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

Hemant Kumar Pathak

doi:10.4236/apm.2011.13009

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.

Highlights

In the last four decades of the last century, there have been a multitude of results on fixed points of nonexpansive and quasi-nonexpansive mappings in Banach spaces (e.g., [5,6,7, 9,10,11]).Our aim in this note is to point out several obscure places in the results of Ahmed and Zeyada [J
The purpose of this note is to point out several obscure places in the results of Ahmed and Zeyada [J
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

Summary

Introduction

In the last four decades of the last century, there have been a multitude of results on fixed points of nonexpansive and quasi-nonexpansive mappings in Banach spaces (e.g., [5,6,7, 9,10,11]). In order to rectify and improve the results of Ahmed and Zeyada, we introduce the concepts of locally quasi-nonexpansive, biased quasinonexpansive and conditionally biased quasi-nonexpansive of a mapping w.r.t. a sequence in metric spaces. The concept of asymptotic regularity was formally introduced by Browder and Petryshyn [3] for mappings in Hilbert spaces. It was defined by Kirk [11] in metric spaces as follows: Definition 1.3. Totically regular if lim d T n x ,T n 1 x 0 for each x D

Main Results

F T Ö at each

Applications to Geometry of Banach Spaces