# On Some New Generalized Difference Sequence Spaces Defined by a Modulus Function

Volume 15, Issue 1, pp 78-87
• 1130 Views

### Authors

Tanweer Jalal - Department of Mathematics, National Institute of Technology, Hazratbal, Srinagar-190006

### Abstract

In this paper we introduces the new generalized difference sequences spaces $\left[\hat{V}, \lambda, f, P\right]_0(\Delta^r_u,E), \left[\hat{V}, \lambda, f, P\right]_1(\Delta^r_u,E), \left[\hat{V}, \lambda, f, P\right]_{\infty}(\Delta^r_u,E), \hat{S}_\lambda(\Delta^r_u,E)$ and $\hat{S}_{\lambda_0}(\Delta^r_u,E)$ (where $E$ is any Banach space) which arise from the notion of generalized de la Vallée- Poussin means and the concept of modulus function. We also give some inclusion relations between these spaces.

### Keywords

• Difference sequence spaces
• modulus function
• paranorm
• statistical convergence.

•  46A45

### References

• [1] F. M. Arani, M. E. Gordji, S. Talebi, Statistical convergence of double sequence in para normed spaces, The J. Math. and Com. Sci., 10 (2014), 47- 53.

• [2] M. Et, M. Basarir, On some new generalized difference sequence spaces, Periodica Mathematica Hungarica, 35 (3 ) (1997), 169-175.

• [3] M. Et, R. Colak, On some generalized difference sequence spaces, Soochow J. of Math. , 21 (1995), 377-386.

• [4] M. Et, F. Nurry, $\Delta^m$- Statistical convergence, Indian J. Pure and Appld. Math, 32 (2001), 961-969.

• [5] H. Fast, Sur la convergence statistique, Colloq. Math., 2 (1951), 241-244.

• [6] J. A. Fridy, On statistical convergence, Analysis , 5 (1985), 301-313.

• [7] H. Kizmaz, On certain sequence spaces, Canad. Math. Bull., 24 (1981), 169-176.

• [8] E. Kolk, The statistical convergence in Banach spaces, Acta. Comment. Univ. Tatru, 928 (1991), 41-52.

• [9] L. Leindler, Über die la Vallee-Pousinsche Summierbarkeit Allgemeiner Orthogonal- reihen, Acta. Math. Acad. Sci. Hungar, 16 (1965), 375-387.

• [10] I. J. Maddox, Elements of Functional Analysis, Cambridge Univ. Press., (1970)

• [11] I. J. Maddox, Sequence spaces defined by a modulus, Math. Proc. Camb. Phil. Soc., 100 (1986), 161-166.

• [12] E. Malkowsky, S. D. Parashar, Matrix transformations in spaces of bounded and convergent difference sequence spaces of order m, Analysis, 17 (1997), 87-97.

• [13] Mursaleen, λ – statistical convergence, Math. Slovaca, 50 (2000), 111-115.

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

• [15] E. Savas, Some sequence spaces and statistical convergence, Int. J. Math. & Math. Sci., 29(5) (2002), 303-306.

• [16] A. Wilansky, Functional Analysis, Blasdell Publishing Company, (1964)