Iterative approximation of fixed points of nonexpansive mappings

• Published in: Journal of mathematical analysis and applications 2006-06, Vol.318 (1), p.288-295
• Main Authors: Chidume, C.E., Chidume, C.O.
• Format: Article
• Language: English
• Subjects: Exact sciences and technology; Functional analysis; Global analysis, analysis on manifolds; Mathematical analysis; Mathematics; Sciences and techniques of general use; Topology. Manifolds and cell complexes. Global analysis and analysis on manifolds; Uniformly Gâteaux differentiable norm; Uniformly smooth real Banach spaces
• ISSN: 0022-247X; 1096-0813
• DOI: 10.1016/j.jmaa.2005.05.023

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:Tx=x}≠∅. For a fixed δ∈(0,1), define S:K→K by Sx:=(1−δ)x+δTx, ∀x∈K. Assume that {zt} converges strongly to a fixed point z of T as t→0, where zt is the unique element of K which satisfies zt=tu+(1−t)Tzt for arbitrary u∈K. Let {αn} be a real sequence in (0,1) which satisfies the following conditions: C1:limαn=0; C2:∑αn=∞. For arbitrary x0∈K, let the sequence {xn} be defined iteratively byxn+1=αnu+(1−αn)Sxn. Then, {xn} converges strongly to a fixed point of T.