Some ideal-convergent generalized difference sequences in a locally convex
space defined by a Musielak-Orlicz function

B. Hazarika; A. Esi

An ideal $I$ is a family of subsets of positive integers $\mathbb{N}$ which is closed under taking finite unions and subsets of its elements. A sequence $(x_k)$ of real numbers is said to be $I$-convergent to a real number $\ell$, if for each $ \varepsilon> 0$ the set $\{k\in \mathbb{N}\colon |x_{k}-\ell|\geq \varepsilon\}$ belongs to $I$. In this article, we introduce a new class of ideal convergent (shortly $I$-convergent) sequence spaces using %an infinite matrix, a Musielak-Orlicz function and the difference operator in locally convex spaces. We investigate some linear topological structures and algebraic properties of these spaces. We also establish some relations between these sequence spaces.

1. C. Aydin, F. Basar, Some new difference sequence spaces, Appl. Math. Comput., 157 (2004), №3, 677–693.

2. Y.A. Cui, On some geometric properties in Musielak-Orlicz sequence spaces, Lecture Notes in Pure and Applied Mathematics, Marcel Dekker. Inc., New York and Basel, 2003, 213 p.

3. R. ё Colak, M. Et, On some generalized difference sequence spaces and related matrix transformation, Hokkaido Math. J., 26 (1997), №3, 483–492.

4. H. Dutta, Characterization of certain matrix classes involving generalized difference summability spaces, Appl. Sci. APPS, 11 (2009), 60–67.

5. M. Et, R. Colak, On generalized difference sequence spaces, Soochow J. Math., 21 (1995), №4, 377–386.

6. M. Et, Y. Altin, B. Choudhary, B.C. Tripathy, On some classes of sequences defined by sequences of Orlicz functions, Math. Ineq. Appl., 9 (2006), №2, 335–342.

7. H. Fast, Sur la convergence statistique, Colloq. Math., 2 (1951), 241–244.

8. J.A. Fridy, On statistical convergence, Analysis, 5 (1985), 301–313.

9. M. Gungor, M. Et, $\delta^m$-strongly almost summable sequences defined by Orlicz functions, Indian J. Pure Appl. Math., 34 (2003), №8, 1141–1151.

10. M. Gurdal, On ideal convergent sequences in 2-normed spaces, Thai J. Math., 4 (2006), №1, 85–91.

11. B. Hazarika, Some lacunary difference sequence spaces defined by Musielak-Orlicz functions, Asia- European Jour. Math., 4 (2011) №4, 613–626.

12. B. Hazarika, On paranormed ideal convergent generalized difference strongly summable sequence spaces defined over n-normed spaces, ISRN Math. Anal., 2011(2011), №17, doi:10.5402/2011/317423.

13. B. Hazarika, E. Savas, Some I-convergent lambda-summable difference sequence spaces of fuzzy real numbers defined by sequence of Orlicz functions, Math. Compu. Modell., 54 (2011), №11–12, 2986–2998.

14. B. Hazarika, On generalized difference ideal convergence in random 2-normed spaces, Filomat, 26 (2012), №6, 1265–1274.

15. B. Hazarika, On fuzzy real valued generalized difference I-convergent sequence spaces defined by Musielak- Orlicz function, Journal of Intelligent and Fuzzy Systems, 25 (2013), №1, 9–15.

16. V.A. Khan, Q.M.D. Lohani, Some new difference sequence spaces defined by Musielak-Orlocz function, Thai J. Math. 6 (2008), №1, 215–223.

17. H. Kizmaz, On certain sequence spaces, Canad. Math. Bull., 24 (1981), №2, 169–176.

18. P. Kostyrko, T. .Salґat, W. Wilczyґnski, I-convergence, Real Analysis Exchange, 26 (2000-2001), №2, 669–686.

19. M.A. Krasnoselski, Y.B. Rutitsky, Convex functions and Orlicz functions, P. Noordhoff, Groningen, Netherlands, 1961.

20. K. Lindberg, On subspaces of Orlicz sequence spaces, Studia Math., 45 (1973), 119–146.

21. J. Lindenstrauss, L. Tzafriri, On Orlicz sequence spaces, Israel J. Math., 10 (1971), 379–390.

22. M. Mursaleen, M.A. Khan, Qamaruddin, Difference sequence spaces defined by Orlicz functions, Demonstratio. Math., 32 (1999), 145–150.

