CONVERGENCE OF ITERATIVE SCHEMES FOR NONEXPANSIVE MAPPINGS

Abstract

Let K be a nonempty closed convex subset of a real uniformly convex Banach space X and suppose T : K → K is a nonexpansive mapping with the nonempty fixed point set Fix(T). Let [Formula: see text], [Formula: see text] and [Formula: see text] be sequences in [0, 1] such that [Formula: see text][Formula: see text][Formula: see text] for some constants a, b, α, β, and γ. Let x0 ∈ K be any initial point. Then it is proved that the implicit iteration [Formula: see text] defined by [Formula: see text] converges weakly to a fixed point of T. Furthermore, it is generalized that if [Formula: see text] is a finite family of nonexpansive self-mappings of K with the nonempty common fixed points set [Formula: see text] and if the parameters [Formula: see text], [Formula: see text], and [Formula: see text] satisfy the conditions (0.1), (0.2), (0.3), and [Formula: see text] for some constant c , then the modified implicit iteration [Formula: see text] defined by [Formula: see text] where Tn = Tn(modN), converges weakly to a common fixed point of the family [Formula: see text]. The results presented in this paper improve and extend the corresponding results of Z. Opial [11], S. Reich [13], H.-K. Xu and R. G. Ori [20], and Zhao et al. [21].