The split common fixed point problem for asymptotically nonexpansive semigroups in Banach spaces

Lin Wang,Zhaoli Ma

doi:10.1186/s13663-016-0588-2

Abstract

In this paper, we propose an iteration method for finding a split common fixed point of asymptotically nonexpansive semigroups in the setting of two Banach spaces, and we obtain some weak and strong convergence theorems of the iteration scheme proposed. The results presented in the paper are new and improve and extend some recent corresponding results.

Highlights

Let E be a real normed linear space and C be a nonempty closed convex subset of E
The mapping T : C → C is said to be asymptotically nonexpansive if there exists a sequence {kn} ⊂ [, ∞) with limn→∞ kn = such that for all x, y ∈ C and each n ≥
The class of asymptotically nonexpansive mappings is an important generalization of the class of nonexpansive mappings, which was introduced by Goebel and Kirk [ ] in

Summary

Introduction

Let E be a real normed linear space and C be a nonempty closed convex subset of E. Cholamjiak et al [ ] obtained a strong convergence theorem of split common fixed point problem involving a uniformly asymptotically regular nonexpansive semigroup and a total asymptotically strict pseudo-contractive mapping in Hilbert spaces. In the setting of two Banach spaces, Tang et al [ ] obtained a weak convergence theorem and a strong convergence theorem of the split common fixed point problem involving a quasi-strict pseudo-contractive mapping and an asymptotically nonexpansive mapping under the following assumptions:. (II) In addition, if = {p ∈ F(S) : Ap ∈ F(T)} = φ and S is semi-compact, {xn} converges strongly to a point x∗ ∈ This naturally brings about the following question: Question Can we obtain the convergence results of split common fixed point problem for asymptotically nonexpansive semigroups in the setting of two Banach spaces?. Under some suitable conditions on parameters, the iteration scheme proposed is shown to converge strongly and weakly to a split common fixed point of asymptotically nonexpansive semigroups in two Banach spaces

Preliminaries

Main results