The general iterative methods for nonexpansive semigroups in Banach spaces

Let E be a real reflexive strictly convex Banach space which has uniformly Gâteaux differentiable norm. Let $𝒮=\{T(s):0\le s\le \mathrm{\infty }\}$ be a nonexpansive semigroup on E such that $\mathrm{Fix}(𝒮):={\cap }_{t\ge 0}\mathrm{Fix}(T(t))\ne \varnothing$ , and f 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 $0<\mathrm{\gamma <!--\; ?\; -->}<min\left\{\frac{\mathrm{\delta <!--\; d\; -->}}{\mathrm{\alpha <!--\; a\; -->}},\frac{1-<!--\; -\; -->\sqrt{\frac{1-<!--\; -\; -->\mathrm{\delta <!--\; d\; -->}}{\mathrm{\lambda <!--\; ?\; -->}}}}{\mathrm{\alpha <!--\; a\; -->}}\right\}$ . When the sequences of real numbers {α n } and {t n } satisfy some appropriate conditions, the three iterative processes given as follows : $\begin{array}{l}{x}_{n+1}={\mathrm{\alpha <!--\; a\; -->}}_n\mathrm{\gamma <!--\; ?\; -->}f(x_n)+(I-<!--\; -\; -->{\mathrm{\alpha <!--\; a\; -->}}_{n}F)T(t_n)x_n,n\ge <!--\; =\; -->0,\\ y_{n+1}={\mathrm{\alpha <!--\; a\; -->}}_n\mathrm{\gamma <!--\; ?\; -->}f(T(t_n)y_n)+(I-<!--\; -\; -->{\mathrm{\alpha <!--\; a\; -->}}_{n}F)T(t_n)y_n,n\ge <!--\; =\; -->0,\end{array}$ and $z_{n+1}=T(t_n)({\alpha }_n\gamma f(z_n)+(I‐{\alpha }_{n}F)z_n),n\ge 0$ converge strongly to $\tilde{x}$ , where $\tilde{x}$ is the unique solution in $\mathrm{Fix}(𝒮)$ of the variational inequality $⟨<!--\; ?\; -->(F-<!--\; -\; -->\mathrm{\gamma <!--\; ?\; -->}f)\stackrel{˜<!--\; ˜\; -->}{x},j(x-<!--\; -\; -->\stackrel{˜<!--\; ˜\; -->}{x})⟩<!--\; ?\; -->\ge <!--\; =\; -->0,x\in <!--\; ?\; -->Fix({S}).$ 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.