Convergence of paths and approximation of fixed points of asymptotically nonexpansive mappings

Let $E$ be a real Banach space with a uniformly Gâteaux differentiable norm possessing uniform normal structure, $K$ be a nonempty closed convex and bounded subset of $E$, $T: K \longrightarrow K$ be an asymptotically nonexpansive mapping with sequence $\{k_n\}_n\subset [1, \infty)$. Let $u\in K$ be fixed, $\{t_n\}_n \subset (0, 1)$ be such that $\lim\limits_{n\to \infty}t_n = 1$, $t_nk_n < 1$, and $\lim\limits_{n\to \infty}\frac{k_n - 1}{k_n-t_n} =0$. Define the sequence $\{z_n\}_n$ iteratively by $z_0\in K$, $z_{n+1}= (1-\frac{t_n}{k_n})u + \frac{t_n}{k_n}T^nz_n, \>n= 0, 1, 2, ..._.$ It is proved that, for each integer $n \geq 0$, there is a unique $x_n \in K$ such that $x_n= (1-\frac{t_n}{k_n})u + \frac{t_n}{k_n}T^nx_n.$If, in addition, $\lim\limits_{n\to \infty}\Vert x_n - Tx_n\Vert = 0$ and $\lim\limits_{n\to \infty}\Vert z_n - Tz_n\Vert = 0$, then $\{z_n\}_n$ converges strongly to a fixed point of $T$.