Hybrid iteration method for common fixed points of an infinite family of nonexpansive mappings in Banach spaces

Wei-Qi Deng

doi:10.1186/1687-1812-2012-22

Abstract

Let E be a real uniformly convex Banach space, and let K be a nonempty closed convex subset of E. Let be a sequence of nonexpansive mappings from K to itself with F :={x ∈ K :T i x = x, ∀i ≥ 1}≠ ∅. For an arbitrary initial point x1 ∈ K, the modified hybrid iteration scheme {x n } is defined as follows: , where A: K → K is an L-Lipschitzian mapping, with i satisfying: n = [(k-i+1)(i+k)/2]+[1+(i-1)(i+2)/2],k ≥ i-1(i = 1,2,...),{λ n } ⊂ [0,1), and {α n } is a sequence in [a, 1 - a] for some a ∈ (0,1). Under some suitable conditions, the strong and weak convergence theorems of {x n } to a common fixed point of the nonexpansive mappings are obtained. The results in this article extend those of the authors whose related researches are restricted to the situation of finite families of nonexpansive mappings. Mathematics Subject Classifications 2000: 47H09; 47J25.

Highlights

Let K be a nonempty closed convex subset of a real uniformly convex Banach space E
A self-mapping T: K ® K is said to be nonexpansive if ||Tx-Ty|| ≤ ||x-y|| for all x,yÎ K.F : K ® K is said to be L-Lipschitzian if there exists a constant L > 0 such that ||FxFy|| ≤ L||x-y|| for all x,y Î K
Theorem 1.1. [12]Let E be a real uniformly convex Banach space endowed with the norm ||·||

Summary

Introduction

Let K be a nonempty closed convex subset of a real uniformly convex Banach space E. Let {Ti}∞ i=1 be a sequence of nonexpansive mappings from K to itself with F :={x Î K :Tix = x, ∀i ≥ 1}≠ ∅. For an arbitrary initial point x1 Î K, the modified hybrid iteration scheme {xn} is defined as follows: xn+1 = αnxn + (1 − αn) Tn∗xn − λn+1μA(Tn∗xn) , n ≥ 1 , where A: K ® K is an LLipschitzian mapping, Tn∗ = Ti with i satisfying: n = [(k-i+1)(i+k)/2]+[1+(i-1)(i+2)/2],k ≥ i-1(i = 1,2,...),{ln} ⊂ [0,1), and {an} is a sequence in [a, 1 - a] for some a Î (0,1).

Results

Conclusion