Strong convergence for total quasi-ϕ-asymptotically nonexpansive semigroups in Banach spaces

Jing Quan,Xiongrui Wang,Shih-Sen Chang

doi:10.1186/1687-1812-2012-142

Abstract

The purpose of this article is to use the modified Halpern-Mann type iteration algorithm for total quasi-ϕ-asymptotically nonexpansive semigroups to prove strong convergence in Banach spaces. The main results presented in this paper extend and improve the corresponding results of many authors.MSC:47H05, 47H09, 49M05.

Highlights

1 Introduction Throughout this article, we assume that E is a real Banach space with norm ·, E∗ is the dual space of E; ·, · is the duality pairing between E and E∗; C is a nonempty closed convex subset of E; N and R denote the natural number set and the set of nonnegative real numbers respectively
One-parameter family T := {T(t) : t ≥ } of mappings from C into itself is said to be a total quasi-φ-asymptotically nonexpansive semigroup on C if conditions (a), (b), (c) in Definition . and following condition (f ) are satisfied: (f ) If F(T) = ∅, there exist sequences {μn}, {νn} with μn, νn → as n → ∞ and a strictly increasing continuous function ψ : R → R with ψ( ) = such that φ p, Tn(t)x ≤ φ(p, x) + μnψ φ(p, x) + νn holds for all x ∈ C, p ∈ F(T) and all n ∈ N
A total quasi-φ-asymptotically nonexpansive semigroup T is said to be uniformly Lipschitzian if there exists a bounded measurable function L : [, ∞) → (, +∞) such that

Summary

Introduction

Throughout this article, we assume that E is a real Banach space with norm · , E∗ is the dual space of E; ·, · is the duality pairing between E and E∗; C is a nonempty closed convex subset of E; N and R denote the natural number set and the set of nonnegative real numbers respectively. T is said to be total asymptotically quasi-nonexpansive if F(T) = ∅, there exist sequences {μn}, {νn} with μn, νn → as n → ∞ and a strictly increasing continuous function ψ : R → R with ψ( ) = such that Tnx – p ≤ x – p + μnψ( x – p ) + νn holds for all x ∈ C, p ∈ F(T) and all n ∈ N.

Objectives

Results

Conclusion