A Generalization of Suzuki's Lemma. (15th June 2011)

• Record Type: Journal Article
• Authors: Panyanak, B.; Cuntavepanit, A.
• Other Names: Anderson D. Academic Editor.
• Is Part Of: Abstract and applied analysis. Volume 2011(2011)
• Publication Date: 2011-06-15
• Subjects: Mathematical analysis -- Periodicals; Mathematical analysis; Applied Mathematics; Mathematical Analysis; Periodicals; 515.05
• DOI: 10.1155/2011/824718
• Languages: English
• ISSNs: 1085-3375

Abstract

Abstract : Let{ z n }, { w n }, and{ v n } be bounded sequences in a metric space of hyperbolic type( X, d ), and let{ α n } be a sequence in[ 0, 1 ] with0 < lim inf n α n ≤ lim sup n α n < 1 . Ifz n + 1 = α n w n ⊕ ( 1 − α n ) v n for alln ∈ ℕ, lim n d ( z n, v n ) = 0, andlim sup n ( d ( w n + 1, w n ) - d ( z n + 1, z n ) ) ≤ 0, thenlim n d ( w n, z n ) = 0 . This is a generalization of Lemma 2.2 in (T. Suzuki, 2005). As a consequence, we obtain strong convergence theorems for the modified Halpern iterations of nonexpansive mappings in CAT(0) spaces.