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

Let EE be a real reflexive Banach space which admits a weakly sequentially continuous duality mapping from EE to E∗E∗, and KK be a nonempty closed convex subset of EE. Suppose that {Tn}(n=1,2,…) is a uniformly asymptotically regular sequence of nonexpansive mappings from KK into itself such that F≔⋂n=1∞F(Tn)≠0̸. For arbitrary initial value x0∈Kx0∈K and fixed contractive mapping f:K→Kf:K→K, define iteratively a sequence {xn}{xn} as follows: xn+1=λn+1f(xn)+(1−λn+1)Tn+1xn,n≥0, where {λn}⊂(0,1){λn}⊂(0,1) satisfies limn→∞λn=0limn→∞λn=0 and ∑n=1∞λn=∞. We prove that {xn}{xn} converges strongly to p∈Fp∈F, as n→∞n→∞, where pp is the unique solution in FF to the following variational inequality: 〈(I−f)p,j(p−u)〉≤0for 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].