The general iterative methods for nonexpansive semigroups in Banach spaces

Let E be a real reflexive strictly convex Banach space which has uniformly Gateaux differentiable norm. Let S = {T(s): 0 <= s < infinity} be a nonexpansive semigroup on E such that Fix(S) :=boolean AND(t >= 0) Fix(T(t)) not equal 0, and f is a contraction on E with coefficient 0 < alpha < 1. Let F be delta-strongly accretive and lambda-strictly pseudo-contractive with delta + lambda > 1 and 0 < gamma < min{delta/alpha, 1-root 1-delta/lambda/2 alpha}. When the sequences of real numbers {alpha(n)} and {t(n)} satisfy some appropriate conditions, the three iterative processes given as follows: x(n+1) = alpha(n)gamma f(x(n)) + (I - alpha F-n)T(t(n))x(n), n >= 0, y(n+1) = alpha(n)gamma f(T(t(n))y(n)) + (I - alpha F-n)T(t(n))y(n), n >= 0, and Z(n+1) = T(t(n))(alpha(n)gamma f(z(n)) + (I - alpha F-n)z(n)), n >= 0 converge strongly to (x)over-tilde, where (x)over-tilde is the unique solution in Fix(S) of the variational inequality <(F - gamma f)(x)over-tilde, j(x-(x)over-tilde)> >= 0, x epsilon 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.