Strong convergence of an iterative method for non‐expansive mappings

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)