Invariant Means and Reversible Semigroup of Relatively Nonexpansive Mappings in Banach Spaces

Abstract

The purpose of this paper is to study modified Halpern type and Ishikawa type iteration for a semigroup of relatively nonexpansive mappingsI={T(s):s∈S}on a nonempty closed convex subsetCof a Banach space with respect to a sequence of asymptotically left invariant means{μn}defined on an appropriate invariant subspace ofl∞(S), whereSis a semigroup. We prove that, given some mild conditions, we can generate iterative sequences which converge strongly to a common element of the set of fixed pointsF(I), whereF(I)=⋂{F(T(s)):s∈S}.