The modified general iterative methods for asymptotically nonexpansive semigroups in Banach spaces

Rabian Wangkeeree,Pakkapon Preechasilp

doi:10.1186/1029-242x-2013-223

Abstract

In this paper, we introduce the modified general iterative methods for finding a common fixed point of asymptotically nonexpansive semigroups, which is a unique solution of some variational inequality. We prove the strong convergence theorems of such iterative scheme in a real Banach space which has a uniformly Gâteaux differentiable norm and admits the duality mapping and uniform normal structure. The main result extends various results existing in the current literature.

Highlights

Let E be a normed linear space, K be a nonempty, closed and convex subset of E
The class of asymptotically nonexpansive maps was introduced by Goebel and Kirk [ ] as an important generalization of the class of nonexpansive maps
In this paper, inspired and motivated by Chang et al [ ], Cholamjiak and Suantai [ ], Li, Li and Su [ ], Wangkeeree and Wangkeeree [ ] and Wangkeeree et al [ ], we introduce the following iterative approximation methods ( . ) and ( . ) for the class of strongly continuous semigroup of asymptotically nonexpansive mappings S = {T(s) : ≤ s < ∞}: yn = αnγ f + (I – αnA)Tn(tn)yn, n ≥, ( . )

Summary

Introduction

Let E be a normed linear space, K be a nonempty, closed and convex subset of E. In , motivated and inspired by Marino and Xu [ ], Li et al [ ] introduced the following general iterative procedures for the approximation of common fixed points of a nonexpansive semigroup {T(s) : s ≥ } on a nonempty, closed and convex subset K in a Hilbert space: tn yn = αnγ f (yn) + (I – αnA) tn T(s)yn ds, n ≥ , tn xn+ = αnγ f (xn) + (I – αnA) tn T(s)xn ds, n ≥ , where {αn} and {tn} are sequences in [ , ] and ( , ∞), respectively, A is a strongly positive bounded linear operator on C and f is a contraction on C Their convergence theorems can be proved under some appropriate control conditions on parameter {αn} and {tn}.

Preliminaries

Main theorem