Proximal and Dynamical Approaches to Equilibrium Problems

Abstract

The theory of equilibrium problems has emerged as an interesting branch of applied mathematics, permitting the general and unified study of a large number of problems arising in mathematical economics, optimization and operations research. Inspired by numerical methods developed for variational inequalities and motivated by recent advances in this field, we propose several ways (including an auxiliary problem principle, a selection method, as well as a dynamical procedure) to solve the following equilibrium problem: $$(GEP)Find\overline x \in CsuchthatF(\overline x ,x) + \left\langle {G(\overline x ),x - \overline x } \right\rangle \geqslant 0\forall x \in C,$$ where C is a nonempty convex closed subset of a real Hilbert space X, F: C × C → ℝ is a given bivariate function with F(x, x) = 0 for all x ∈ C and G: C → ℝ is a continuous mapping. This problem has useful applications in nonlinear analysis, including as special cases optimization problems, variation al inequalities, fixed-point problems and problems of Nash equilibria. Throughtout the paper, X is a real Hilbert space, <· , ·> denotes the associated inner product and | · | stands for the corresponding norm. From now on, we assume that the solution set, S, of problem (GEP) is nonempty. This corresponds to some important situations such as linear programming and semi-coercive minimization problems.