%0 Journal Article
%T A Note on Convergence of a Sequence and Its Applications to Geometry of Banach Spaces
%A Hemant Kumar Pathak
%J Advances in Pure Mathematics
%P 33-41
%@ 2160-0384
%D 2011
%I Scientific Research Publishing
%R 10.4236/apm.2011.13009
%X 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.
%K Locally Quasi-Nonexpansive
%K Biased Quasi-Nonexpansive
%K Conditionally Biased 
Quasi-Nonexpansive
%K Drop
%K Super Drop
%U http://www.scirp.org/journal/PaperInformation.aspx?PaperID=5186