Iterative Algorithm for Common Fixed Points of Infinite Family of Nonexpansive Mappings in Banach Spaces

Let C be a nonempty closed convex subset of a real uniformly smooth Banach space X, {Tk}k 1 : C → C an infinite family of nonexpansive mappings with the nonempty set of common fixed points ⋂∞ k 1 Fix Tk , and f : C → C a contraction. We introduce an explicit iterative algorithm xn 1 αnf xn 1 − αn Lnxn, where Ln ∑n k 1 ωk/sn Tk, Sn ∑n k 1 ωk, and wk > 0 with ∑∞ k 1 ωk 1. Under certain appropriate conditions on {αn}, we prove that {xn} converges strongly to a common fixed point x∗ of {Tk}k 1, which solves the following variational inequality: 〈x∗ − f x∗ , J x∗ − p 〉 ≤ 0, p ∈ ⋂∞k 1 Fix Tk , where J is the normalized duality mapping of X. This algorithm is brief and needs less computational work, since it does not involve W-mapping.