DOI QR코드

DOI QR Code

ON VECTOR VALUED DIFFERENCE SEQUENCE SPACES

  • Manoj Kumar (Department of Mathematics, Baba Mastnath University) ;
  • Ritu (Department of Mathematics, Baba Mastnath University) ;
  • Sandeep Gupta (Department of Mathematics, Arya P.G. College)
  • Received : 2024.08.03
  • Accepted : 2024.08.29
  • Published : 2024.09.30

Abstract

In the present paper, using the notion of difference sequence spaces, we introduce new kind of Cesàro summable difference sequence spaces of vector valued sequences with the aid of paranorm and modulus function. In addition, we extend the notion of statistical convergence to introduce a new sequence space SC1(∆, q) which coincides with C11(X, ∆, φ, λ, q) (one of the above defined Cesàro summable difference sequence spaces) under the restriction of bounded modulus function.

Keywords

Acknowledgement

Authors would like to thank the referees for helpful comments and suggestions, which improved the presentation of this paper.

References

  1. A. Aizpuru, M. C. Listan-Garcia and F. Rambla-Barreno, Density by moduli and statistical convergence, Quaest. Math. 37 (4) (2014), 525-530. https://doi.org/10.2989/16073606.2014.981683 
  2. Y. Altin and M. Et, Generalized difference sequence spaces defined by a modulus function in a locally convex space, Soochow J. Math. 31 (2) (2005), 233-243. https://www.researchgate.net/publication/228659065 
  3. Y. Altin, M. Isik and R. Colak, A new sequence space defined by a modulus, Stud. Univ. Babes-Bolyai Math. 53 (2008), 3-13. https://zbmath.org/1212.46010 
  4. Y. Altin, Properties of some sets of sequences defined by a modulus function, Acta Math. Sci. 29 (2) (2009), 427-434. https://doi.org/10.1016/s0252-9602(09)60042-4 
  5. C. A. Bektas, M. Et and R. Colak, Generalized difference sequence spaces and their dual spaces, J. Math. Anal. Appl. 292 (2) (2004), 423-432. https://doi.org/10.1016/j.jmaa.2003.12.006 
  6. V. K. Bhardwaj and I. Bala, On lacunary generalized difference sequence spaces defined by orlicz functions in a seminormed space and ∆mq-lacunary statistical convergence, Demonstr. Math. 41 (2) (2008), 415-424. https://doi.org/10.1515/dema-2008-0217 
  7. V. K. Bhardwaj and S. Gupta, Cesaro summable difference sequence space, J. Inequal. Appl. 2013 (1), 1-9. https://doi.org/10.1186/1029-242x-2013-315 
  8. V. K. Bhardwaj and N. Singh, Some sequence spaces defined by Orlicz functions, Demonstr. Math. 33 (3) (2000), 571-582. https://doi.org/10.1515/dema-2000-0314 
  9. R. C. Buck, Generalized asymptotic density, Amer. J. Math. 75 (2) (1953), 335-346. https://doi.org/10.2307/2372456 
  10. M. Burgin and O. Duman, Statistical convergence and convergence in statistics, arXiv preprint. math/0612179 (2006). https://doi.org/10.48550/arXiv.math/0612179 
  11. R. Colak, On some generalized sequence spaces, Commun. Fac. Sci. Uni. Ank. Ser. A1 Math. Stat. 38 (1989) 35-46. https://doi.org/10.1501/commua1_0000000301 
  12. J. Connor, The statistical and strong p-Cesaro convergence of sequences, Analysis. 8 (1-2) (1988), 47-64. https://doi.org/10.1524/anly.1988.8.12.47 
  13. J. Connor, On strong matrix summability with respect to a modulus and statistical convergence, Canad. Math. Bull. 32 (2) (1989), 194-198. https://doi.org/10.4153/cmb-1989-029-3 
  14. M. Et, V. K. Bhardwaj and S. Gupta On deferred statistical boundedness of order α, Comm. Statist. Theory Methods. 51 (24) (2022), 8786-8798. https://doi.org/10.1080/03610926.2021.1906434 
  15. H. Fast, Sur la convergence statistique, Colloq Math. 2 (3-4) (1951), 241-244. https://doi.org/10.4064/cm-2-3-4-241-244 
