New Lacunary sequence spaces defined by fractional difference operator

Document Type : Research Paper

Author

Department of Mathematics, Cluster University of Jammu, Jammu 180001, J& K, India

Abstract

In the present paper, we introduce new lacunary strong convergent vector-valued sequence spaces defined by fractional difference operator and Musielak-Orlicz function. We make an effort to study some topological properties and also prove some inclusion relations between these spaces.

Keywords

[1] P. Baliarsingh, Some new difference sequence spaces of fractional order and their dual spaces, Appl. Math. Comput.
219 (2013), 9737–9742.
[2] T. Bilgin, Lacunary strong A-convergence with respect to a modulus, Studia Univ. Babes-Bolyai Math. 48 (2001),
39–46.
[3] T. Bilgin, Lacunary strong A-convergence with respect to a sequence of modulus functions, Appl. Math. Comput.
151 (2004), 595–600.
[4] J.S. Connor, A topological and functional analytic approach to statistical convergence, Analysis of divergence.
Birkh¨auser Boston, 1999.
[5] M. Et and R. C¸ olak, On some generalized difference sequence spaces , Soochow J. Math. 21 (1995), 377–386.
[6] H. Fast, Sur la convergence statistique, Colloq. Math. 2 (1951), 241–244.
[7] A.R. Freedman, J.J. Sember and M. Raphael, Some Cesaro-type summability spaces, Proc. London Math. Soc.
37 (1978), 508–520.
[8] J.A. Fridy, On the statistical convergence, Anal. 5 (1985), 301–303.
[9] M. Isık, On statistical convergence of generalized difference sequence spaces, Soochow J. Math. 30 (2004), 197–205.
[10] H. Kızmaz, On certain sequence spaces , Cand. Math. Bull. 24 (1981), 169–176.
[11] E. Kolk, The statistical convergence in Banach spaces, Acta. Comment. Univ. Tartu 928 (1991), 41–52.
[12] J. Lindenstrauss and L. Tzafriri, On Orlicz sequence spaces, Israel J. Math. 10 (1971), 379–390.
[13] I.J. Maddox, Statistical convergence in a locally convex space, Math. Proc. Cambridge Phil. Soc. 104 (1988),
141–145.
[14] E. Malkowsky and E. Savas, Some λ-sequence spaces defined by a modulus, Arch. Math. 36 (2000), 219–228.
[15] L. Maligranda, Orlicz spaces and interpolation, Seminars in Mathematics 5, Polish Academy of Science, 1989.[16] M. Mursaleen, λ-statistical convergence, Math. Slovaca 50 (2000), 111–115.
[17] M. Mursaleen, S. K. Sharma, S. A. Mohiuddine and A. Kili¸cman, New difference sequence spaces defined by
Musielak-Orlicz function, Abstr. Appl. Anal. 2014 (2014), 9 pages.
[18] M. Mursaleen and S. K. Sharma, Entire sequence spaces defined on locally convex Hausdorff topological space,
Iran. J. Sci. Technol. 38 (2014), 105–109.
[19] M. Mursaleen, A. Alotaibi and S.K. Sharma, New classes of generalized seminormed difference sequence spaces,
Abstr. Appl. Anal. 2014 (2014), 11 pages.
[20] M. Mursaleen, A. Alotaibi and S.K. Sharma, Some new lacunary strong convergent vector-valued sequence spaces,
Abstr. Appl. Anal. 2014 (2014), 9 pages.
[21] J. Musielak, Orlicz spaces and modular spaces, Lecture Notes in Mathematics, 1034 (1983).
[22] K. Raj, A. K. Sharma and S. K. Sharma, A Sequence space defined by Musielak-Orlicz functions , Int. J. Pure
Appl. Math. 67 (2011), 475–484.
[23] T. Salat, On statictical convergent sequences of real numbers, Math. Slovaca 30 (1980), 139–150.
[24] E. Savas, Strong almost convergence and almost λ-statistical convergence, Hokkaido Math. J. 29 (2000), 531–566.
[25] I.J. Schoenberg, The integrability of certain functions and related summability methods, Amer. Math. Month. 66
(1959), 361–375.
Volume 13, Issue 2
July 2022
Pages 2413-2424
  • Receive Date: 23 March 2021
  • Revise Date: 08 July 2021
  • Accept Date: 21 August 2021