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A Note on Convergence of a Sequence and Its Applications to Geometry of Banach Spaces

Author(s): Hemant Kumar Pathak

Journal: Advances in Pure Mathematics
ISSN 2160-0368

Volume: 01;
Issue: 03;
Start page: 33;
Date: 2011;
Original page

Keywords: Locally Quasi-Nonexpansive | Biased Quasi-Nonexpansive | Conditionally Biased Quasi-Nonexpansive | Drop | Super Drop

The purpose of this note is to point out several obscure places in the results of Ahmed and Zeyada [J. Math. Anal. Appl. 274 (2002) 458-465]. In order to rectify and improve the results of Ahmed and Zeyada, we introduce the concepts of locally quasi-nonexpansive, biased quasi-nonexpansive and conditionally biased quasi-nonexpansive of a mapping w.r.t. a sequence in metric spaces. In the sequel, we establish some theorems on convergence of a sequence in complete metric spaces. As consequences of our main result, we obtain some results of Ghosh and Debnath [J. Math. Anal. Appl. 207 (1997) 96-103], Kirk [Ann. Univ. Mariae Curie-Sklodowska Sec. A LI.2, 15 (1997) 167-178] and Petryshyn and Williamson [J. Math. Anal. Appl. 43 (1973) 459-497]. Some applications of our main results to geometry of Banach spaces are also discussed.
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