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

Author(s)
Hemant Kumar Pathak

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.

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.

KEYWORDS

Locally Quasi-Nonexpansive, Biased Quasi-Nonexpansive, Conditionally Biased Quasi-Nonexpansive, Drop, Super Drop

Locally Quasi-Nonexpansive, Biased Quasi-Nonexpansive, Conditionally Biased Quasi-Nonexpansive, Drop, Super Drop

Cite this paper

nullH. 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.

nullH. Pathak, "A Note on Convergence of a Sequence and Its Applications to Geometry of Banach Spaces,"

References

[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

[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