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

Abstract EN

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.