I-Pre-Cauchy Double Sequences and Orlicz Functions ()

Vakeel A. Khan, Nazneen Khan, Ayhan Esi, Sabiha Tabassum

Department of Applied Mathematics, Zakir Hussain College of Engineering and Technology, Aligarh Muslim University, Aligarh, India.

Department of Mathematics, Aligarh Muslim University, Aligarh, India.

Department of Mathematics, Science and Art Faculty, Adiyaman University, Adiyaman, Turkey.

**DOI: **10.4236/eng.2013.55A008
PDF HTML XML
3,353
Downloads
5,165
Views
Citations

Department of Applied Mathematics, Zakir Hussain College of Engineering and Technology, Aligarh Muslim University, Aligarh, India.

Department of Mathematics, Aligarh Muslim University, Aligarh, India.

Department of Mathematics, Science and Art Faculty, Adiyaman University, Adiyaman, Turkey.

Let be a double sequence and let *M* be a bounded Orlicz function. We prove that *x* is I-pre-Cauchy if and only if This implies a theorem due to Connor, Fridy and Klin [1], and Vakeel A. Khan and Q. M. Danish Lohani [2]

Share and Cite:

V. Khan, N. Khan, A. Esi and S. Tabassum, "I-Pre-Cauchy Double Sequences and Orlicz Functions," *Engineering*, Vol. 5 No. 5A, 2013, pp. 52-56. doi: 10.4236/eng.2013.55A008.

Conflicts of Interest

The authors declare no conflicts of interest.

[1] | J. Connor, J. A. Fridy and J. Kline, “Statistically PreCauchy Sequence,” Analysis, Vol. 14, 1994, pp. 311-317. |

[2] | A. K. Vakeel and Q. M. Danish Lohani, “Statistically Pre-Cauchy Sequences and Orlicz Functions,” Southeast Asian Bulletin of Mathematics, Vol. 31, No. 6, 2007, pp. 1107-1112. |

[3] | H. Steinhaus, “Sur la Convergence Ordinaire et la Convergence Asymptotique,” Colloquium Mathematicum, Vol. 2, 1951, pp. 73-74. |

[4] | H. Fast, “Sur la Convergence Statistique,” Colloquium Mathematicum, Vol. 2, 1951, pp. 241-244. |

[5] | R. C. Buck, “Generalized Asymptotic Density,” American Journal of Mathematics, Vol. 75, No. 2, 1953, pp. 335346. |

[6] | I. J. Schoenberg, “The Integrability of Certain Functions and Related Summability Methods,” The American Mathematical Monthly, Vol. 66, 1959, pp. 361-375. |

[7] | T. Salat, “On Statistically Convergent Sequences of Real Numbers,” Mathematica Slovaca, Vol. 30, 1980, pp. 139150. |

[8] | J. A. Fridy, “On Statistical Convergence,” Analysis, Vol 5, 1985, pp. 301-311. |

[9] | J. S. Connor, “The Statistical and Strong P-Cesaro Convergence of Sequences,” Analysis, Vol. 8, 1988, pp. 4763. |

[10] | M. Gurdal, “Statistically Pre-Cauchy Sequences and Bounded Moduli,” Acta et Commentationes Universitatis Tarytensis de Mathematica, Vol. 7, 2003, pp. 3-7. |

[11] | T. J. I. Bromwich, “An Introduction to the Theory of Infinite Series,” MacMillan and Co. Ltd., New York, 1965. |

[12] | B. C. Tripathy, “Statistically Convergent Double Sequences,” Tamkang Journal of Mathematics, Vol. 32, No. 2, 2006, pp. 211-221. |

[13] | M. Basarir and O. Solancan, “On Some Double Seuence Spaces,” The Journal of The Indian Academy of Mathematics, Vol. 21, No. 2, 1999, pp. 193-200. |

[14] | I. J. Maddox, “Elements of Functional Analysis,” Cambridge University Press, Cambridge, Cambridge, 1970. |

[15] | J. Lindenstrauss and L. Tzafriri, “On Orlicz Sequence Spaces,” Israel Journal of Mathematics, Vol. 10, No. 3, 1971, pp. 379-390. doi:10.1007/BF02771656 |

[16] | M. Et, “On Some New Orlicz Sequence Spaces,” Journal of Analysis, Vol. 9, 2001, pp. 21-28. |

[17] | S. D. Parashar and B. Choudhary, “Sequence Spaces Defined by Orlicz Function,” Indian Journal of Pure and Applied Mathematics, Vol. 25, 1994, pp. 419-428. |

[18] | B. C. Tripathy and Mahantas, “On a Class of Sequences Related to the lp Space Defined by the Orlicz Functions,” Soochow Journal of Mathematics, Vol. 29, No. 4, 2003, pp. 379-391. |

[19] | A. K. Vakeel and S. Tabassum, “Statistically Pre-Cauchy Double Sequences and Orlicz Functions,” Southeast Asian Bulletin of Mathematics, Vol. 36, No. 2, 2012, pp. 249-254. |

[20] | A. K. Vakeel, K. Ebadullah and A Ahmad, “I-Pre-Cauchy Sequences and Orlicz Functions,” Journal of Mathematical Analysis, Vol. 3, No. 1, 2012, pp. 21-26. |

[21] | P. Kostyrko, T. Salat and W. Wilczynski, “I-Convergence,” Real Analysis Exchange, Vol. 26, No. 2, 2000, pp. 669-686. |

[22] | T. Salat, B. C. Tripathy and M. Ziman, “On Some Properties of I-Convergence,” Tatra Mountains Mathematical Publications, Vol. 28, 2004, pp. 279-286. |

[23] | K. Demirci, “I-Limit Superior and Limit Inferior,” Mathematical Communications, Vol. 6, 2001, pp. 165-172. |

[24] | B. C. Tripathy and B. Hazarika, “Paranorm I-Convergent Sequence Spaces,” Mathematica Slovaca, Vol. 59, No. 4, 2009, pp. 485-494. doi:10.2478/s12175-009-0141-4 |

[25] | B. C. Tripathy and B. Hazarika, “Some I-Convergent Sequence Spaces Defined by Orlicz Function,” Acta Mathematica Applicatae Sinica, Vol. 27, No. 1, 2011, pp. 149154. doi:10.1007/s10255-011-0048-z |

[26] | B. C. Tripathy and B. Hazarika, “I-Monotonic and I-Convergent Sequences,” Kyungpook Mathematical Journal, Vol. 51, No. 2, 2011, pp. 233-239. doi:10.5666/KMJ.2011.51.2.233 |

[27] | A. K. Vakeel, K. Ebadullah and S. Suthep, “On a New I-Convergent Sequence Spaces,” Analysis, Vol. 32, No. 3, 2012, pp. 199-208. doi:10.1524/anly.2012.1148 |

[28] | M. Gurdal and M. B. Huban, “On I-Convergence of Double Sequences in the Topology induced by Random 2Norms,” Matematicki Vesnik, Vol. 65, No. 3, 2013, pp. 1-13. |

Journals Menu

Contact us

customer@scirp.org | |

+86 18163351462(WhatsApp) | |

1655362766 | |

Paper Publishing WeChat |

Copyright © 2022 by authors and Scientific Research Publishing Inc.

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.