A hybrid projection method for solving a common solution of a system of equilibrium problems and fixed point problems for asymptotically strict pseudocontractions in the intermediate sense in Hilbert spaces

Chatchawan Watchararuangwit,Poom Kumam,Pongrus Phuangphoo

doi:10.1186/1029-242x-2012-252

Abstract

Abstract In this paper, we introduce a new iterative algorithm which is constructed by using the hybrid projection method for finding a common solution of a system of equilibrium problems of bifunctions satisfying certain conditions and a common solution of fixed point problems of a family of uniformly Lipschitz continuous and asymptotically λ i -strict pseudocontractive mappings in the intermediate sense. We prove the strong convergence theorem for a new iterative algorithm under some mild conditions in Hilbert spaces. Finally, we also give a numerical example which supports our results. MSC:47H05, 47H09, 47H10.

Highlights

Let C be a closed and convex subset of a real Hilbert space H with the inner product ·, · and the norm ·
In this paper, motivated and inspired by the previously mentioned above results, we introduce a new iterative algorithm by the hybrid projection method for finding a common solution of a system of equilibrium problems of bifunctions satisfying certain conditions and a common solution of fixed point problems of a family of uniformly Lipschitz continuous and asymptotically λi-strict pseudocontractive mappings in the intermediate sense in a real Hilbert space
Lemma . ([ ]) Let C be a nonempty closed and convex subset of a real Hilbert space H and S : C → C be a uniformly L-Lipschitz continuous and asymptotically λ-strict pseudocontraction in the intermediate sense

Summary

Introduction

Let C be a closed and convex subset of a real Hilbert space H with the inner product ·, · and the norm ·. In this paper, motivated and inspired by the previously mentioned above results, we introduce a new iterative algorithm by the hybrid projection method for finding a common solution of a system of equilibrium problems of bifunctions satisfying certain conditions and a common solution of fixed point problems of a family of uniformly Lipschitz continuous and asymptotically λi-strict pseudocontractive mappings in the intermediate sense in a real Hilbert space. ([ ]) Let C be a nonempty closed and convex subset of a real Hilbert space H and S : C → C be a uniformly L-Lipschitz continuous and asymptotically λ-strict pseudocontraction in the intermediate sense. From Lemma . , we get SEP(Fm) is closed and convex

From the assumption of

It is obvious that