A Novel Generalized Difference Spaces Constructed by the Modulus Function

Abstract

A major role of this document is to present a generalized difference spaces denoted by $w(\Delta ^{r},\hat{A},p,f,q,s),$ $w_{0}(\Delta ^{r},\hat{A},p,f,q,s),$ and $w_{\infty}(\Delta ^{r},\hat{A},p,f,q,s)$, of which arguments are defined as follows, and also to investigate some algebraic and topological characteristics of the spaces. Here; $\hat{A}$ is an infinite matrix, $p=(p_{k})$ is a bounded sequence of strictly positive real numbers, $f$ is any modulus function, $q$ is a semi norm, and $s$ is any non-negative real number. Besides these, the relationship between the spaces obtained by various values of those arguments is going to be considered. Finally, the newly obtained results are going to be compared with those of other studies.

Keywords

• Difference sequence spaces,modulus function

References

1. [1] S. Banach, Theorie des Operations Lin´eaires, Monografie Matematyczne, vol. 1, Warszawa, 1932.
2. [2] F. Başar, Summability Theory and Its Applications, ˙Istanbul 2011.
3. [3] E. Bulut, Ö . Çakar, The sequence space l (p; s) and related matrix tranformations, Commun. Fac. Sci. Univ. Ankara, Ser. A1, Math. Stat. 28, (1979), 33–44.
4. [4] M. Candan, Some new sequence spaces defined by a modulus function and infinite matrix in a seminormed space, J. Math. Anal., 3(2) (2012) ; 1–9.
5. [5] R. Çolak, Ç . CÇakar. Banach limits and related matrix transformations, Stud. Sci. Math. Hung. 24 (1989) ; 429–436.
6. [6] G. Das, Banach and other limits, J. London Math. Soc. 7 (1973), 501–507.
7. [7] A. Esi, Moduülüs fonksiyonu yardımıyla tanımlanmıs¸ bazı yeni dizi uzayları ve istatistiksel yakınsaklık, Doktora tezi, Fırat U¨ niversitesi Elazığ, 1995.
8. [8] M. Et, R. C¸ olak, On generalized difference sequence spaces, Soochow J. Math. 21 (4) (1995) 377–386.

9. [9] B. Kuttner, Note on strong summability, J. London Math. Soc. 21 (1946), 118–122.
10. [10] J. Connor, On strong Matrix summability with respect to a modulus and statistical convergence, Canad. Mat. Bull. 32 (2) (1989), 194–198.
11. [11] G.G. Lorentz, A contribution to the theory of divergent sequences, Acta Math. 80 (1948) 167–190.
12. [12] I. J. Maddox, Spaces of strongly summable sequences, Quart. J. Math. 18(1967), 345–355.
13. [13] I. J. Maddox, Sequence spaces defined by a modulus, Math. Proc. Camb. Philos Soc. 100(1986), 161–166.
14. [14] I. J. Maddox, A New Type of Convergence, Math. Proc. Cambridge Philos. Soc 83 (1978), 61–64.
15. [15] I. J. Maddox, On Strong Almost Convergence, Math. Proc. Cambridge Philos. Soc. 85 (1979), 345–350.
16. [16] H. Nakano, Concave modulares, J. Math. Loc. Japan, Vol.5, (1953), 29–49.
17. [17] S. Nanda, Strongly Almost Convergent Sequences, Bull Calcutta Math. Soc. 76 (1984) 236–40.
18. [18] S. Pehlivan, A sequence space defined by a modulus, Erciyes Univ. Journal of Science, 5(1-2)(1989), 875–880.
19. [19] W. H. Ruckle, FK spaces in which the sequence of coordinate vectors is bounded, Canad. J. Math. 25(1973), 973–978.
20. [20] E. Savas¸, On Strong Almost A-Summability with respect to a Modulus and Statistical Convergence, Indian J. Pure Appl. Math. 23 (1992), 217–22.
21. [21] E. Savas¸, On Some Generalized Sequence Spaces Defined by a Modulus, Indian J. Pure Appl. Math., 30 (5) (1999) 459–464.
22. [22] S.Simons, The sequence spaces l (pv) and m(pv) ; Proc. London Math. Soc. (3)15:
23. [23] B. Altay, F. Bas¸ar, The matrix domain and the fine spectrum of the difference operator D on the sequence space `p, (0 < p < 1), Commun. Math. Anal. 2(2)(2007), 1–11.
24. [24] H. Kızmaz, On certain sequence spaces. Canad. Math. Bull. 24(2)(1981), 169–176.
25. [25] F. Bas¸ar, B. Altay, On the space of sequences of p-bounded variation and related matrix mappings, Ukrainian Math. J. 55(1)(2003), 136–147.
26. [26] R. C¸ olak, M. Et, E. Malkowsky, Some Topics of Sequence Spaces, in : Lecture Notes in Mathematics, Fırat Univ. Press, 2004, pp. 1-63. ISBN:975–394–0386–6.
27. [27] Z.U. Ahmad, Mursaleen, K¨othe-Toeplitz duals of some new sequence spaces and their matrix maps, Publ. Inst. Math. (Beograd) 42(1987), 57–61.
28. [28] C¸ . Asma, R. C¸ olak, On the K¨othe-Toeplitz duals of some generalized sets of difference sequences, Demonstratio Math. 33(2000), 797–803.
29. [29] M.A. Sarıg¨ol, On difference sequence spaces, J. Karadeniz Tech. Univ. Fac. Arts Sci. Ser. Math.-Phys. 10(1987), 63–71.
30. [30] E. Malkowsky, Absolute and ordinary K¨othe-Toeplitz duals of some sets of sequences and matrix transformations, Pulb. Inst. Math (Beograd), (NS), 46(60)(1989), 97–103.
31. [31] B. Choudhary, S.K. Mishra, A note on certain sequence spaces, J. Anal. 1(1993), 139–148.
32. [32] S.K. Mishra, Matrix maps involving certain sequence spaces, Indian J. Pure Appl. Math. 24(2)(1993), 125–132.
33. [33] M. Mursaleen, A.K. Gaur, A.H. Saifi, Some new sequence spaces and their duals and matrix transformations, Bull. Calcutta Math. Soc. 88(3)(1996), 207–212.
34. [34] C. Gnanaseelan, P.D. Srivastava, The a􀀀;b􀀀 and g􀀀 duals of some generalised difference sequence spaces, Indian J. Math. 38(2)(1996), 111–120.
35. [35] E. Malkowsky, A note on the K¨othe-Toeplitz duals of generalized sets of bounded and convergent difference sequences, J. Anal. 4(1996), 81–91.
36. [36] A.K. Gaur, M. Mursaleen, Difference sequence spaces, Int. J. Math. Sci. 21(4)(1998), 701–706.
37. [37] E. Malkowsky, M. Mursaleen, Qamaruddin, Generalized sets of difference sequences, their duals and matrix transformations, in : equence Spaces and Applications, Narosa, New Delhi, 1999, pp. 68–83.
38. [38] E. Malkowsky, M. Mursaleen, Some matrix transformations between the difference sequence spaces Dc0(p), Dc(p) and D`¥(p), Filomat 15(2001), 353–363.
39. [39] T. Bilgin, The sequence spaces `(p; f ;q; s) on seminormed spaces, Bul. of Calcutta Math. Soc. 86(4)(1994), 295–304.
40. [40] A S¸ ahiner, Some new paranormed space defined by modulus function, Indian J. Pure Appl. Math. 33(12)(2002), 1877–1888.