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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 [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 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 involving nonlinear operators
47H10 Fixed-point theorems
54H25 Fixed-point and coincidence theorems (topological aspects)
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