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Iterative approximation of fixed points of nonexpansive mappings. (English) Zbl 1095.47034

Let K be a nonempty closed convex subset of a real Banach space E which has a uniformly Gâteaux differentiable norm and let T:KK be a nonexpansive mapping with nonempty set of all fixed points F(T). For a sequence {α n } of real numbers in [0,1] and an arbitrary uK, the sequence {x n } in K defined by x 0 K and

x n+1 =α n u+(1-α n )Tx n ,n0,

was introduced by B. Halpern [Bull. Am. Math. Soc. 73, 957–961 (1967; Zbl 0177.19101)] and subsequently studied by several authors in order to approximate the fixed points of T, see, e.g., the reviewer’s recent monograph [V. Berinde, “Iterative approximation of fixed points” (Efemeride, Baia Mare) (2002; Zbl 1036.47037)].

In the present paper, the authors use the same kind of iterative scheme but with the averaged map S, given by Sx:=(1-δ)x+δTx,δ(0,1), instead of T, and prove a strong convergence theorem for approximating the fixed points of T.

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