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

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.

___

  • [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.