On the convergence of the Ishikawa iteration in the class of quasi contractive operators. (English) Zbl 1100.47054

Berinde, V.

Acta Math. Univ. Comen., New Ser. 73, No. 1, 119-126 (2004).

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.

Reviewer: Michal Zajac (Bratislava)

MSC:

47J25	Iterative procedures involving nonlinear operators
47H10	Fixed-point theorems
54H25	Fixed-point and coincidence theorems (topological aspects)

Keywords:

Banach space; quasi-contraction; fixed point; Ishikawa iteration; convergence theorem

Citations:

Zbl 0286.47036; Zbl 0353.47029