Generalized Equilibrium Problem with Mixed Relaxed Monotonicity

Abstract

We extend the concept of relaxed α-monotonicity to mixed relaxed α-β-monotonicity. The concept of mixed relaxed α-β-monotonicity is more general than many existing concepts of monotonicities. Finally, we apply this concept and well known KKM-theory to obtain the solution of generalized equilibrium problem.

Existence and iterative approximation method for solving mixed equilibrium problem under generalized monotonicity in Banach spaces

“…By adding the inequalities (24) and (25), and using the skew symmetry assumption of b and condition (i)', we obtain…”

“…The appellation equilibrium problems appeared in the seminal paper by Blum-Oettli [3] where they provided the evidence of unifying aspect of (EP) and gave some fundamental notions and results. Later on these problems were studied extensively by various authors in the context of both Hilbert and Banach spaces under different kinds of generalized monotonicities [2,13,14,20,25]. All these papers are based on bifunction equilibrium problems.…”

“…But for finding solutions of mixed equilibrium problems involving variational-like inequalities, projection type methods can not be applied, due to the nature of these problems. To overcome such difficulties, auxiliary principle technique was proposed in a substantial number of papers on existence and uniqueness, formulation of iterative methods and convergence analysis for bifunction equilibrium problems and variational inequalities, see for instance [4,7,15,20,22,25,27,16,21,23] and the references therein.…”

On the existence of solutions of generalized equilibrium problems with α−β−η -monotone mappings

“…If we choose f (x, y) = −2, in this case f is α-β-η-monotone with respect to Definition 2.7, but f is not α-β-monotone with respect to Definition 6 in [16].…”