The general iterative methods for nonexpansive semigroups in Banach spaces

Let E be a real reflexive Banach space which admits a weakly sequentially continuous duality mapping from E to E∗. Let S {T s : 0 ≤ s < ∞} be a nonexpansive semigroup on E such that Fix S : ⋂t≥0Fix T t / ∅, and f is a contraction on E with coefficient 0 < α < 1. Let F be δ-strongly accretive and λ-strictly pseudocontractive with δ λ > 1 and γ a positive real number such that γ < 1/α 1 − √ 1 − δ/λ . When the sequences of real numbers {αn} and {tn} satisfy some appropriate conditions, the three iterative processes given as follows: xn 1 αnγf xn I − αnF T tn xn, n ≥ 0, yn 1 αnγf T tn yn I − αnF T tn yn, n ≥ 0, and zn 1 T tn αnγf zn I − αnF zn , n ≥ 0 converge strongly to x̃, where x̃ is the unique solution in Fix S of the variational inequality 〈 F − γf x̃, j x − x̃ 〉 ≥ 0, x ∈ Fix S . Our results extend and improve corresponding ones of Li et al. 2009 Chen and He 2007 , and many others.