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

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. |

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