Abstract and Applied Analysis

A Generalization of Suzuki's Lemma

B. Panyanak and A. Cuntavepanit

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Abstract

Let { z n } , { w n } , and { v n } be bounded sequences in a metric space of hyperbolic type ( X , ) , and let { α n } be a sequence in [ 0 , 1 ] with 0 < lim inf n α n lim sup n α n < 1 . If z n + 1 = α n w n ( 1 - α n ) v n for all n , lim n ( z n , v n ) = 0 , and lim sup n ( ( w n + 1 , w n ) - ( z n + 1 , z n ) ) 0 , then lim n ( 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.

Article information

Source
Abstr. Appl. Anal., Volume 2011 (2011), Article ID 824718, 14 pages.

Dates
First available in Project Euclid: 12 August 2011

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1313171184

Digital Object Identifier
doi:10.1155/2011/824718

Mathematical Reviews number (MathSciNet)
MR2802832

Zentralblatt MATH identifier
1221.54065

Citation

Panyanak, B.; Cuntavepanit, A. A Generalization of Suzuki's Lemma. Abstr. Appl. Anal. 2011 (2011), Article ID 824718, 14 pages. doi:10.1155/2011/824718. https://projecteuclid.org/euclid.aaa/1313171184


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