## 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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