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Iterative sequences for asymptotically quasi-nonexpansive mappings with error member. (English) Zbl 1001.47034

The paper concerns convergence of Ishikawa iterative sequences \((x_n)_n\) with error members for asymptotically quasi-nonexpansive mappings \(T:E\to E\). Here \(E\) is a nonempty closed convex set of a Banach space and the set \(F(T)\) of fixed points of \(T\) is nonempty. The main results state \((x_n)_n\) converges to a fixed point of \(T\) if and only if \(\liminf_{n\to\infty}d(x_n,F(T))=0\).

MSC:

47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
47J25 Iterative procedures involving nonlinear operators
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References:

[1] Petryshyn, W.V.; Williamson, T.E., Strong and weak convergence of the sequence of successive approximations for quasi-nonexpansive mappings, J. math. anal. appl., 43, 459-497, (1973) · Zbl 0262.47038
[2] Ghosh, M.K.; Debnath, L., Convergence of Ishikawa iterates of quasi-nonexpansive mappings, J. math. anal. appl., 207, 96-103, (1997) · Zbl 0881.47036
[3] Liu, Qihou, Iterative sequences for asymptotic quasi-nonexpansive mapping, J. Math. Anal. Appl, in press. · Zbl 1033.47047
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