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On the convergence of the Ishikawa iteration in the class of quasi contractive operators. (English) Zbl 1100.47054
Let $E$ be a Banach space, $K\subset E$ a closed convex subset, and $x_0\in K$. Let $\{\alpha _n\},\{\beta _n\}\subset [0,1]$, and let $T\: K\to K$. The Ishikawa iteration procedure [{\it S. Ishikawa}, Proc. Am. Math. Soc. 44, 147--150 (1974; Zbl 0286.47036)] is given by $x_{n+1}=(1-\alpha _n)x_n+\alpha _nTy_n$, $y_n=(1-\beta _n)x_n+\beta _nTx_n$. In the main result of this paper, a condition under which $x_n$ converges to a fixed point of $T$ is proved. This improves a result of {\it B. E. Rhoades} [J. Math. Anal. Appl. 56, 741--750 (1976; Zbl 0353.47029)], where $E$ was supposed to be uniformly convex and the assumptions on the sequence $\{\alpha _n\}$ were stronger than in the paper under review.

##### MSC:
 47J25 Iterative procedures (nonlinear operator equations) 47H10 Fixed-point theorems for nonlinear operators on topological linear spaces 54H25 Fixed-point and coincidence theorems in topological spaces
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