Fixed Point Theory and Applications
Volume 2008 (2008), Article ID 363257, 17 pages
doi:10.1155/2008/363257
Research Article
Approximating Common Fixed Points of Lipschitzian Semigroup in Smooth Banach Spaces
Department of Mathematics, University of Kurdistan, Sanandaj 416, Kurdistan 66196-64583, Iran
Received 16 August 2008; Accepted 10 December 2008
Academic Editor: Mohamed Khamsi
Copyright © 2008 Shahram Saeidi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
Let S be a left amenable semigroup, let 𝒮={T(s):s∈S} be a representation of S as Lipschitzian mappings from a nonempty compact convex
subset C of a smooth Banach space E into C with a uniform Lipschitzian condition, let {μn} be a strongly left regular sequence of means defined on an 𝒮-stable subspace of l∞(S), let f be a contraction on C, and let {αn}, {βn}, and {γn} be sequences in (0, 1) such that αn+βn+γn=1, for all n. Let xn+1=αnf(xn)+βnxn+γnT(μn)xn, for all n≥1. Then, under suitable hypotheses on the constants, we show that {xn} converges strongly to some z in F(𝒮), the set of common fixed points of 𝒮, which is the unique solution of the variational inequality 〈(f−I)z,J(y−z)〉≤0, for all y∈F(𝒮).