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

Abstract EN

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=∑k=1n(ωk/sn)Tk,Sn=∑k=1nωk, 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.