Some idealconvergent generalized difference sequences in a locally convex space defined by a MusielakOrlicz function 

Author 
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 MusielakOrlicz 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; MusielakOrlicz function

Reference 
1. C. Aydin, F. Basar, Some new difference sequence spaces, Appl. Math. Comput., 157 (2004), ¹3, 677–693.
2. Y.A. Cui, On some geometric properties in MusielakOrlicz sequence spaces, Lecture Notes in Pure and Applied Mathematics, Marcel Dekker. Inc., New York and Basel, 2003, 213 p. 3. R. ¸ Colak, M. Et, On some generalized difference sequence spaces and related matrix transformation, Hokkaido Math. J., 26 (1997), ¹3, 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), ¹4, 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), ¹2, 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), ¹8, 1141–1151. 10. M. Gurdal, On ideal convergent sequences in 2normed spaces, Thai J. Math., 4 (2006), ¹1, 85–91. 11. B. Hazarika, Some lacunary difference sequence spaces defined by MusielakOrlicz functions, Asia European Jour. Math., 4 (2011) ¹4, 613–626. 12. B. Hazarika, On paranormed ideal convergent generalized difference strongly summable sequence spaces defined over nnormed spaces, ISRN Math. Anal., 2011(2011), ¹17, doi:10.5402/2011/317423. 13. B. Hazarika, E. Savas, Some Iconvergent lambdasummable difference sequence spaces of fuzzy real numbers defined by sequence of Orlicz functions, Math. Compu. Modell., 54 (2011), ¹11–12, 2986–2998. 14. B. Hazarika, On generalized difference ideal convergence in random 2normed spaces, Filomat, 26 (2012), ¹6, 1265–1274. 15. B. Hazarika, On fuzzy real valued generalized difference Iconvergent sequence spaces defined by Musielak Orlicz function, Journal of Intelligent and Fuzzy Systems, 25 (2013), ¹1, 9–15. 16. V.A. Khan, Q.M.D. Lohani, Some new difference sequence spaces defined by MusielakOrlocz function, Thai J. Math. 6 (2008), ¹1, 215–223. 17. H. Kizmaz, On certain sequence spaces, Canad. Math. Bull., 24 (1981), ¹2, 169–176. 18. P. Kostyrko, T. .Sal´at, W. Wilczy´nski, Iconvergence, Real Analysis Exchange, 26 (20002001), ¹2, 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), ¹4, 419–428. 24. W.H. Ruckle, FKspaces 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 2normed 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 Iconvergent sequence spaces, Math. Slovaca, 59 (2009), ¹4, 485–494. 28. B.C. Tripathy, B. Hazarika, Some Iconvergent sequence spaces defined by Orlicz functions, Acta Math. Appl. Sinica, 27 (2011), ¹1, 149–154. 29. B.C. Tripathy, B. Hazarika, Imonotonic and Iconvergent sequences, Kyungpook Mathematical Journal, 51 (2011), ¹2, 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), ¹1, 175–192. 
Pages 
195208

Volume 
42

Issue 
2

Year 
2014

Journal 
Matematychni Studii

Full text of paper  
Table of content of issue 