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Approximation of fixed points of strongly pseudocontractive maps without Lipschitz assumption. (English) Zbl 0871.47045

Summary: The following result is shown: Let X be a real Banach space with a uniformly convex dual X * , and let K be a nonempty closed convex and bounded subset of X. Assume that T:KK is a continuous strong pseudocontraction. Let {α n } n=1 and {β n } n=1 be two real sequences satisfying (i) 0<α n ,β n <1 for all n1; (ii) n=1 α n =; and (iii) α n 0, β n 0 as n. Then the Ishikawa iterative sequence {x n } n=1 generated by

(I)x 1 K,x n+1 =(1-α n )x n +α n Ty n ,y n =(1-β n )x n +β n Tx n ,n1,

converges strongly to the unique fixed point of T.

47J25Iterative procedures (nonlinear operator equations)