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

Songnian He and Jun Guo

Journal of Applied Mathematics, 2012, vol. 2012, issue 1

Abstract: 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.

Date: 2012
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https://doi.org/10.1155/2012/787419

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Persistent link: https://EconPapers.repec.org/RePEc:wly:jnljam:v:2012:y:2012:i:1:n:787419

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