Statistically Convergent Double Sequence Spaces in 2-Normed Spaces Defined by Orlicz Function

ABSTRACT

The concept of statistical convergence was introduced by Stinhauss [1] in 1951. In this paper, we study con- vergence of double sequence spaces in 2-normed spaces and obtained a criteria for double sequences in 2-normed spaces to be statistically Cauchy sequence in 2-normed spaces.

The concept of statistical convergence was introduced by Stinhauss [1] in 1951. In this paper, we study con- vergence of double sequence spaces in 2-normed spaces and obtained a criteria for double sequences in 2-normed spaces to be statistically Cauchy sequence in 2-normed spaces.

Cite this paper

nullV. Khan and S. Tabassum, "Statistically Convergent Double Sequence Spaces in 2-Normed Spaces Defined by Orlicz Function,"*Applied Mathematics*, Vol. 2 No. 4, 2011, pp. 398-402. doi: 10.4236/am.2011.24048.

nullV. Khan and S. Tabassum, "Statistically Convergent Double Sequence Spaces in 2-Normed Spaces Defined by Orlicz Function,"

References

[1] H. Stinhaus, “Sur la Convergence Ordinarie et la Convergence Asymptotique,” Colloqium Mathematicum, Vol. 2, No. 1, 1951, pp. 73-74.

[2] H. Fast, “Sur la Convergence Statistique,” Colloqium Mathematicum, Vol. 2, No. 1, 1951, pp. 241-244.

[3] I. J. Schoenberg, “The Integrability of Certain Functions and Related Summability Methods,” American Mathematical Monthly, Vol. 66, No. 5, 1959, pp. 361-375. doi:10.2307/2308747

[4] J. A. Fridy and C. Orhan, “Statistical Limit Superior and Limit Inferior,” Proceedings of the American Mathematical Society, Vol. 125, No. 12, 1997, pp. 3625-3631. doi:10.1090/S0002-9939-97-04000-8

[5] S. G?hler, “2-Merische R?me und Ihre Topological Struktur,” Mathematische Nachrichten, Vol. 26, No. 1-2, 1963, pp. 115-148.

[6] S. G?hler, “Linear 2-Normietre R?me,” Mathematische Nachrichten, Vol. 28, No. 1-2, 1965, pp. 1-43.

[7] S. G?hler, “Uber der Uniformisierbarkeit 2-Merische R?me,” Mathematische Nachrichten, Vol. 28, No. 3-4, 1964, pp. 235-244.

[8] H. Gunawan and Mashadi, “On Finite Dimensional 2-Normed Spaces,” Soochow Journal of Mathematics, Vol. 27, No. 3, 2001, pp. 631-639.

[9] M. Gurdal and S. Pehlivan, “Statistical Convergence in 2-Normed Spaces,” Southeast Asian Bulletin of Mathematics, Vol. 33, No. 2, 2009, pp. 257-264.

[10] A. R. Freedman and I. J. Sember, “Densities and Summability,” Pacific Journal of Mathematics, Vol. 95, 1981, pp. 293-305.

[11] J. A. Fridy, “On Statistical Convergence,” Analysis, Vol. 5, No. 4, 1985, pp. 301-313.

[12] I. J. Maddox, “Sequence Spaces Defined by a Modulus,” Mathematical Proceedings of the Cambridge Philosophical Society, Vol. 100, No. 1, 1986, pp. 161-166. doi:10.1017/S0305004100065968

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

[14] V. A. Khan and Q. M. D. Lohani, “Statistically Pre-Cauchy Sequence and Orlicz Functions,” Southeast Asian Bulletin of Mathematics, Vol. 31, No. 6, 2007, pp. 1107- 1112.

[15] V. A. Khan, “On a New Sequence Space Defined by Orlicz Functions,” Communication, Faculty of Science, University of Ankara, Series Al, Vol. 57, No. 2, 2008, pp. 25-33.

[16] V. A. Khan, “On a New Sequence Space Related to the Orlicz Sequence Space,” Journal of Mathematics and Its Applications, Vol. 30, 2008, pp. 61-69.

[17] V. A. Khan, “On a New Sequence Spaces Defined by Musielak Orlicz Functions,” Studia Mathematica, Vol. 55 No. 2, 2010, pp. 143-149.

[18] V. A. Khan, “Quasi almost Convergence in a Normed Space for Double Sequences,” Thai Journal of Mathematics, Vol. 8, No. 1, 2010, pp. 227-231.

[19] V. A. Khan and S. Tabassum, “Statistically Pre-Cauchy Double Sequences and Orlicz Functions,” Accepted by Southeast Asian Bulletin of Mathematics.

