Fixed Point Theory and Applications
Volume 2008 (2008), Article ID 363257, 17 pages
doi:10.1155/2008/363257

Approximating common fixed points of Lipschitzian semigroup in smooth Banach spaces

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).