# zbMATH — the first resource for mathematics

Weak and strong convergence to fixed points of asymptotically nonexpansive mappings. (English) Zbl 0709.47051
One of the main results is: Let E be a uniformly convex Banach space satisfying Opial’s condition, $$\emptyset \neq A\subset E$$ closed bounded and convex and T: $$A\to A$$ asymptotically nonexpansive with sequence $$(k_ n)\in [1,\infty)^ N$$ for which $$\sum^{\infty}_{n=1}(k_ n- 1)<\infty$$. Suppose that $$x_ 1\in A$$ and $$(\alpha_ n)\in [0,1]^ N$$ is bounded away. Then the sequence $$(x_ n)$$ given by $$x_{n+1}=\alpha_ nT^ n(x_ n)+(1-\alpha_ n)x_ n$$ converges weakly to some fixed point of T.
Two similar results are also obtained concerning the strong convergence of the sequence $$(x_ n)$$ to a fixed point of T.
Reviewer: S.L.Singh

##### MSC:
 47H10 Fixed-point theorems 47H09 Contraction-type mappings, nonexpansive mappings, $$A$$-proper mappings, etc. 47J25 Iterative procedures involving nonlinear operators
Full Text:
##### References:
 [1] Zeidler, Nonlinear Functional Analysis and its Applications I, Fixed-Point Theorems (1986) · Zbl 0583.47050 [2] Samanta, J. Indian Math. Soc. 45 pp 251– (1981) [3] DOI: 10.1016/0022-247X(73)90087-5 · Zbl 0262.47038 [4] DOI: 10.2307/2043110 · Zbl 0377.47037 [5] Górnicki, Comment. Math. Univ. Carolin. 30 pp 249– (1989) [6] DOI: 10.2307/2038462 · Zbl 0256.47045 [7] DOI: 10.2307/2043667 · Zbl 0489.47035
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.