Let be a nonempty closed convex subset of a real Banach space which has a uniformly Gâteaux differentiable norm and let be a nonexpansive mapping with nonempty set of all fixed points . For a sequence of real numbers in and an arbitrary , the sequence in defined by and
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 , 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 , given by , instead of , and prove a strong convergence theorem for approximating the fixed points of .