New Iterative Manner Involving Sunny Nonexpansive Retractions for Pseudocontractive Mappings

Yaqin Zheng,Jinwei Shi

doi:10.1155/2012/207896

Abstract

Iterative methods for pseudocontractions have been studied by many authors in the literature. In the present paper, we firstly propose a new iterative method involving sunny nonexpansive retractions for pseudocontractions in Banach spaces. Consequently, we show that the suggested algorithm converges strongly to a fixed point of the pseudocontractive mapping which also solves some variational inequality.

Highlights

Let C be a nonempty closed convex subset of a real Banach space E
We assume that V I Fix T, A / ∅
6 we show lim sup δθ x† − Ax†, j xn 1 − x† ≤ 0, 3.13 n→∞

Summary

Introduction

Let C be a nonempty closed convex subset of a real Banach space E. We assume the following: C1 E is a uniformly convex and 2-uniformly smooth Banach space which admits a weakly sequentially continuous duality mapping j from E to E∗; C2 C is a nonempty closed convex subset of E; C3 QC is a sunny nonexpansive retraction from E onto C; C4 T : C → C is a λ-strict pseudocontraction; C5 θ : E → E is a ρ-contraction; C6 A : E → E is strongly positive i.e., Ax, j x ≥ γ x 2 for some 0 < γ < 1 and linear bounded operator with 1 − ζ I − θA ≤ 1 − ζ − θ for all ζ > 0, θ > 0 and 0 < ζ θ < 1; C7 Fix T / ∅.

Results

Conclusion