Strong convergence of an iterative algorithm on an infinite countable family of nonexpansive mappings

摘要

Let C be a nonempty closed convex subset of a real strictly convex and reflexive Banach space E which has a uniformly Gâteaux differentiable norm. Let f:C→C be a given contractive mapping and {Tn}n=1∞:C→C be an infinite family of nonexpansive mappings such that the common fixed point sets F:=⋂n=1∞F(Tn)≠∅. Let {αn} and {βn} be two real sequences in [0, 1]. For given x0∈C arbitrarily, let the sequence {xn} be generated iteratively byxn+1=αnf(xn)+βnxn+(1-αn-βn)Wnxn,whereWn is the W-mapping generated by the mappings Tn,Tn-1,…,T1 and ξn,ξn-1,…,ξ1. Suppose the iterative parameters {αn} and {βn} satisfy the following control conditions:(C1)limn→∞αn=0;(C2)∑n=0∞αn=∞;(B5)limsupn→∞βn<1.Then the sequence {xn} converges strongly to p∈F where p is the unique solution in F to the following variational inequality:〈(I-f)p,j(p-x∗)〉⩽0for allx∗∈F.