Convergence Theorems of Finite-Step Iteration With Errors for Non-Self Asymptotically Nonexpansive in the Intermediate Sense Mappings

G. S. Saluja

Let $K$ be a nonempty closed convex nonexpansive retract of a uniformly convex Banach space $E$ with $P$ as a nonexpansive retraction. Let $T:K\to E$ be non-self asymptotically nonexpansive in the intermediate sense mapping with $F(T)\neq\emptyset$. Let $\{\alpha_n^{(i)}\}$, $\{\beta_n^{(i)}\}$ and $\{\gamma_n^{(i)}\}$ are sequences in $[0,1]$ with $\alpha_n^{(i)}+\beta_n^{(i)}+\gamma_n^{(i)}=1$ for all $i=1,2,\dots,N$. From arbitrary $x_1\in K$, define the sequence $\{x_n\}$ iteratively by (8), where $\{u_n^{(i)}\}<\infty$ for all $i = 1,2,\dots,N$ are bounded sequences in $K$ with $\sum_{n=1}^{\infty}u_n^{(i)} <\infty$. (i) If the dual $E^*$ of $E$ has the Kadec-Klee property, then $\{x_n\}$ converges weakly to a fixed point of $T$; (ii) if $T$ satisfies condition (A), then $\{x_n\}$ converges strongly to a fixed point of $T$. The results presented in this paper extend and improve the corresponding results of Rhoades [1], Chidume et al. [4, 6], Schu [11, 12], Osilike and Aniagbosor [18], Tan and Xu [17], Plubtieng and Wangkeeree [21], Su and Qin [31] and many others.