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We introduce a new class of asymptotically nonexpansive mappings and study approximating methods for finding their fixed points. We deal with the Krasnosel'skii-Mann-type iterative process. The strong and weak convergence results for self-mappings in normed spaces are presented. We also consider the asymptotically weakly contractive mappings.
Let K be a nonempty closed convex subset of a real Banach space E which has a uniformly Gâteaux differentiable norm and T : K → K be a nonexpansive mapping with F (T ) := {x ∈ K:Assume that {z t } converges strongly to a fixed point z of T as t → 0, where z t is the unique element of K which satisfies z t = tu + (1 − t)T z t for arbitrary u ∈ K. Let {α n } be a real sequence in (0, 1) which satisfies the following conditions: C1: lim α n = 0; C2: α n = ∞. For arbitrary x 0 ∈ K, let the sequence {x n } be defined iteratively byThen, {x n } converges strongly to a fixed point of T .
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