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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, KE a closed convex subset, and x 0 K. Let {α n },{β n }[0,1], and let TKK. The Ishikawa iteration procedure [S. Ishikawa, Proc. Am. Math. Soc. 44, 147–150 (1974; Zbl 0286.47036)] is given by x n+1 =(1-α n )x n +α n Ty n , y n =(1-β n )x n +β n Tx 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 {α n } were stronger than in the paper under review.

MSC:
47J25Iterative procedures (nonlinear operator equations)
47H10Fixed point theorems for nonlinear operators on topological linear spaces
54H25Fixed-point and coincidence theorems in topological spaces