A New Type Generalized Difference Sequence Space $m\left( \phi,p\right) \left( \Delta_{m}^{n}\right) $

Year 2019, Volume: 2 Issue: 3, 194 - 197, 30.12.2019

Mikail Et , Rifat Colak

Abstract

Let $\left( \phi_{n}\right) $ be a non-decreasing sequence of positive numbers such that $n\phi_{n+1}\leq \left( n+1\right) \phi_{n}$ for all $n\in \mathbb{N}$. The class of all sequences $\left( \phi_{n}\right) $ is denoted by $\Phi$. The sequence space $m\left( \phi \right) $ was introduced by Sargent [1] and he studied some of its properties and obtained some relations with the space $\ell_{p}$. Later on it was investigated by Tripathy and Sen [2] and Tripathy and Mahanta [3]. In this work, using the generalized difference operator $\Delta_{m}^{n}$, we generalize the sequence space $m\left( \phi \right) $ to sequence space $ m\left( \phi,p\right) \left( \Delta _{m}^{n}\right) ,$ give some topological properties about this space and show that the space $m\left( \phi,p\right) \left( \Delta_{m}^{n}\right) $ is a $BK-$space by a suitable norm$.$ The results obtained are generalizes some known results.

Keywords

Difference sequence, BK-space, Symmetric space, Normal space.

References

• [1] W. L. C. Sargent, Some sequence spacess related to $\ell_{p}$ spaces, J. London Math. Soc. 35 (1960), 161-171.
• [2] B. C. Tripathy, M. Sen, On a new class of sequences related to the space $\ell_{p},$; Tamkang J. Math. 33(2) (2002), 167-171.
• [3] B. C. Tripathy, S. Mahanta, On a class of sequences related to the $\ell_{p}$ space defined by Orlicz functions, Soochow J. Math. 29(4) (2003), 379–391.
• [4] H. Kızmaz, On certain Sequence spaces, Canad. Math. Bull. 24(2) (1981), 169-176.
• [5] M. Et, R. Çolak, On generalized difference sequence spaces, Soochow J. Math. 21(4) (1995), 377-386.
• [6] A. Esi, B. C. Tripathy, B. Sarma, On some new type generalized difference sequence spaces, Math. Slovaca 57(5) (2007), 475–482.
• [7] A. Esi, B. C. Tripathy, A New Type Of Difference Sequence Spaces, International Journal of Science & Technology 1(1) (2006), 11-14.
• [8] B . C. Tripathy, A. Esi, B. K. Tripathy, On a new type of generalized difference Cesaro Sequence spaces, Soochow J. Math. 31(3) (2005), 333-340.
• [9] Y. Altin, Properties of some sets of sequences defined by a modulus function, Acta Math. Sci. Ser. B Engl. Ed. 29(2) (2009), 427–434.
• [10] H. Altinok, M. Et, R. Çolak, Some remarks on generalized sequence space of bounded variation of sequences of fuzzy numbers, Iran. J. Fuzzy Syst. 11(5) (2014), 39–46, 109.
• [11] S. Demiriz, C. Çakan, Some topological and geometrical properties of a new difference sequence space. Abstr. Appl. Anal. 2011, Art. ID 213878, 14 pp.
• [12] S. Erdem, S. Demiriz, On the new generalized block difference sequence spaces, Appl. Appl. Math., Special Issue No. 5 (2019), 68–83.
• [13] M. Et, A. Alotaibi, S. A. Mohiuddine, On $(\Delta^{m},I)$-statistical convergence of order $\alpha,$, The Scientific World Journal, 2014, 535419 DOI: 10.1155/2014/535419.
• [14] M. Et, M. Mursaleen, M. Işık, On a class of fuzzy sets defined by Orlicz functions, Filomat 27(5) (2013), 789–796.
• [15] M. Et, V. Karakaya, A new difference sequence set of order Alpha and its geometrical properties, Abstr. Appl. Anal. 2014, Art. ID 278907, 4 pp.
• [16] M. Karaka¸s, M. Et, V. Karakaya, Some geometric properties of a new difference sequence space involving lacunary sequences, Acta Math. Sci. Ser. B (Engl. Ed.) 33(6) (2013), 1711–1720.
• [17] M. A. Sarıgöl, On difference sequence spaces, J. Karadeniz Tech. Univ. Fac. Arts Sci. Ser. Math.-Phys. 10 (1987), 63–71.
• [18] E. Sava¸s, M. Et, On $(\Delta_{\lambda}^{m},I)-$statistical convergence of order $\alpha$, Period. Math. Hungar. 71(2) (2015), 135–145.
• [19] R. Çolak, M. Et, On some difference sequence sets and their topological properties, Bull. Malays. Math. Sci. Soc. 28(2) (2005), 125–130.