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 mappings $\mathfrak{I}=\{T(s):s\in S\}$ on a nonempty closed convex subset $C$ of a Banach space with respect to a sequence of asymptotically left invariant means $\{{\mu }_n\}$ defined on an appropriate invariant subspace of $l^{\mathrm{\infty }}(S)$, where $S$ is 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 points $F(\mathfrak{I})$, where $F(\mathfrak{I})=\bigcap \{F(T(s)):s\in S\}$.