Shrinking Projection Method of Common Fixed Point Problems for Discrete Asymptotically Strictly Pseudocontractive Semigroups and Mixed Equilibrium Problems in Hilbert Spaces

Pattanapong Tianchai

doi:10.5402/2011/529407

Abstract

This paper is concerned with a common element of the set of common fixed point for a discrete asymptotically strictly pseudocontractive semigroup and the set of solutions of the mixed equilibrium problems in Hilbert spaces. The strong convergence theorem for the above two sets is obtained by a general iterative scheme based on the shrinking projection method which extends and improves the corresponding ones due to Kim [Proceedings of the Asian Conference on Nonlinear Analysis and Optimization (Matsue, Japan, 2008), 139–162].

Highlights

Throughout this paper, we always assume that C is a nonempty closed convex subset of a real Hilbert space H with inner product and norm denoted by ·, · and ·, respectively
The set of solutions of problem 1.2 is denoted by MEP Φ, φ ; that is, MEP Φ, φ x ∈ C : Φ x, y φ y − φ x ≥, ∀y ∈ C
Inspired and motivated by the works mentioned above, in this paper, we introduce a general iterative scheme 3.1 below to find a common element of the set of common fixed point for a discrete asymptotically κ-SPC semigroup and the set of solutions of the mixed equilibrium problems in Hilbert spaces

Summary

Introduction

Throughout this paper, we always assume that C is a nonempty closed convex subset of a real Hilbert space H with inner product and norm denoted by ·, · and · , respectively. And he introduced an iterative scheme to find a common fixed point of a discrete asymptotically κ-SPC semigroup and a bounded sequence {Ln} ⊂ 1, ∞ such that limn → ∞Ln 1 as follows: x0 x ∈ C chosen arbitrarily, yn αnxn 1 − αn Tnxn, Cn z ∈ C : yn − z 2 ≤ xn − z 2 1 − αn θn κ − αn xn − Tnxn 2 , Qn {z ∈ C : xn − z, x0 − xn ≥ 0}, xn 1 PCn∩Qn x0 , ∀n ∈ N ∪ {0}, where θn Ln − 1 · sup{ xn − z 2 : z ∈ F S } < ∞ He proved that under the parameter 0 ≤ αn < 1 for all n ∈ N ∪ {0}, if F S is a nonempty bounded subset of C, the sequence {xn} generated by 1.16 converges strongly to PF S x0. The strong convergence theorem for the above two sets is obtained based on the shrinking projection method which extend and improve the corresponding ones due to Kim 16

Preliminaries

Main Results

Applications