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

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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.Conflicts of Interest

The authors declare no conflicts of interest.

[1] | M. A. Ahmed and F. M. Zeyad, “On Convergence of a Se-quence in Complete Metric Spaces and its Applications to Some Iterates of Quasi-Nonexpansive Mappings,” Journal of Mathematical Analysis and Applications, Vol. 274, No. 1, 2002, pp. 458-465. doi:10.1016/S0022-247X(02)00242-1 |

[2] | J.-P. Aubin, “Ap-plied Abstract Analysis,” Wiley-Inter- science, New York, 1977. |

[3] | F. E. Browder and W. V. Petryshyn, “The Solution by Iteration of Nonlinear Functional Equations in Banach Spaces,” Bulletin of the American Mathematical Society, Vol. 272, 1966, pp. 571-575. doi:10.1090/S0002-9904-1966-11544-6 |

[4] | J. Caristi, “Fixed Point Theorems for Mappings Satisfying Inwardness Condi-tions,” Transaction of the American Mathematical Society, Vol. 215, 1976, pp. 241-251. doi:10.1090/S0002-9947-1976-0394329-4 |

[5] | J. B. Diaz and F. T. Metcalf, “On the Set of Sequencial Limit Points of Suc-cessive Approximations,” Transactions of the American Mathematical Society, Vol. 135, 1969, pp. 459-485. |

[6] | W. G. Dotson Jr., “On the Mann Iteration Process,” Transaction of the American Mathematical Society, Vol. 149, 1970, pp. 65-73. doi:10.1090/S0002-9947-1970-0257828-6 |

[7] | W. G. Dotson Jr., “Fixed Points of Quasinon-Expansive Mappings,” Journal of the Australian Mathematical Society, Vol. 13, 1972, pp. 167-170. |

[8] | M. K. Ghosh and L. Debnath, “Convergence of Ishikawa Iterates of Quasi-Nonexpansive Mappings,” Journal of Mathematical Analysis and Applications, Vol. 207, No. 1, 1997, pp. 96-103. doi:10.1006/jmaa.1997.5268 |

[9] | S. Ishi-kawa, “Fixed Points by a New Iteration Method,” Proceedings of the American Mathematical Society, Vol. 44, No. 1, 1974, pp. 147-150. doi:10.1090/S0002-9939-1974-0336469-5 |

[10] | W. A. Kirk, “Remarks on Approximationand Approximate Fixed Points in Metric Fixed Point Theory,” Annales Universitatis Mariae Curie-Sk?odowska, Section A, Vol. 51, No. 2, 1997, pp. 167-178. |

[11] | W. A. Kirk, “Nonexpansive Mappings And Asymptotic Regularity,” Ser. A: Theory Methods, Nonlinear Analysis, Vol. 40, No. 1-8, 2000, pp. 323-332. |

[12] | W. R. Mann, “Mean Valued Methods In Iteration,” Proceedings of the American Mathematical Society, Vol. 4, No. 3, 1953, pp. 506-510. doi:10.1090/S0002-9939-1953-0054846-3 |

[13] | W. V. Pety-shyn and T. E. Williamson Jr., “Strong and Weak Convergence of The Sequence of Successive Approximations for Quasi-Nonexpansive Mappings,” Journal of Mathematical Analysis and Applications, Vol. 43, 1973, pp. 459-497. doi:10.1016/0022-247X(73)90087-5 |

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