Mann and Ishikawa iterations with errors for asymptotically nonexpansive mappings (Q1962960)

The paper concerns some generalizations of convergence of special type of iterations of asymptotically nonexpansive mappings in uniformly convex Banach spaces to a fixed point. A map \(T: E\to E\) defined on a subset of a Banach space is said to be asymptotically nonexpansive iff \[ \|T^nx- T^nq\|\leq k_n\|x-y\| \] for all \(x,y\in E\) and \(n\in\mathbb{N}\), where \((k_n)\) is a sequence of real numbers such that \[ k_n\geq 1\quad\text{and}\quad \lim_{n\to\infty} k_n= 1. \] The author considers the convergence of the following iteration process with errors \[ x_{n+1}= (1-\alpha_n) x_n+ \alpha_n T^n y_n+ u_n, \] \[ y_n= (1- \beta_n) x_n+ \beta_n T^nx_n+ v_n, \] where \(\{u_n\}\) and \(\{v_n\}\) are sequences in \(E\) satisfying \[ \sum^\infty_{n= 1}\|u_n\|<\infty\quad\text{and} \quad \sum^\infty_{n=1}\|v_n\|< \infty, \] and \(\{\alpha_n\}\) and \(\{\beta_n\}\) are sequences of real numbers in \([0,1]\). Under some additional assumptions it has been proved that the sequence of iterations \(\{x_n\}\) converges strongly to a fixed point of \(T\). The results presented here are some generalizations of the results obtained in 1994 by B. E. Rhoades. Let's note that some facts (see e.g. Lemma 1, Lemma 6) proved in the paper are very obvious.