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Year 2015, Volume: 3 Issue: 2, 84 - 98, 30.10.2015

R. Babu N. Subramanıan P. Thırunavukkarasu

https://doi.org/10.36753/mathenot.421335

Abstract

References

  • [1] B. Altay and F. Başar, Some new spaces of double sequences, J. Math. Anal. Appl., 309(1) (2005), 70-90.
  • [2] F. Başar and Y. Sever, The space Lp of double sequences, Math. J. Okayama Univ, 51 (2009), 149-157.
  • [3] M. Başarır and O. Solancan, On some double sequence spaces, J. Indian Acad. Math., 21(2) (1999), 193-200.
  • [4] T. J. I’A. Bromwich, An introduction to the theory of infinite series Macmillan and Co.Ltd., New York, (1965).
  • [5] J. Cannor, On strong matrix summability with respect to a modulus and statistical convergence, Canad. Math. Bull., 32(2) (1989), 194-198.
  • [6] P. Chandra and B. C. Tripathy, On generalized Kothe-Toeplitz duals of some sequence spaces, Indian Journal of Pure and Applied Mathematics, 33(8) (2002), 1301-1306.
  • [7] A. Gökhan and R. C¸ olak, The double sequence spaces C_2^P(p) and c_2^PB(p), Appl. Math. Comput., 157(2) (2004), 491-501.
  • [8] A. Gökhan and R. C¸ olak, Double sequence spaces L_2^∞, ibid., 160(1) (2005), 147-153.
  • [9] M. Gupta and S. Pradhan, On Certain Type of Modular Sequence space, Turk J. Math., 32 (2008), 293-303.
  • [10] G. Goes and S. Goes. Sequences of bounded variation and sequences of Fourier coefficients, Math. Z., 118 (1970), 93-102.
  • [11] G. H. Hardy, On the convergence of certain multiple series, Proc. Camb. Phil. Soc., 19 (1917), 86-95.
  • [12] H. J. Hamilton, Transformations of multiple sequences, Duke Math. J., 2 (1936), 29-60.
  • [13] H. J. Hamilton, A Generalization of multiple sequences transformation, Duke Math. J., 4 (1938), 343-358.
  • [14] H. J. Hamilton, Preservation of partial Limits in Multiple sequence transformations, Duke Math. J., 4 (1939), 293-297.
  • [15] M. A. Krasnoselskii and Y. B. Rutickii, Convex functions and Orlicz spaces, Gorningen, Netherlands, (1961).
  • [16] P. K. Kamthan and M. Gupta, Sequence spaces and series, Lecture notes, Pure and Applied Mathematics, 65 Marcel Dekker, In c., New York, (1981).
  • [17] J. Lindenstrauss and L. Tzafriri, On Orlicz sequence spaces, Israel J. Math., 10 (1971), 379-390.
  • [18] I. J. Maddox, Sequence spaces defined by a modulus, Math. Proc. Cambridge Philos. Soc., 100(1) (1986), 161-166.
  • [19] F. Moricz, Extentions of the spaces c and c0 from single to double sequences, Acta. Math. Hung., 57(1-2) (1991), 129-136.
  • [20] F. Moricz and B. E. Rhoades, Almost convergence of double sequences and strong regularity of summability matrices, Math. Proc. Camb. Phil. Soc., 104 (1988), 283-294.
  • [21] M. Mursaleen and O. H. H. Edely, Statistical convergence of double sequences, J. Math. Anal. Appl., 288(1) (2003), 223-231.
  • [22] M. Mursaleen, Almost strongly regular matrices and a core theorem for double sequences, J. Math. Anal. Appl., 293(2) (2004), 523-531.
  • [23] M. Mursaleen and O. H. H. Edely,Almost convergence and a core theorem for double sequences, J. Math. Anal. Appl., 293(2) (2004), 532-540.
  • [24] H. Nakano, Concave modulars, J. Math. Soc. Japan, 5 (1953), 29-49.
  • [25] A. Pringsheim, Zurtheorie derzweifach unendlichen zahlenfolgen, Math. Ann., 53 (1900), 289-321.
  • [26] W. H. Ruckle, FK spaces in which the sequence of coordinate vectors is bounded, Canad. J. Math., 25 (1973), 973-978.
  • [27] G. M. Robison, Divergent double sequences and series, Amer. Math. Soc. Trans., 28 (1926), 50-73.
  • [28] N. Subramanian and U. K. Misra, The semi normed space defined by a double gai sequence of modulus function, Fasciculi Math., 45 (2010), 111-120.
  • [29] B. C. Tripathy, On statistically convergent double sequences, Tamkang J. Math., 34(3) (2003), 231-237.
  • [30] A. Turkmenoglu, Matrix transformation between some classes of double sequences, J. Inst. Math. Comp. Sci. Math. Ser., 12(1) (1999), 23-31.
  • [31] B. C. Tripathy and S. Mahanta, On a class of vector valued sequences associated with multiplier sequences, Acta Math. Applicata Sinica (Eng. Ser.), 20(3) (2004), 487-494.
  • [32] B. C. Tripathy and M. Sen, Characterization of some matrix classes involving paranormed sequence spaces, Tamkang Journal of Mathematics, 37(2) (2006), 155-162.
  • [33] B. C. Tripathy and A. J. Dutta, On fuzzy real-valued double sequence spaces 2L_p^F, Mathematical and Computer Modelling, 46 (9-10) (2007), 1294-1299.
  • [34] B. C. Tripathy and B. Sarma, Statistically convergent difference double sequence spaces, Acta Mathematica Sinica, 24(5) (2008), 737-742.
  • [35] B. C. Tripathy and B. Sarma, Vector valued double sequence spaces defined by Orlicz function, Mathematica Slovaca, 59(6) (2009), 767-776.
  • [36] B. C. Tripathy and A. J. Dutta, Bounded variation double sequence space of fuzzy real numbers, Computers and Mathematics with Applications, 59(2) (2010), 1031-1037.
  • [37] B. C. Tripathy and B. Sarma, Double sequence spaces of fuzzy numbers defined by Orlicz function, Acta Mathematica Scientia, 31 B(1) (2011), 134-140.
  • [38] B. C. Tripathy and P. Chandra, On some generalized difference paranormed sequence spaces associated with multiplier sequences defined by modulus function, Anal. Theory Appl. , 27(1) (2011), 21-27.
  • [39] B. C. Tripathy and A. J. Dutta, Lacunary bounded variation sequence of fuzzy real numbers, Journal in Intelligent and Fuzzy Systems, 24(1) (2013), 185-189.
  • [40] J. Y. T. Woo, On Modular Sequence spaces, Studia Math., 48 (1973), 271-289.
  • [41] A. Wilansky, Summability through Functional Analysis, North-Holland Mathematical Studies, North-Holland Publishing, Amsterdam, Vol.85 (1984).
  • [42] M. Zeltser, Investigation of Double Sequence Spaces by Soft and Hard Analitical Methods, Dissertationes Mathematicae Universitatis Tartuensis 25, Tartu University Press, Univ. of Tartu, Faculty of Mathematics and Computer Science, Tartu, 2001.

