ON SOME NEW DIFFERENCE SEQUENCE SPACES DERIVED BY USING RIESZ MEAN AND A MUSIELAK-ORLICZ FUNCTION

Year 2016, Volume: 4 Issue: 2, 56 - 69, 01.10.2016

Kuldip Raj , Renu Anand

Abstract

In this paper we introduce new difference sequence spaces $r^{q}(\mathcal{M},\\ \Delta^{m}_{n},u,p)$ by using Riesz mean and Musielak-Orlicz function. We also make an effort to study some topological properties and compute $\alpha-,\beta-$ and $ \gamma- $ duals of these spaces. Finally, we study matrix transformations on newly formed spaces.

Keywords

sequence space of non-absolute type, Musielak-Orlicz function, paranorm space, matrix transformations

References

• [1] A. Esi, Some new sequence spaces dened by Orlicz Functions, Bull. Inst. Math. Acad. Sinica, 27 (1999), 71-76.
• [2] M. Et and A. Esi, On Kothe-Toeplitz duals of generalized dierence sequence spaces, Bull. Malays. Math. Sci. Soc., 23 (2000), 25-32.
• [3] A. Esi, B. C. Tripathy and B. Sharma, On some new type generalized difference sequence spaces, Math. Slovaca, 57 (2007), 1-8.
• [4] A. Esi and Isik Mahmut, Some generalized difference sequence spaces, Thai J. Math., 3 (2005) 241-247.
• [5] M. Et and R. Colak, On some generalized sequence spaces, Soochow. J. Math., 21 (1995), 377-386.
• [6] K. G. Gross Erdmann, Matrix transformations between the sequence spaces of Maddox, J. Math. Anal. Appl., 180 (1993), 223- 238.
• [7] E. Herawati, M. Mursaleen and I. E. Supama Wijayanti, Order matrix transformations on some Banach lattice valued sequence spaces, Appl. Math. Comput., 247 (2014), 1122-1128.
• [8] H. Kzmaz, On certain sequence spaces, Canad. Math-Bull., 24 (1981), 169-176.
• [9] J. Lindenstrauss and L. Tzafriri, On Orlicz sequence spaces, Israel J. Math., 10 (1971), 379-390.
• [10] C. G. Lascarides and I. J. Maddox, Matrix transformations between some classes of sequences, Proc. Camb. Phil. Soc., 68 (1970), 99-104.
• [11] I. J. Maddox, Elements of Functional Analysis, The University Press, Cambridge, 1988.
• [12] I. J. Maddox, Paranormed sequence spaces generated by innite matrices, Proc. Camb. Phil. Soc., 64 (1968), 335-340.
• [13] I. J. Maddox, Spaces of strongly summable sequences, Quart. J. Math. Oxford, 18 (1967), 345-355.
• [14] L. Maligranda, Orlicz spaces and interpolation, Seminars in Mathematics 5, Polish Academy of Science, 1989.
• [15] M. Mursaleen, K. Raj and S. K.Sharma, Some spaces of difference sequences and Lacunary statistical convergence in n-normed spaces dened by a sequence of Orlicz functions, Miskolc Math. Notes, 16 (2015), 283-304.
• [16] M. Mursaleen, S. K. Sharma, A. Kilicman, Sequence spaces dened by Musielak-Orlicz function over n-normed spaces, Abstr. Appl. Anal., 27 (2013), 47-58.
• [17] M. Mursaleen, S. K Sharma, A. Kilicman, New class of generalized seminormed sequence spaces, Abstr. Appl. Anal., 2014, Article ID 461081, 7 pages.
• [18] M. Mursaleen, S. K Sharma, S. A. Mohiuddine and A. Kilicman, New difference sequence spaces dened by Musielak-Orlicz function, Abstr. Appl. Anal. 2014.
• [19] J. Musielak, Orlicz spaces and modular spaces, Lecture notes in Mathematics, 1034 (1983). [20] S. A. Mohiuddine, K. Raj and A. Alotaibi, Generalized spaces of double sequences for Orlicz functions and bounded regular matrices over n-normed spaces, J. Inequal. Appl., 2014, 2014:332.
• [21] S. A. Mohiuddine, M. Mursaleen and A. Alotaibi, Compact operators for almost conservative and strongly conservative matrices, Abstr. Appl. Anal. 2014, Art. ID 567317, 6 pp.
• [22] G. M. Petersen, Regular matrix transformations, McGraw-Hill, London, 1966.
• [23] K. Raj, S. K. Sharma and A. Gupta, Some difference paranormed sequence spaces over n-normed spaces dened by Musielak-Orlicz function, Kyungpook Math. J., 54 (2014), 73-86.
• [24] K. Raj and S.K.Sharma, Some seminormed diffrence sequence spaces dened by Musielak Orlicz function over n-normed spaces, J. Math. Appl., 38 (2015), 125-141.
• [25] K. Raj and M. Arsalan Khan, Some spaces of double sequences their duals and matrix transformations , Azerb. J. Math., 6 (2016), 19pp.
• [26] N. A. Sheikh and A. H. Ganie, A new paranormed sequence space and some matrix transformations , Acta Math. Acad. Paedago. Nyregy., 28 (2012), 47-58.
• [27] N. A. Sheikh and A. H. Ganie, On the sequence space l(p, s) and some matrix transformations , Nonlinear func. Anal. Appl., 18 (2013), 253-258.
• [28] B. C. Tripathy, A. Esi and T. Balakrushna, On a new type of generalized difference Cesaro sequence spaces, Soochow J. Math., 31 (2005), 333-340.
• [29] O. Toeplitz, Uberallegemeine Lineare mittelbildungen, Prace Math. Fiz., 22 (1991), 113-119.
• [30] C. S. Wang, On Norlund sequence spaces, Tamkang J. Math., 9 (1978), 269-274.
• [31] A. Wilansky, Summability through Functional Analysis, North-Holland Math. Stud., 85 (1984).