Some Paranormed Difference Sequence Spaces Derived by Using Generalized Means

• Journal title : Kyungpook mathematical journal
• Volume 55, Issue 4,  2015, pp.909-931
• Publisher : Department of Mathematics, Kyungpook National University
• DOI : 10.5666/KMJ.2015.55.4.909
Title & Authors
Some Paranormed Difference Sequence Spaces Derived by Using Generalized Means
MANNA, ATANU; MAJI, AMIT; SRIVASTAVA, PARMESHWARY DAYAL;

Abstract
This paper presents some new paranormed sequence spaces $\small{X(r,s,t,p;{\Delta})}$ where $\small{X{\in}\{l_{\infty}(p),c(p),c_0(p),l(p)\}}$ defined by using generalized means and difference operator. It is shown that these are complete linear metric spaces under suitable paranorms. Furthermore, the $\small{{\alpha}}$-, $\small{{\beta}}$-, $\small{{\gamma}}$-duals of these sequence spaces are computed and also obtained necessary and sufficient conditions for some matrix transformations from $\small{X(r,s,t,p;{\Delta})}$ to X. Finally, it is proved that the sequence space $\small{l(r,s,t,p;{\Delta})}$ is rotund when $\small{p_n}$ > 1 for all n and has the Kadec-Klee property.
Keywords
Sequence spaces;Difference operator;Generalized means;$\small{{\alpha}}$-, $\small{{\beta}}$-, $\small{{\gamma}}$-duals;Matrix transformations;Rotundity;Kadec-Klee property;
Language
English
Cited by
References
1.
A. Kananthai, M. Mursaleen, W. Sanhan and S. Suantai, On property (H) and rotundity of difference sequence spaces, J. Nonlinear Convex Anal., 3(3)(2002), 401-409.

2.
A. M. Jarrah and E. Malkowsky, Ordinary, absolute and strong summability and matrix transformations, Filomat, 17(2003), 59-78.

3.
A. Wilansky, Summability through Functional Analysis, North-Holland Math. Stud., 85, Else-vier Science Publishers, Amsterdam, New York, Oxford, 1984

4.
B. Altay and F. Basar, Some paranormed Riesz sequence spaces of non-absolute type, Southeast Asian Bull. Math., 30(4)(2006), 591-608.

5.
B. Altay and F. Basar, On the paranormed Riesz sequence spaces of non-absolute type, Southeast Asian Bull. Math., 26(5)(2003), 701-715.

6.
B. Altay and F. Basar, Some paranormed sequence spaces of non-absolute type derived by weighted mean, J. Math. Anal. Appl., 319(2)(2006), 494-508.

7.
B. Altay and F. Basar, Generalization of the sequence space l(p) derived by weighted mean, J. Math. Anal. Appl., 330(1)(2007), 174-185.

8.
B. Choudary and S. K. Mishra, On Kothe-Toeplitz duals of certain sequence spaces and their matrix transformations, Indian J. Pure Appl. Math., 24(5)(1993), 291-301.

9.
C. Aydin and F. Basar, Some new difference sequence spaces, Appl. Math. Comput., 157(3)(2004), 677-693.

10.
C. Wu, S. Chen and Y. Wang, H-property of sequence Orlicz spaces, J. Harbin Inst. Tech. Math Issue, (1985), 6-11.

11.
F. Basar and B. Altay, Matrix mappings on the space bs(p) and its ${\alpha}-,\;{\beta}-,\;{\gamma}-$ duals, Aligarh Bull. Math., 21(1)(2002), 79-91.

12.
H. Hudzik and D. Pallaschke, On some convexity property of Orlicz sequence spaces, Math. Nachr., 186(1997), 167-185.

13.
H. Kizmaz, On certain sequence spaces, Canad. Math. Bull., 24(2)(1981), 169-176.

14.
I. J. Maddox, Paranormed sequence spaces generated by infinite matrices, Proc. Cambridge Philos. Soc., 64(1968), 335-340.

15.
K. G. Grosse-Erdmann, Matrix transformations between the sequence spaces of Maddox, J. Math. Anal. Appl., 180(1)(1993), 223-238.

16.
M. Basarir, Paranormed Cesaro difference sequence space and related matrix transformations, Doga Mat., 15(1)(1991), 14-19.

17.
M. Mursaleen and A. K. Noman, On generalized means and some related sequence spaces, Comput. Math. Appl., 61(4)(2011), 988-999.

18.
R. Colak, M. Et, On some generalized difference sequence spaces and related matrix transformations, Hokkaido Math. J., 26(3)(1997), 483-492.

19.
S. T. Chen, Geometry of Orlicz spaces, Dissertationes Math. (1996), p. 356.

20.
S. Demiriz and C. Cakan, On some new paranormed sequence spaces, Gen. Math. Notes, 1(2)(2010), 26-42.

21.
S. Demiriz and C. Cakan, Some new paranormed difference sequence spaces and weighted core, Comput. Math. Appl., 64(6)(2012), 1726-1739.

22.
V. Karakaya and H. Polat, Some new paranormed sequence spaces defined by Euler and difference operators, Acta Sci. Math. (Szeged), 76(1-2)(2010), 87-100.

23.
W. Orlicz, A note on modular spaces I, Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys., 9(1961), 157-162.

24.
Y. Cui and H. Hudzik, On the Uniform Opial property in some modular sequence spaces, Func. Approx. Comment., 26(1998), 93-102.

25.
Z. U. Ahmad and M. Mursaleen, Kothe-Toeplitz duals of some new sequence spaces and their matrix maps, Publ. Inst. Math. (Beograd), 42(56)(1987), 57-61.