On existence of equilibria of set-valued maps Grzegorz Gabor 1,2 and Marc Quincampoix 1 August 25, 2004 1 Laboratoire de Math´ ematiques, Unit´ e CNRS FRE 2218 Universit´ e de Bretagne Occidentale 6, avenue Victor Le Gorgeu B.P. 809 29285 Brest Cedex, France 2 Faculty of Mathematics and Computer Science N. Copernicus Univ. of Toru´ n Chopina 12/18, 87–100 Toru´ n, Poland e-mail: ggabor@mat.uni.torun.pl, Marc.Quincampoix@univ-brest.fr Abstract The present paper is devoted to sufficient conditions for existence of equilibria of Lipschitz multivalued maps in prescribed subsets of finite-dimensional spaces. The main improvement of the present study lies in the fact that we do not suppose any regular assumptions on the boundary of the subset. Our approach is based on behaviour of trajectories to the corresponding differential inclusion. R´ esum´ e Cette note est consacr´ ee ` a l’´ etude de conditions suffisantes pour l’existence d’´ equilibres d’une application multivoque lipschitzienne dans un sous-ensemble ferm´ e donn´ e d’un espace de dimension finie. L’apport principal du pr´ esent article est qu’il n’est nul besoin d’imposer des conditions de r´ egularit´ e sur le bord du sous-ensemble pour pouvoir mener cette ´ etude. Notre approche se fonde sur le comportement des trajectoires de l’inclusion diff´ erentielle associ´ ee ` a l’application multivoque. Riasunto L’articolo fornisce delle condizioni sufficienti per l’esistenza di punti di equilibrio di applicazioni multivoche Lipschitziane in assegnati sottoinsiemi di spazi finito-dimensionali. Il principale contributo del presente articolo consiste nel fatto che non si danno condizioni di regolarit sulla frontiera degli insiemi considerati. L’approccio basato sullo studio del comportamento delle traiettorie della corrispondente inclusione differenziale. 2000 Mathematics Subject Classification: Primary: 34A60, Secondary: 54C60, 54C65, 47H04. Keywords and phrases: equilibria, viability, differential inclusions, multi- valued maps, strong deformation retracts, Lipschitz selections, L-retracts 1