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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
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References:

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