Viscosity approximation methods for nonexpansive mapping sequences in Banach spaces

Let E be a real strictly convex Banach space with a uniformly Gâteaux differentiable norm, and K be a nonempty closed convex subset of E. Suppose that {Tn}(n=1,2,…) is a uniformly asymptotically regular sequence of nonexpansive mappings from K to itself such that F≔⋂n=1∞F(Tn)≠0̸. For arbitrary initial value x0∈K and fixed contractive mapping f:K→K, define iteratively a sequence {xn} as follows: xn+1=λn+1f(xn)+(1−λn+1)Tn+1xn,n≥0, where {λn}⊂(0,1) satisfies limn→∞λn=0 and ∑n=1∞λn=∞. Suppose for any nonexpansive mapping T:K→K, {zt} strongly converges to a fixed point z of T as t→0, where {zt} is the unique element of K which satisfies zt=tf(zt)+(1−t)Tzt. Then as n→∞, xn→z. Our results extend and improve the corresponding ones of 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] and J.S. Jung [Iterative approaches to common fixed points of nonexpansive mappings in Banach spaces, J. Math. Anal. Appl. 302 (2005) 509-520] and H.K. Xu [Viscosity approximation methods for nonexpansive mappings, J. Math. Anal. Appl. 298 (2004) 279-291].