THE RANDOM OF LACUNARY STATISTICAL ON χ^2 OVER p− METRIC SPACES DEFINED BY MUSIELAK

Year 2015, Volume: 3 Issue: 2, 84 - 98, 30.10.2015

R. Babu N. Subramanıan P. Thırunavukkarasu

https://doi.org/10.36753/mathenot.421335

Abstract

Mursaleen introduced the concepts of statistical convergence in
random 2-normed spaces. Recently Mohiuddine and Aiyup defined the notion
of lacunary statistical convergence and lacunary statistical Cauchy in random
2-normed spaces. In this paper, we define and study the notion of lacunary
statistical convergence and lacunary of statistical Cauchy sequences in random
on χ2 over p− metric spaces defined by Musielak and prove some theorems
which generalizes Mohiuddine and Aiyup results.

Keywords

analytic sequence double sequences χ 2 space Musielak - modulus function Random p− metric space Lacunary sequence Statistical convergence

References

  • [1] B. Altay and F. Başar, Some new spaces of double sequences, J. Math. Anal. Appl., 309(1) (2005), 70-90.
  • [2] F. Başar and Y. Sever, The space Lp of double sequences, Math. J. Okayama Univ, 51 (2009), 149-157.
  • [3] M. Başarır and O. Solancan, On some double sequence spaces, J. Indian Acad. Math., 21(2) (1999), 193-200.
  • [4] T. J. I’A. Bromwich, An introduction to the theory of infinite series Macmillan and Co.Ltd., New York, (1965).
  • [5] J. Cannor, On strong matrix summability with respect to a modulus and statistical convergence, Canad. Math. Bull., 32(2) (1989), 194-198.
  • [6] P. Chandra and B. C. Tripathy, On generalized Kothe-Toeplitz duals of some sequence spaces, Indian Journal of Pure and Applied Mathematics, 33(8) (2002), 1301-1306.
  • [7] A. Gökhan and R. C¸ olak, The double sequence spaces C_2^P(p) and c_2^PB(p), Appl. Math. Comput., 157(2) (2004), 491-501.
  • [8] A. Gökhan and R. C¸ olak, Double sequence spaces L_2^∞, ibid., 160(1) (2005), 147-153.
  • [9] M. Gupta and S. Pradhan, On Certain Type of Modular Sequence space, Turk J. Math., 32 (2008), 293-303.
  • [10] G. Goes and S. Goes. Sequences of bounded variation and sequences of Fourier coefficients, Math. Z., 118 (1970), 93-102.
  • [11] G. H. Hardy, On the convergence of certain multiple series, Proc. Camb. Phil. Soc., 19 (1917), 86-95.
  • [12] H. J. Hamilton, Transformations of multiple sequences, Duke Math. J., 2 (1936), 29-60.
  • [13] H. J. Hamilton, A Generalization of multiple sequences transformation, Duke Math. J., 4 (1938), 343-358.
  • [14] H. J. Hamilton, Preservation of partial Limits in Multiple sequence transformations, Duke Math. J., 4 (1939), 293-297.
  • [15] M. A. Krasnoselskii and Y. B. Rutickii, Convex functions and Orlicz spaces, Gorningen, Netherlands, (1961).
  • [16] P. K. Kamthan and M. Gupta, Sequence spaces and series, Lecture notes, Pure and Applied Mathematics, 65 Marcel Dekker, In c., New York, (1981).
  • [17] J. Lindenstrauss and L. Tzafriri, On Orlicz sequence spaces, Israel J. Math., 10 (1971), 379-390.
  • [18] I. J. Maddox, Sequence spaces defined by a modulus, Math. Proc. Cambridge Philos. Soc., 100(1) (1986), 161-166.
  • [19] F. Moricz, Extentions of the spaces c and c0 from single to double sequences, Acta. Math. Hung., 57(1-2) (1991), 129-136.
  • [20] F. Moricz and B. E. Rhoades, Almost convergence of double sequences and strong regularity of summability matrices, Math. Proc. Camb. Phil. Soc., 104 (1988), 283-294.
