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
iusp@impa.br
Gábor Kassay
Faculty of Mathematics and Computer Sciences, Babes-Bolyai University, 1 Kogalniceanu Street, 400084 Cluj-Napoca, Romania
kassay@math.ubbcluj.ro
Wilfredo Sosa
Universidad Nacional de Ingeniería, Instituto de Matemática y Ciencias Afines, Calle de los Biólogos 245, Lima 12, Perú
sosa@uni.edu.pe
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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