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 \(\liminf_{n\to\infty}d(x_n,F(T))=0\).


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