On convergence of a sequence in complete metric spaces and its applications to some iterates of quasi-nonexpansive mappings

The purpose of this paper is to establish some theorems on convergence of a sequence in complete metric spaces. As applications, some results of Ghosh and Debnath [J. Math. Anal. Appl. 207 (1997) 96–103], Kirk [Ann. Univ. Mariae Curie-Skłodowska Sect. A LI.2, 15 (1997) 167–178] and Petryshyn and Williamson [J. Math. Anal. Appl. 43 (1973) 459– 497] are obtained from our results as special cases. Also, we give comments on some results in [J. Math. Anal. Appl. 207 (1997) 96–103, J. Math. Anal. Appl. 43 (1973) 459– 497]. Some examples are introduced to support our comments.  2002 Elsevier Science (USA). All rights reserved.