23. S.D. Parashar, B. Choudhary, Sequence spaces defined by Orlicz functions, Indian J. Pure Appl. Math., 25 (1994), №4, 419–428.

24. W.H. Ruckle, FK-spaces in which the sequence of coordinate vectors is bounded, Canad. J. Math., 25 (1973), 973–978.

25. E. Savas, $\delta^m$-strongly summable sequence spaces in 2-normed spaces defined by ideal convergence and an Orlicz function, Appl. Math. Comput., 217 (2010), 271–276.

26. I.J. Schoenberg, The integrability of certain functions and related summability methods, Amer. Math. Monthly, 66 (1959), 361–375.

27. B.C. Tripathy, B. Hazarika, Paranorm I-convergent sequence spaces, Math. Slovaca, 59 (2009), №4, 485–494.

28. B.C. Tripathy, B. Hazarika, Some I-convergent sequence spaces defined by Orlicz functions, Acta Math. Appl. Sinica, 27 (2011), №1, 149–154.

29. B.C. Tripathy, B. Hazarika, I-monotonic and I-convergent sequences, Kyungpook Mathematical Journal, 51 (2011), №2, 233–239.

30. B.C. Tripathy, M. Et, Y. Altin, Generalized difference sequence spaces defined by Orlicz function in a locally convex space, J. Anal. Appl., 3 (2003), №1, 175–192.

