Approximation of common fixed points of an infinite family of nonexpansive mappings in Banach spaces

Let CC be a closed convex subset of a real uniformly smooth and strictly convex Banach space EE. Consider the following iterative algorithm given by {x0=x∈Carbitrarily chosen ,yn=βnxn+(1−βn)Wnxn,xn+1=αnf(xn)+(1−αn)yn,∀n≥0, where ff is a contraction on CC and WnWn is a mapping generated by an infinite family of nonexpansive mappings {Ti}i=1∞. Assume that the set of common fixed points of this infinite family of nonexpansive mappings is not empty.In this paper, we prove that the sequence {xn}{xn} generated by the above iterative algorithm converges strongly to a common fixed point of {Ti}i=1∞, which solves some variational inequality. Our results improve and extend the results announced by many others.