[20] V. A. Khan and S. Tabassum, “Some Vector Valued Mul- tiplier Difference Double Sequence Spaces in 2-Normed Spaces Defined by Orlicz Function,” Submitted to Journal of Mathematics and Applications.

[21] F. Moricz and B. E. Rhoades, “Almost Convergence of Double Sequences and Strong Regularity of Summability Matrices,” Mathematical Proceedings of the Cambridge Philosophical Society, Vol. 104, No. 2, 1988, pp. 283-294. doi:10.1017/S0305004100065464

[1] H. Stinhaus, “Sur la Convergence Ordinarie et la Convergence Asymptotique,” Colloqium Mathematicum, Vol. 2, No. 1, 1951, pp. 73-74.

[2] H. Fast, “Sur la Convergence Statistique,” Colloqium Mathematicum, Vol. 2, No. 1, 1951, pp. 241-244.

[3] I. J. Schoenberg, “The Integrability of Certain Functions and Related Summability Methods,” American Mathematical Monthly, Vol. 66, No. 5, 1959, pp. 361-375. doi:10.2307/2308747

[4] J. A. Fridy and C. Orhan, “Statistical Limit Superior and Limit Inferior,” Proceedings of the American Mathematical Society, Vol. 125, No. 12, 1997, pp. 3625-3631. doi:10.1090/S0002-9939-97-04000-8

[5] S. G?hler, “2-Merische R?me und Ihre Topological Struktur,” Mathematische Nachrichten, Vol. 26, No. 1-2, 1963, pp. 115-148.

[6] S. G?hler, “Linear 2-Normietre R?me,” Mathematische Nachrichten, Vol. 28, No. 1-2, 1965, pp. 1-43.

[7] S. G?hler, “Uber der Uniformisierbarkeit 2-Merische R?me,” Mathematische Nachrichten, Vol. 28, No. 3-4, 1964, pp. 235-244.

[8] H. Gunawan and Mashadi, “On Finite Dimensional 2-Normed Spaces,” Soochow Journal of Mathematics, Vol. 27, No. 3, 2001, pp. 631-639.

[9] M. Gurdal and S. Pehlivan, “Statistical Convergence in 2-Normed Spaces,” Southeast Asian Bulletin of Mathematics, Vol. 33, No. 2, 2009, pp. 257-264.

[10] A. R. Freedman and I. J. Sember, “Densities and Summability,” Pacific Journal of Mathematics, Vol. 95, 1981, pp. 293-305.

[11] J. A. Fridy, “On Statistical Convergence,” Analysis, Vol. 5, No. 4, 1985, pp. 301-313.

[12] I. J. Maddox, “Sequence Spaces Defined by a Modulus,” Mathematical Proceedings of the Cambridge Philosophical Society, Vol. 100, No. 1, 1986, pp. 161-166. doi:10.1017/S0305004100065968

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

[14] V. A. Khan and Q. M. D. Lohani, “Statistically Pre-Cauchy Sequence and Orlicz Functions,” Southeast Asian Bulletin of Mathematics, Vol. 31, No. 6, 2007, pp. 1107- 1112.

[15] V. A. Khan, “On a New Sequence Space Defined by Orlicz Functions,” Communication, Faculty of Science, University of Ankara, Series Al, Vol. 57, No. 2, 2008, pp. 25-33.

[16] V. A. Khan, “On a New Sequence Space Related to the Orlicz Sequence Space,” Journal of Mathematics and Its Applications, Vol. 30, 2008, pp. 61-69.

[17] V. A. Khan, “On a New Sequence Spaces Defined by Musielak Orlicz Functions,” Studia Mathematica, Vol. 55 No. 2, 2010, pp. 143-149.

[18] V. A. Khan, “Quasi almost Convergence in a Normed Space for Double Sequences,” Thai Journal of Mathematics, Vol. 8, No. 1, 2010, pp. 227-231.

[19] V. A. Khan and S. Tabassum, “Statistically Pre-Cauchy Double Sequences and Orlicz Functions,” Accepted by Southeast Asian Bulletin of Mathematics.

[20] V. A. Khan and S. Tabassum, “Some Vector Valued Mul- tiplier Difference Double Sequence Spaces in 2-Normed Spaces Defined by Orlicz Function,” Submitted to Journal of Mathematics and Applications.

[21] F. Moricz and B. E. Rhoades, “Almost Convergence of Double Sequences and Strong Regularity of Summability Matrices,” Mathematical Proceedings of the Cambridge Philosophical Society, Vol. 104, No. 2, 1988, pp. 283-294. doi:10.1017/S0305004100065464