Strong convergence theorems for a semigroup of asymptotically nonexpansive mappings

Let $K$ be a nonempty closed convex subset of a real Banach space $E$ . Let $\mathcal{T}:=\{T\left(t\right):t\ge 0\}$ be a strongly continuous semigroup of asymptotically nonexpansive mappings from $K$ into $K$ with a sequence $\left\{L_t\right\}\subset [1,\infty )$ . Suppose $F\left(\mathcal{T}\right)\ne \varnothing$ . Then, for a given $u\in K$ there exists a sequence $\left\{u_n\right\}\subset K$ such that $u_n=(1‐{\alpha }_n)\frac{1}{t_n}{\int }_0^{t_n}T\left(s\right)u_{n}ds+{\alpha }_{n}u$ , for $n\in ‐{N}$ , where $t_n\in R^{+}$ , $\left\{a_n\right\}\subset (0,1)$ and $\left\{L_t\right\}$ satisfy certain conditions. Suppose, in addition, that $E$ is reflexive strictly convex with a Gâteaux differentiable norm. Then, the sequence $\left\{u_n\right\}$ converges strongly to a point of $F\left(\mathcal{T}\right)$ . Furthermore, an explicit sequence $\left\{x_n\right\}$ which converges strongly to a fixed point of $\mathcal{T}$ is proved.