### Gabor Kassay (Babes-Bolyai University, Romania)

#### On equilibrium problems and surjectivity

*Tuesday 26 May 2009 at 15.30, JCMB 6206*

##### Abstract

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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