Fixed Point Theory and Applications
Volume 2008 (2008), Article ID 363257, 17 pages
Research Article

Approximating Common Fixed Points of Lipschitzian Semigroup in Smooth Banach Spaces

Shahram Saeidi

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.


Let S be a left amenable semigroup, let 𝒮={T(s):sS} 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 n1. 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 (fI)z,J(yz)0, for all yF(𝒮).