*(English)*Zbl 0942.47046

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

for all $x,y\in E$ and $n\in \mathbb{N}$, where $\left({k}_{n}\right)$ is a sequence of real numbers such that

The author considers the convergence of the following iteration process with errors

where $\left\{{u}_{n}\right\}$ and $\left\{{v}_{n}\right\}$ are sequences in $E$ satisfying

and $\left\{{\alpha}_{n}\right\}$ and $\left\{{\beta}_{n}\right\}$ are sequences of real numbers in $[0,1]$.

Under some additional assumptions it has been proved that the sequence of iterations $\left\{{x}_{n}\right\}$ 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.

##### MSC:

47H10 | Fixed point theorems for nonlinear operators on topological linear spaces |

47J25 | Iterative procedures (nonlinear operator equations) |

47H09 | Mappings defined by “shrinking” properties |