	Some ideal-convergent generalized difference sequences in a locally convex space defined by a Musielak-Orlicz function
Author	B. Hazarika, A. Esi bh_rgu@yahoo.co.in, aesi23@hotmail.com Department of Mathematics, Rajiv Gandhi University, Rono Hills, India; Department of Mathematics, Science and Art Faculty, Adiyaman University, Turkey
Abstract	An ideal $I$ is a family of subsets of positive integers $\mathbb{N}$ which is closed under taking finite unions and subsets of its elements. A sequence $(x_k)$ of real numbers is said to be $I$-convergent to a real number $\ell$, if for each $ \varepsilon> 0$ the set $\{k\in \mathbb{N}\colon \|x_{k}-\ell\|\geq \varepsilon\}$ belongs to $I$. In this article, we introduce a new class of ideal convergent (shortly $I$-convergent) sequence spaces using %an infinite matrix, a Musielak-Orlicz function and the difference operator in locally convex spaces. We investigate some linear topological structures and algebraic properties of these spaces. We also establish some relations between these sequence spaces.
Keywords	$I$-convergence; difference space; Musielak-Orlicz function
Reference	1. C. Aydin, F. Basar, Some new difference sequence spaces, Appl. Math. Comput., 157 (2004), №3, 677–693. 2. Y.A. Cui, On some geometric properties in Musielak-Orlicz sequence spaces, Lecture Notes in Pure and Applied Mathematics, Marcel Dekker. Inc., New York and Basel, 2003, 213 p. 3. R. ё Colak, M. Et, On some generalized difference sequence spaces and related matrix transformation, Hokkaido Math. J., 26 (1997), №3, 483–492. 4. H. Dutta, Characterization of certain matrix classes involving generalized difference summability spaces, Appl. Sci. APPS, 11 (2009), 60–67. 5. M. Et, R. Colak, On generalized difference sequence spaces, Soochow J. Math., 21 (1995), №4, 377–386. 6. M. Et, Y. Altin, B. Choudhary, B.C. Tripathy, On some classes of sequences defined by sequences of Orlicz functions, Math. Ineq. Appl., 9 (2006), №2, 335–342. 7. H. Fast, Sur la convergence statistique, Colloq. Math., 2 (1951), 241–244. 8. J.A. Fridy, On statistical convergence, Analysis, 5 (1985), 301–313. 9. M. Gungor, M. Et, $\delta^m$-strongly almost summable sequences defined by Orlicz functions, Indian J. Pure Appl. Math., 34 (2003), №8, 1141–1151. 10. M. Gurdal, On ideal convergent sequences in 2-normed spaces, Thai J. Math., 4 (2006), №1, 85–91. 11. B. Hazarika, Some lacunary difference sequence spaces defined by Musielak-Orlicz functions, Asia- European Jour. Math., 4 (2011) №4, 613–626. 12. B. Hazarika, On paranormed ideal convergent generalized difference strongly summable sequence spaces defined over n-normed spaces, ISRN Math. Anal., 2011(2011), №17, doi:10.5402/2011/317423. 13. B. Hazarika, E. Savas, Some I-convergent lambda-summable difference sequence spaces of fuzzy real numbers defined by sequence of Orlicz functions, Math. Compu. Modell., 54 (2011), №11–12, 2986–2998. 14. B. Hazarika, On generalized difference ideal convergence in random 2-normed spaces, Filomat, 26 (2012), №6, 1265–1274. 15. B. Hazarika, On fuzzy real valued generalized difference I-convergent sequence spaces defined by Musielak- Orlicz function, Journal of Intelligent and Fuzzy Systems, 25 (2013), №1, 9–15. 16. V.A. Khan, Q.M.D. Lohani, Some new difference sequence spaces defined by Musielak-Orlocz function, Thai J. Math. 6 (2008), №1, 215–223. 17. H. Kizmaz, On certain sequence spaces, Canad. Math. Bull., 24 (1981), №2, 169–176. 18. P. Kostyrko, T. .Salґat, W. Wilczyґnski, I-convergence, Real Analysis Exchange, 26 (2000-2001), №2, 669–686. 19. M.A. Krasnoselski, Y.B. Rutitsky, Convex functions and Orlicz functions, P. Noordhoff, Groningen, Netherlands, 1961. 20. K. Lindberg, On subspaces of Orlicz sequence spaces, Studia Math., 45 (1973), 119–146. 21. J. Lindenstrauss, L. Tzafriri, On Orlicz sequence spaces, Israel J. Math., 10 (1971), 379–390. 22. M. Mursaleen, M.A. Khan, Qamaruddin, Difference sequence spaces defined by Orlicz functions, Demonstratio. Math., 32 (1999), 145–150. 23. S.D. Parashar, B. Choudhary, Sequence spaces defined by Orlicz functions, Indian J. Pure Appl. Math., 25 (1994), №4, 419–428. 24. W.H. Ruckle, FK-spaces in which the sequence of coordinate vectors is bounded, Canad. J. Math., 25 (1973), 973–978. 25. E. Savas, $\delta^m$-strongly summable sequence spaces in 2-normed spaces defined by ideal convergence and an Orlicz function, Appl. Math. Comput., 217 (2010), 271–276. 26. I.J. Schoenberg, The integrability of certain functions and related summability methods, Amer. Math. Monthly, 66 (1959), 361–375. 27. B.C. Tripathy, B. Hazarika, Paranorm I-convergent sequence spaces, Math. Slovaca, 59 (2009), №4, 485–494. 28. B.C. Tripathy, B. Hazarika, Some I-convergent sequence spaces defined by Orlicz functions, Acta Math. Appl. Sinica, 27 (2011), №1, 149–154. 29. B.C. Tripathy, B. Hazarika, I-monotonic and I-convergent sequences, Kyungpook Mathematical Journal, 51 (2011), №2, 233–239. 30. B.C. Tripathy, M. Et, Y. Altin, Generalized difference sequence spaces defined by Orlicz function in a locally convex space, J. Anal. Appl., 3 (2003), №1, 175–192.
Pages	195-208
Volume	42
Issue	2
Year	2014
Journal	Matematychni Studii
Full text of paper	pdf
Table of content of issue	html

Some ideal-convergent generalized difference sequences in a locally convex space defined by a Musielak-Orlicz function