## 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)

### Citations:

Zbl 0286.47036; Zbl 0353.47029
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