  • [21] M. Mursaleen and O. H. H. Edely, Statistical convergence of double sequences, J. Math. Anal. Appl., 288(1) (2003), 223-231.
  • [22] M. Mursaleen, Almost strongly regular matrices and a core theorem for double sequences, J. Math. Anal. Appl., 293(2) (2004), 523-531.
  • [23] M. Mursaleen and O. H. H. Edely,Almost convergence and a core theorem for double sequences, J. Math. Anal. Appl., 293(2) (2004), 532-540.
  • [24] H. Nakano, Concave modulars, J. Math. Soc. Japan, 5 (1953), 29-49.
  • [25] A. Pringsheim, Zurtheorie derzweifach unendlichen zahlenfolgen, Math. Ann., 53 (1900), 289-321.
  • [26] W. H. Ruckle, FK spaces in which the sequence of coordinate vectors is bounded, Canad. J. Math., 25 (1973), 973-978.
  • [27] G. M. Robison, Divergent double sequences and series, Amer. Math. Soc. Trans., 28 (1926), 50-73.
  • [28] N. Subramanian and U. K. Misra, The semi normed space defined by a double gai sequence of modulus function, Fasciculi Math., 45 (2010), 111-120.
  • [29] B. C. Tripathy, On statistically convergent double sequences, Tamkang J. Math., 34(3) (2003), 231-237.
  • [30] A. Turkmenoglu, Matrix transformation between some classes of double sequences, J. Inst. Math. Comp. Sci. Math. Ser., 12(1) (1999), 23-31.
  • [31] B. C. Tripathy and S. Mahanta, On a class of vector valued sequences associated with multiplier sequences, Acta Math. Applicata Sinica (Eng. Ser.), 20(3) (2004), 487-494.
  • [32] B. C. Tripathy and M. Sen, Characterization of some matrix classes involving paranormed sequence spaces, Tamkang Journal of Mathematics, 37(2) (2006), 155-162.
  • [33] B. C. Tripathy and A. J. Dutta, On fuzzy real-valued double sequence spaces 2L_p^F, Mathematical and Computer Modelling, 46 (9-10) (2007), 1294-1299.
  • [34] B. C. Tripathy and B. Sarma, Statistically convergent difference double sequence spaces, Acta Mathematica Sinica, 24(5) (2008), 737-742.
  • [35] B. C. Tripathy and B. Sarma, Vector valued double sequence spaces defined by Orlicz function, Mathematica Slovaca, 59(6) (2009), 767-776.
  • [36] B. C. Tripathy and A. J. Dutta, Bounded variation double sequence space of fuzzy real numbers, Computers and Mathematics with Applications, 59(2) (2010), 1031-1037.
  • [37] B. C. Tripathy and B. Sarma, Double sequence spaces of fuzzy numbers defined by Orlicz function, Acta Mathematica Scientia, 31 B(1) (2011), 134-140.
  • [38] B. C. Tripathy and P. Chandra, On some generalized difference paranormed sequence spaces associated with multiplier sequences defined by modulus function, Anal. Theory Appl. , 27(1) (2011), 21-27.
  • [39] B. C. Tripathy and A. J. Dutta, Lacunary bounded variation sequence of fuzzy real numbers, Journal in Intelligent and Fuzzy Systems, 24(1) (2013), 185-189.
  • [40] J. Y. T. Woo, On Modular Sequence spaces, Studia Math., 48 (1973), 271-289.
  • [41] A. Wilansky, Summability through Functional Analysis, North-Holland Mathematical Studies, North-Holland Publishing, Amsterdam, Vol.85 (1984).
  • [42] M. Zeltser, Investigation of Double Sequence Spaces by Soft and Hard Analitical Methods, Dissertationes Mathematicae Universitatis Tartuensis 25, Tartu University Press, Univ. of Tartu, Faculty of Mathematics and Computer Science, Tartu, 2001.
There are 42 citations in total.

