Generalized Mixed Equilibria, Variational Inclusions, and Fixed Point Problems

A E Al-Mazrooei,J.-C Yao,A Latif,A S M Alofi

doi:10.1155/2014/251065

A E Al-Mazrooei, J.-C Yao + Show 2 more

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https://doi.org/10.1155/2014/251065

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Abstract

We propose two iterative algorithms for finding a common element of the set of solutions of finite generalized mixed equilibrium problems, the set of solutions of finite variational inclusions for maximal monotone and inverse strong monotone mappings, and the set of common fixed points of infinite nonexpansive mappings and an asymptoticallyκ-strict pseudocontractive mapping in the intermediate sense in a real Hilbert space. We prove some strong and weak convergence theorems for the proposed iterative algorithms under suitable conditions.

Highlights

Let H be a real Hilbert space with inner product ⟨⋅, ⋅⟩ and norm ‖⋅‖, let C be a nonempty closed convex subset of H, and let PC be the metric projection of H onto C
We propose two iterative algorithms for finding a common element of the set of solutions of finite generalized mixed equilibrium problems, the set of solutions of finite variational inclusions for maximal monotone and inverse strong monotone mappings, and the set of common fixed points of infinite nonexpansive mappings and an asymptotically κ-strict pseudocontractive mapping in the intermediate sense in a real Hilbert space
Let C be a nonempty subset of a Hilbert space H

Summary

Introduction

Let H be a real Hilbert space with inner product ⟨⋅, ⋅⟩ and norm ‖⋅‖, let C be a nonempty closed convex subset of H, and let PC be the metric projection of H onto C. A mapping S : C → C is said to be an asymptotically κstrict pseudocontractive mapping in the intermediate sense with sequence {γn}, if there exist a constant κ ∈ [0, 1) and a sequence {γn} in [0, ∞) with limn → ∞γn = 0 such that lim sup sup n → ∞ x,y∈C. In this paper, inspired by the research work mentioned above, we introduce two iterative algorithms for finding a common element of the set of solutions of finite generalized mixed equilibrium problems, the set of solutions of finite variational inclusions for maximal monotone and inverse strong monotone mappings, and the set of common fixed points of infinite nonexpansive mappings and an asymptotically κstrict pseudocontractive mapping in the intermediate sense in a real Hilbert space. We prove some strong and weak convergence theorems for the proposed iterative algorithms under mild conditions

Preliminaries

Strong Convergence Theorem

Weak Convergence Theorem