Viscosity approximation methods for nonexpansive mapping sequences in Banach spaces Original Research Article

Let EE be a real strictly convex Banach space with a uniformly Gâteaux differentiable norm, and KK be a nonempty closed convex subset of EE. Suppose that View the MathML source{Tn}(n=1,2,…) is a uniformly asymptotically regular sequence of nonexpansive mappings from KK to itself such that View the MathML sourceF≔⋂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: View the MathML sourcexn+1=λn+1f(xn)+(1−λn+1)Tn+1xn,n≥0, Turn MathJax on where {λn}⊂(0,1){λn}⊂(0,1) satisfies limn→∞λn=0limn→∞λn=0 and View the MathML source∑n=1∞λn=∞. Suppose for any nonexpansive mapping T:K→KT:K→K, {zt}{zt} strongly converges to a fixed point zz of TT as t→0t→0, where {zt}{zt} is the unique element of KK which satisfies zt=tf(zt)+(1−t)Tztzt=tf(zt)+(1−t)Tzt. Then as n→∞n→∞, xn→zxn→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].