# zbMATH — the first resource for mathematics

Mann and Ishikawa iterations with errors for asymptotically nonexpansive mappings. (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 $\|T^nx- T^nq\|\leq k_n\|x-y\|$ 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.

##### MSC:
 47H10 Fixed-point theorems 47J25 Iterative procedures involving nonlinear operators 47H09 Contraction-type mappings, nonexpansive mappings, $$A$$-proper mappings, etc.
Full Text:
##### References:
 [1] Rhoades, B.E., Fixed point iterations for certain nonlinear mappings, J. math. anal. appl., 183, 118-120, (1994) · Zbl 0807.47045 [2] Schu, J., Iterative construction of fixed points of asymptotically nonexpansive mappings, J. math. anal. appl., 158, 407-413, (1991) · Zbl 0734.47036 [3] Liu, L.S., Ishikawa and Mann iterative process with errors for nonlinear strongly accretive mappings in Banach spaces, J. math. anal. appl., 194, 114-125, (1995) · Zbl 0872.47031 [4] Ishikawa, S., Fixed points by a new iteration method, (), 147-150 · Zbl 0286.47036 [5] Mann, W.R., Mean value methods in iteration, (), 506-510 · Zbl 0050.11603 [6] Xu, H.K., Inequalities in Banach spaces with applications, Nonlinear anal., 16, 1127-1138, (1991) · Zbl 0757.46033 [7] Tan, K.K.; Xu, H.K., Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process, J. math. anal. appl., 178, 301-308, (1993) · Zbl 0895.47048 [8] Liu, Q.H., Convergence theorems of the sequence of iterates for asymptotically demicontractive and hemicontractive mappings, Nonlinear analysis, 26, 1835-1842, (1996) · Zbl 0861.47047 [9] Goebel, K.; Kirk, W.A., A fixed point theorem for asymptotically nonexpansive mappings, (), 171-174 · Zbl 0256.47045
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.