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>0 F(T(t))≠∅,f:C→C, be a fixed contractive mapping. With some appropriate conditions on {α n } and {t n }, two strongly convergent theorem for the following implicit and explicit viscosity iterative schemes {x n } are proved: x n =α n f(x n )+(1−α n )T(t n )x n , for n∈N, x n+1=α n f(x n )+(1−α n )T(t n )x n , for n∈N, and the cluster point of {x n } 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).