Abstract and Applied Analysis
Volume 2011 (2011), Article ID 824718, 14 pages
http://dx.doi.org/10.1155/2011/824718
Research Article

A Generalization of Suzuki's Lemma

B. Panyanak and A. Cuntavepanit

Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand

Received 4 February 2011; Accepted 27 April 2011

Academic Editor: D. Anderson

Copyright © 2011 B. Panyanak and A. Cuntavepanit. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 ] with 0 < l i m i n f 𝑛 𝛼 𝑛 ≤ l i m s u p 𝑛 𝛼 𝑛 < 1 . If 𝑧 𝑛 + 1 = 𝛼 𝑛 𝑤 𝑛 ⊕ ( 1 − 𝛼 𝑛 ) 𝑣 𝑛 for all 𝑛 ∈ ℕ , l i m 𝑛 𝑑 ( 𝑧 𝑛 , 𝑣 𝑛 ) = 0 , and l i m s u p 𝑛 ( 𝑑 ( 𝑤 𝑛 + 1 , 𝑤 𝑛 ) − 𝑑 ( 𝑧 𝑛 + 1 , 𝑧 𝑛 ) ) ≤ 0 , then l i m 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.