Nonlinear Analysis 52 (2003) 621–635 www.elsevier.com/locate/na New existence results for equilibrium problems Alfredo N. Iusem a ; * , Wilfredo Sosa b a Instituto de Matem atica Pura e Aplicada, Estrada Dona Castorina 110, Jardim Botˆ anico, CEP 22460-320 RJ, Rio de Janeiro, Brazil b Instituto de Matem atica y Ciencias Anes, Jir on Ancash 536, Lima 1, Lima, Peru Received 23 October 2001; accepted 20 February 2002 Abstract We consider equilibrium problems in the framework of the formulation proposed by Blum and Oettli, which includes variational inequalities, Nash equilibria in noncooperative games, and vector optimization problems, for instance, as particular cases. We establish new sucient and= or necessary conditions for existence of solutions of such problems. Our results are based upon the relation between equilibrium problems and certain auxiliary convex feasibility problems, together with extensions to equilibrium problems of gap functions for variational inequalities. Then we apply our results to some particular instances of equilibrium problems, obtaining results which include, among others, a new lemma of the alternative for convex optimization problems. ? 2002 Elsevier Science Ltd. All rights reserved. Keywords: Equilibrium problems; Convex feasibility problems; Variational inequalities; Convex optimization 1. Introduction The problem of interest, which we call equilibrium problem, abbreviated (EP), is dened as follows EP : Find x ∈ K such that f(x;y) ¿ 0 for all y ∈ K; (1) where 1. X is a real Hausdor topological vector space, 2. K is a nonempty closed convex subset of X , 3. f : K × K → R is a function which satises the following properties: * Corresponding author. Fax: +51-1-426-3734. E-mail address: iusp@impa.br (A.N. Iusem). 0362-546X/02/$ - see front matter ? 2002 Elsevier Science Ltd. All rights reserved. PII:S0362-546X(02)00154-2