Viscosity Approximation Method for Nonexpansive Semigroups in Banach Spaces

Abstract

Let C be a nonempty closed convex subset of a reflexive Banach space E which admits a weakly sequentially continuous duality mapping from E to E∗ and {T(t):t>0} be a nonexpansive semigroup on C such that F=⋂t>0F(T(t))≠∅,f:C→C, be a fixed contractive mapping. With some appropriate conditions on {αn} and {tn}, two strongly convergent theorem for the following implicit and explicit viscosity iterative schemes {xn} are proved: xn=αnf(xn)+(1−αn)T(tn)xn, for \(n\in \mathbb{N} \), xn+1=αnf(xn)+(1−αn)T(tn)xn, for \(n\in \mathbb{N} \), and the cluster point of {xn} is the unique solution in F to the following variational inequality: 〈(I−f)p,j(p−x)〉≤0, ∀x∈F. The idea and method presented in this paper are especially based on the ones of Chen and He (Appl. Math. Lett. 20:751–757, 2007).