks is closed and that the Euler characteristic x Ks is well defined. If x K x Ks then there exists an equilibria in K that is a point x e K such that f x 0. In the above Theorem the set Ks or Ks f is the set of elements x ofthe boundary of K such that the solution to the Cauchy problem x t f x t t 0 x 0 x leaves K immediately that is there exists Ơ 0 such that x 0 ơ n K 0 . Epilipschitz sets are sets which are locally the epigraph of a Lipschitz function an equivalent formulation is given in 25 . It is worth pointing out that when f x is inward for any x e dK we have that K is invariant by the differential equation x t f x t t 0 Copyright 2005 Hindawi Publishing Corporation Fixed Point Theory and Applications 2005 3 2005 267-279 DOI 268 Fixed point without invariance and consequently Ks 0. So our theorem contains for example the famous fixed point Brouwer theorem viewed as an existence result for equilibria of the map x x - g x for convex compact closed sets. It contains also several results of existence of equilibria which impose inwardness conditions of the type Vx e dK f x e Ck x .