Details

Primary Language English
Journal Section Articles
Authors

R. Babu This is me

N. Subramanıan This is me

P. Thırunavukkarasu This is me

Publication Date October 30, 2015
Submission Date July 2, 2014
Published in Issue Year 2015 Volume: 3 Issue: 2

Cite

APA Babu, R., Subramanıan, N., & Thırunavukkarasu, P. (2015). THE RANDOM OF LACUNARY STATISTICAL ON χ^2 OVER p− METRIC SPACES DEFINED BY MUSIELAK. Mathematical Sciences and Applications E-Notes, 3(2), 84-98. https://doi.org/10.36753/mathenot.421335
AMA Babu R, Subramanıan N, Thırunavukkarasu P. THE RANDOM OF LACUNARY STATISTICAL ON χ^2 OVER p− METRIC SPACES DEFINED BY MUSIELAK. Math. Sci. Appl. E-Notes. October 2015;3(2):84-98. doi:10.36753/mathenot.421335
Chicago Babu, R., N. Subramanıan, and P. Thırunavukkarasu. “THE RANDOM OF LACUNARY STATISTICAL ON χ^2 OVER p− METRIC SPACES DEFINED BY MUSIELAK”. Mathematical Sciences and Applications E-Notes 3, no. 2 (October 2015): 84-98. https://doi.org/10.36753/mathenot.421335.
EndNote Babu R, Subramanıan N, Thırunavukkarasu P (October 1, 2015) THE RANDOM OF LACUNARY STATISTICAL ON χ^2 OVER p− METRIC SPACES DEFINED BY MUSIELAK. Mathematical Sciences and Applications E-Notes 3 2 84–98.
IEEE R. Babu, N. Subramanıan, and P. Thırunavukkarasu, “THE RANDOM OF LACUNARY STATISTICAL ON χ^2 OVER p− METRIC SPACES DEFINED BY MUSIELAK”, Math. Sci. Appl. E-Notes, vol. 3, no. 2, pp. 84–98, 2015, doi: 10.36753/mathenot.421335.
ISNAD Babu, R. et al. “THE RANDOM OF LACUNARY STATISTICAL ON χ^2 OVER p− METRIC SPACES DEFINED BY MUSIELAK”. Mathematical Sciences and Applications E-Notes 3/2 (October 2015), 84-98. https://doi.org/10.36753/mathenot.421335.
JAMA Babu R, Subramanıan N, Thırunavukkarasu P. THE RANDOM OF LACUNARY STATISTICAL ON χ^2 OVER p− METRIC SPACES DEFINED BY MUSIELAK. Math. Sci. Appl. E-Notes. 2015;3:84–98.
MLA Babu, R. et al. “THE RANDOM OF LACUNARY STATISTICAL ON χ^2 OVER p− METRIC SPACES DEFINED BY MUSIELAK”. Mathematical Sciences and Applications E-Notes, vol. 3, no. 2, 2015, pp. 84-98, doi:10.36753/mathenot.421335.
Vancouver Babu R, Subramanıan N, Thırunavukkarasu P. THE RANDOM OF LACUNARY STATISTICAL ON χ^2 OVER p− METRIC SPACES DEFINED BY MUSIELAK. Math. Sci. Appl. E-Notes. 2015;3(2):84-98.

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