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

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→Cf: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}{αn} and {βn}{βn} be two real sequences in [0, 1]. For given x0∈Cx0∈C arbitrarily, let the sequence {xn}{xn} be generated iteratively byxn+1=αnf(xn)+βnxn+(1-αn-βn)Wnxn,xn+1=αnf(xn)+βnxn+(1-αn-βn)Wnxn,whereWnWn is the W  -mapping generated by the mappings Tn,Tn-1,…,T1Tn,Tn-1,…,T1 and ξn,ξn-1,…,ξ1ξn,ξn-1,…,ξ1. Suppose the iterative parameters {αn}{αn} and {βn}{βn} satisfy the following control conditions:(C1)limn→∞αn=0limn→∞αn=0;(C2)∑n=0∞αn=∞;(B5)limsupn→∞βn<1limsupn→∞βn<1.Then the sequence {xn}{xn} converges strongly to p∈Fp∈F where p is the unique solution in F to the following variational inequality:〈(I-f)p,j(p-x∗)〉⩽0for allx∗∈F.