Strong Convergence Theorems of the General Iterative Methods for Nonexpansive Semigroups in Banach Spaces

Abstract

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 . 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 , where is the unique solution in Fix(𝒮) of the variational inequality , x ∈ Fix(𝒮). Our results extend and improve corresponding ones of Li et al. (2009) Chen and He (2007), and many others.