A New Iterative Method for Equilibrium Problems and Fixed Point Problems

Abdul Latif,Mohammad Eslamian

doi:10.1155/2013/178053

Abstract

Introducing a new iterative method, we study the existence of a common element of the set of solutions of equilibrium problems for a family of monotone, Lipschitz-type continuous mappings and the sets of fixed points of two nonexpansive semigroups in a real Hilbert space. We establish strong convergence theorems of the new iterative method for the solution of the variational inequality problem which is the optimality condition for the minimization problem. Our results improve and generalize the corresponding recent results of Anh (2012), Cianciaruso et al. (2010), and many others.

Highlights

Let H be a real Hilbert space with inner product ⟨⋅, ⋅⟩ and norm ‖ ⋅ ‖
Introducing a new iterative method, we study the existence of a common element of the set of solutions of equilibrium problems for a family of monotone, Lipschitz-type continuous mappings and the sets of fixed points of two nonexpansive semigroups in a real Hilbert space
Let C be a nonempty closed convex subset of H, and let ProjC be a nearest point projection of H into C; that is, for x ∈ H, ProjCx is the unique point in C with the property ‖x − ProjCx‖ := inf{‖x − y‖ : y ∈ C}