Advances in Pure Mathematics

Volume 1, Issue 3 (May 2011)

ISSN Print: 2160-0368   ISSN Online: 2160-0384

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

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DOI: 10.4236/apm.2011.13009    5,751 Downloads   13,812 Views  Citations

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ABSTRACT

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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H. Pathak, "A Note on Convergence of a Sequence and Its Applications to Geometry of Banach Spaces," Advances in Pure Mathematics, Vol. 1 No. 3, 2011, pp. 33-41. doi: 10.4236/apm.2011.13009.

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