The problem of interest, called "Equilibrim Problem", abbreviated EP, is defined as follows. Given two nonempty sets A and B, and a function f:A×B ⇒ R, EP consists of finding a ∈ A such that f(a,b) ≥ 0 for all b ∈ B.
To emphasize the importance of this problem in nonlinear analysis and in several applied fields, we first mention its most important particular cases as optimization, Kirszbraun's problem, saddlepoint (minimax) problems and variational inequalities. Then we study sufficient and/or necessary conditions for the existence of solutions of equilibrium problems.
We show that in finite dimensional spaces, our conditions will be sufficient for the existence of solutions without making any monotonicity assumption on the bifunction f which defines the problem. As a consequence we establish surjectivity of set-valued operators of the form T + λ I, 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.
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