Equilibrium Problems

The mathematical formulation of the equilibrium problem is to find an element x̄ of a set K such that

where FK × K →is a bifunction such that F(x, x) = 0 for all x ∈ K. It is an unified model of several fundamental mathematical problems, namely, optimization problems, saddle point problems, fixed point problems, minimax inequality problems, Nash equilibrium problem, complementarity problems, variational inequality problems, etc. In 1955, Nikaido and Isoda [134] first considered equilibrium problem (5.1) as an auxiliary problem to establish the existence results for Nash's equilibrium points in noncooperative games. In the theory of equilibrium problems, the key contribution was made by Ky Fan [79], whose new existence results contained the original techniques which became a basis for most further existence theorems in the setting of topological vector spaces. That is why equilibrium problem (5.1) is also known as Ky Fan type inequality. Within the context of calculus of variations, motivated mainly by the works of Stampacchia [160], there arises the work of Brézis, Niremberg, and Stampacchia [45] establishing a more general result than that in [79]. In the last three decades, the theory of equilibrium problems emerges as a new direction of research in nonlinear analysis, optimization, optimal control, game theory, mathematical economics, etc. Most of the results on the existence of solutions for equilibrium problems are studied in the setting of topological vector spaces by using some kind of fixed point (Fan-Browder type fixed point) theorem or KKM type theorem. The term “equilibrium problem” was first used by Muu and Oettli [130] and later adopted by Blum and Oettli [38]. For further details, we refer to [3–5, 7–10, 12, 13, 32– 36, 38, 54–57, 62–64, 83, 84, 96, 97, 106, 107, 125, 130, 135] and the references therein. In most of the existence results for a solution of equilibrium problems, the convexity of the underlying set K and the bifunction F is assumed; see, for example, [10, 13, 35, 36, 45, 56, 57, 83, 84, 107] and the references therein. Inspired by the work of Blum and Oettli [38] and Oettli and Théra [135], the existence theory for solutions of equilibrium problems has been developed by many researchers in the setting of metric spaces and without any convexity assumption on the underlying set K and bifunction F; see, for example, [4, 7–9, 12, 32, 38, 54, 55, 63, 107, 106, 111, 135] and the references therein.