Approximation methods for common fixed points of infinite countable family of nonexpansive mappings

Let EE be a real strictly convex and reflexive Banach space with a uniformly Gâteaux differentiable norm and CC be a nonempty closed convex subset of EE. Consider the iterative sequence xn+1=λn+1f(xn)+βxn+(1−β−λn+1)Wnxn,xn+1=λn+1f(xn)+βxn+(1−β−λn+1)Wnxn, where WnWn is the WW-mapping generated by an infinite countable family of nonexpansive mappings Tn,Tn−1,…,T1Tn,Tn−1,…,T1 and αn,αn−1,…,α1αn,αn−1,…,α1 such that the common fixed point sets F≔⋂n=1∞F(Tn)≠0̸ and f:C→Cf:C→C is a given contractive mapping. Under very mild conditions on the parameters, we prove that {xn}{xn} converges strongly to p∈Fp∈F where pp is the unique solution in FF to the following variational inequality: 〈(I−f)p,j(p−x∗)〉≤0for all x∗∈F.