Iterative approximation to common fixed points of nonexpansive mapping sequences in reflexive Banach spaces

✍ Scribed by Yisheng Song; Rudong Chen

• Publisher: Elsevier Science
• Year: 2007
• Tongue: English
• Weight: 199 KB
• Volume: 66
• Category: Article
• ISSN: 0362-546X
• DOI: 10.1016/j.na.2005.12.004

✦ Synopsis

Let E be a real reflexive Banach space which admits a weakly sequentially continuous duality mapping from E to E * , and K be a nonempty closed convex subset of E. Suppose that {T n } (n = 1, 2, . . .) is a uniformly asymptotically regular sequence of nonexpansive mappings from K into itself such that F := ∞ n=1 F(T n ) = ∅. For arbitrary initial value x 0 ∈ K and fixed contractive mapping f : K → K , define iteratively a sequence {x n } as follows:

where {λ n } ⊂ (0, 1) satisfies lim n→∞ λ n = 0 and ∞ n=1 λ n = ∞. We prove that {x n } converges strongly to p ∈ F, as n → ∞, where p is the unique solution in F to the following variational inequality:

Our results extend and improve the corresponding ones given by O' Hara et al. [