  16. A. R. Freedman, J. J. Sember and M. Raphael, Some Cesaro-type summability spaces, Proc. Lond. math. Soc. 37 (3) (1978), 508-520. https://doi.org/10.1112/plms/s3-37.3.508 
  17. J. A. Fridy, On statistical convergence, Analysis. 5 (4) (1985), 301-314. https://doi.org/10.1524/anly.1985.5.4.301 
  18. D. Ghosh and P. D. Srivastava, On some vector valued sequence spaces defined using a modulus function, Indian J. pure appl. Math. 30 (8) (1999), 819-826. 
  19. S. Gupta and V. K. Bhardwaj, On deferred f-statistical convergence, Kyungpook Math. J. 58 (2018), 91-103. https://doi.org/10.5666/KMJ.2018.58.1.91 
  20. M. Isik, On statistical convergence of generalized difference sequences Soochow J. Math. 30 (2) (2004), 197-206. https://www.researchgate.net/publication/267671338 
  21. G. Karabacak and A. Or, Rough Statistical Convergence for Generalized Difference Sequences, Electron J. Mathe. Anal. Appl. 11 (1) (2023), 222-230. 
  22. H. Kizmaz, On certain sequence spaces, Canad. Math. bull. 24 (2) (1981), 169-176. https://doi.org/10.4153/CMB-1981-027-5 
  23. E. Kolk, The statistical convergence in Banach spaces, Acta Comment. Univ. Tartu. Math. 928 (1991), 41-52. http://www.ams.org/mathscinet-getitem?mr=1150232 
  24. I. J. Maddox, Elements of functional analysis, Camb. Univ. Press. 1970. https://doi.org/10.1002/bimj.19700120325 
  25. I. J. Maddox, Spaces of strongly summable sequences, Q. J. Math. 18 (1) (1967), 345-355. https://doi.org/10.1093/qmath/18.1.345 
  26. M. Mursaleen, λ-statistical convergence, Math. Slovaca. 50 (1) (2000), 111-115. 
  27. H. Nakano, Concave modulars, J. Math. Soc. Japan. 5 (1) (1953), 29-49. https://doi.org/10.2969/jmsj/00510029 
  28. I. Niven and H. S. Zuckerman, An Introduction to Theory of Numbers, Fourth.Ed., New York John Willey and Sons, 1980 
  29. E. Ozturk and T. Bilgin, Strongly summable sequence spaces defined by a modulus, Indian J. Pure Appl. Math. 25 (6) (1994), 621-621. 
  30. D. Rath and B. C. Tripathy, On statistically convergent and statistically Cauchy sequences, Indian J. Pure Appl Math. 25 (1994), 381-381. 
  31. W. H. Ruckle, Sequence Spaces, Pitman Advanced Publishing Program, 1981. 
  32. T. Salat, On statistically convergent sequences of real numbers Math. slovaca. 30 (2) (1980), 139-150. http://dml.cz/dmlcz/136236 
  33. H. Sengul and M. Et, f-lacunary statistical convergence and strong f-lacunary summability of order α, Filomat. 32 (13) (2018), 4513-4521. https://doi.org/10.2298/fil1813513s 
  34. N. Sharma and S. Kumar, Statistical Convergence and Cesaro Summability of Difference Sequences relative to Modulus Function, J. Class. Anal. 23 (1) (2024), 51-62. https://doi.org/10.7153/jca-2024-23-05 
  35. H. Steinhaus, Sur la convergence ordinaireet la convergence asymptotique, Colloq. Math. 2 (1) (1951), 73-74. 
  36. B. C. Tripathy, On statistically convergent and statistically bounded sequences, Bull. Malays. Math. Soc. 20 (1) (1997), 31-33. 
  37. B. C. Tripathy and H. Dutta, On some lacunary difference sequence spaces defined by a sequence of Orlicz functions and q-lacunary ∆-statistical convergence, An. stiint. Univ. "Ovidius" Constanta. Ser. Mat. 20 (1) (2012), 417-430. https://doi.org/10.2478/v10309-012-0028-1 
  38. B. C. Tripathy, A. Esi and B. K. Tripathy, On a new type of generalized difference Cesaro sequence spaces, Soochow J. Math. 31 (3) (2005), 333-340. https://www.researchgate.net/publication/266984781 
  39. A. K. Verma and L. K. Singh, (∆mv, f)-lacunary statistical convergence of order α, Proyecciones. 41 (4) (2022), 791-804. https://doi.org/10.22199/issn.0717-6279-4757 
  40. A. Zygmund, Trigonometric Series, Cambridge Univ. Press, UK, 1979.