Iterative approximation to common fixed points of nonexpansive mapping sequences in reflexive Banach spaces

Abstract

Let E be a real reflexive Banach space which admits a weakly sequentially continuous duality mapping from E to E ∗ , and K be a nonempty closed convex subset of E . Suppose that { T n } ( n = 1 , 2 , … ) is a uniformly asymptotically regular sequence of nonexpansive mappings from K into itself such that F ≔ ⋂ n = 1 ∞ F ( T n ) ≠ 0̸ . For arbitrary initial value x 0 ∈ K and fixed contractive mapping f : K → K , define iteratively a sequence { x n } as follows: x n + 1 = λ n + 1 f ( x n ) + ( 1 − λ n + 1 ) T n + 1 x n , n ≥ 0 , where { λ n } ⊂ ( 0 , 1 ) satisfies lim n → ∞ λ n = 0 and ∑ n = 1 ∞ λ n = ∞ . We prove that { x n } converges strongly to p ∈ F , as n → ∞ , where p is the unique solution in F to the following variational inequality: 〈 ( I − f ) p , j ( p − u ) 〉 ≤ 0 for all u ∈ F ( T ) . Our results extend and improve the corresponding ones given by O’Hara et al. [J.G. O’Hara, P. Pillay, H.-K. Xu, Iterative approaches to finding nearest common fixed point of nonexpansive mappings in Hilbert spaces, Nonlinear Anal. 54 (2003) 1417–1426], J.S. Jung [Iterative approaches to common fixed points of nonexpansive mappings in Banach spaces, J. Math. Anal. Appl. 302 (2005) 509–520], H.K. Xu [Viscosity approximation methods for nonexpansive mappings, J. Math. Anal. Appl. 298 (2004) 279–291] and O’Hara et al. [J.G. O’Hara, P. Pillay, H.-K. Xu, Iterative approaches to convex feasibility problem in Banach space, Nonlinear Anal. Available online 20 October 2005. doi:10.1016/j.na.2005.07.36].