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Journal of Convex Analysis 16 (2009), No. 3, 807--826
Copyright Heldermann Verlag 2009

An Existence Result for Equilibrium Problems with Some Surjectivity Consequences

Alfredo N. Iusem
Instituto de Matemática Pura e Aplicada, Estrada Dona Castorina 110, Jardim Botânico, CEP 22460-320 Rio de Janeiro, Brazil

Gábor Kassay
Faculty of Mathematics and Computer Sciences, Babes-Bolyai University, 1 Kogalniceanu Street, 400084 Cluj-Napoca, Romania

Wilfredo Sosa
Universidad Nacional de Ingeniería, Instituto de Matemática y Ciencias Afines, Calle de los Biólogos 245, Lima 12, Perú

We present conditions for existence of solutions of equilibrium problems, which are sufficient in finite dimensional spaces, without making any monotonicity assumption on the bifunction which defines the problem. As a consequence we establish surjectivity of set-valued operators of the form T + λI, with λ > 0, where T satisfies a property weaker than monotonicity, which we call pre-monotonicity. We study next the notion of maximal pre-monotonicity. Finally we adapt our condition for non-convex optimization problems, obtaining as a by-product an alternative proof of Frank-Wolfe's Theorem.

Keywords: Equilibrium problems, convex feasibility problems, variational inequalities, convex optimization.

MSC: 90C47, 49J35

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