The general iterative methods for nonexpansive semigroups in Banach spaces

摘要

Let E be a real reflexive strictly convex Banach space which has uniformly Gateaux differentiable norm. Let S = {T(s) : 0 < s < 00} be a nonexpansive semigroup on E such that Fix(S) := ∩_t≥0Fix(T(t)) ≠ 0, and / is a contraction on E with coefficient 0 < α < 1. Let F be δ-strongly accretive and λ-strictly pseudo-contractive with δ + λ > 1 and 0 < γ < min {δ/α 1-1/2 1-δ/λ/2α} When the sequences of real numbers {a_n} and {t_n} satisfy some appropriate conditions, the three iterative processes given as follows : x_(n+1) = α_nγ f(x_n)+ (I - α_nF)T(t_n)x_n, n ≥ 0, y_(n+1)= α_nγ f(T(t_n)y_n) + (I - α_nF)T(t_n)y_n, n ≥ 0, and Z_(n+1) = T(t_n)(a_nγf(z_n) + (I - α_nF)z_n), n ≥ 0 converge strongly to x, where x is the unique solution in Fix(S) of the variational inequality F-γf)x,j(x-x))≥0, x∈Fix(S). Our results extend and improve corresponding ones of Li et al. (Nonlinear Anal 70:3065-3071, 2009) and Chen and He (Appl Math Lett 20:751-757, 2007) and many others.