Abstract
Let E be a real reflexive Banach space having a weakly continuous duality mapping J~φ~ with a gauge function φ, and let K be a nonempty closed convex subset of E. Suppose that T is a non‐expansive mapping from K into itself such that F (T) ≠ ∅︁. For an arbitrary initial value x~0~ ∈ K and fixed anchor u ∈ K, define iteratively a sequence {x~n~ } as follows:
x~n +1~ = α~n~ u + β~n~ x~n~ + γ~n~ Tx~n~ , n ≥ 0,
where {α~n~ }, {β~n~ }, {γ~n~ } ⊂ (0, 1) satisfies α~n~ +β~n~ + γ~n~ = 1, (C 1) lim~n →∞~ α~n~ = 0, (C 2) ∑^∞^~n =1~ α~n~ = ∞ and (B) 0 < lim inf~n →∞~ β~n~ ≤ lim sup~n →∞~ β~n~ < 1. We prove that {x~n~ } converges strongly to Pu as n → ∞, where P is the unique sunny non‐expansive retraction of K onto F (T). We also prove that the same conclusions still hold in a uniformly convex Banach space with a uniformly Gâteaux differentiable norm or in a uniformly smooth Banach space. Our results extend and improve the corresponding ones by C. E. Chidume and C. O. Chidume [Iterative approximation of fixed points of non‐expansive mappings, J. Math. Anal. Appl. 318, 288–295 (2006)], and develop and complement Theorem 1 of T. H. Kim and H. K. Xu [Strong convergence of modified Mann iterations, Nonlinear Anal. 61, 51–60 (2005)]. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)