Generalized Statistical Convergence of Double Sequences in Paranormed Spaces

Main Article Content

Abdullah Alotaibi
Alaa Mohammed Aljahili
S.A. Mohiuddine

Abstract

We introduce the notion of (λ, µ)-statistical convergence of double sequences in a setting of paranormed space and prove that every convergent sequence is (λ, µ)-statistically convergent but not conversely by supporting an illustrative example. We also define the notions of (λ, µ)-statistical Cauchy and strongly (λ, µ)p-summable double sequences in a paranormed space and obtain their relationship with (λ, µ)- statistical convergence.

Article Details

References

  1. T. Acar and S. A. Mohiuddine, Statistical (C, 1)(E, 1) summability and Korovkin's theorem, Filomat 30(2) (2016), 387-393.
  2. A. Alotaibi and A. M. Alroqi, Statistical convergence in a paranormed space, J. Inequal. Appl. 2012 (2012), 39.
  3. A. Alotaibi, M. Mursaleen, S. A. Mohiuddine, Korovkin type approximation theorems for σ-convergence of double sequences, J. Nonlinear Convex Anal. 16(1) (2015), 183-192.
  4. M. A. Alghamdi and M. Mursaleen, λ-Statistical convergence in paranormed space, Abstr. Appl. Anal. 2013 (2013), Article ID 264520.
  5. F. A. Arani, M. E. Gordji and S. Talebi, Statistical convergence of double sequence in paranormed spaces, J. Math. Comput. Sci. 10 (2014), 47-53.
  6. C. Belen and S. A. Mohiuddine, Generalized weighted statistical convergence and application, Appl. Math. Comput. 219 (2013), 9821-9826.
  7. N. L. Braha, H. M. Srivastava and S. A. Mohiuddine, A Korovkin's type approximation theorem for periodic functions via the statistical summability of the generalized de la Vall ´ee Poussin mean, Appl. Math. Comput. 228 (2014), 162-169.
  8. H. C ¸ akalli, On statistical convergence in topological groups, Pure Appl. Math. Sci. 43 (1996), 27-31.
  9. J. S. Connor, The statistical and strong p-Cesaro convergence of sequences, Analysis 8 (1988), 47-63.
  10. O. H. H. Edely, S. A. Mohiuddine and A. K. Noman, Korovkin type approximation theorems obtained through generalized statistical convergence, Appl. Math. Lett. 23 (2010), 1382-1387.
  11. H. Fast, Sur la convergence statistique, Coll. Math. 2 (1951), 241-244.
  12. J. A. Fridy, On statistical convergence, Analysis 5(4) (1985), 301-313.
  13. B. Hazarika and V. Kumar, On asymptotically double lacunary statistical equivalent sequences in ideal context, J. Inequal. Appl. 2013 (2013), 543.
  14. U. Kadak and S. A. Mohiuddine, Generalized statistically almost convergence based on the difference operator which includes the (p, q)-Gamma function and related approximation theorems, Results Math. 73 (2018), 9.
  15. E. Kolk, The statistical convergence in Banach spaces, Tartu Ul. Toime. 928 (1991), 41-52.
  16. I. J. Maddox, Statistical convergence in a locally convex space, Math. Camb. Phil. Soc. 104 (1988), 141-145.
  17. S. A. Mohiuddine, Statistical weighted A-summability with application to Korovkin's type approximation theorem, J. Inequal. Appl. 2016 (2016), 101.
  18. S. A. Mohiuddine and B. A. S. Alamri, Generalization of equi-statistical convergence via weighted lacunary sequence with associated Korovkin and Voronovskaya type approximation theorems, Rev. R. Acad. Cienc. Exactas Fs. Nat., Ser. A Mat., RACSAM 113 (3) (2019), 1955-1973.
  19. S. A. Mohiuddine and M. Aiyub, Lacunary statistical convergence in random 2-normed spaces, Appl. Math. Inf. Sci. 6(3) (2012), 581-585.
  20. S. A. Mohiuddine, A. Alotaibi and M. Mursaleen, Statistical convergence of double sequences in locally solid Riesz spaces, Abstr. Appl. Anal. 2012 (2012), 719729.
  21. S. A. Mohiuddine, A. Asiri and B. Hazarika, Weighted statistical convergence through difference operator of sequences of fuzzy numbers with application to fuzzy approximation theorems, Int. J. Gen. Syst. 48(5) (2019), 492-506.
  22. S. A. Mohiuddine and B. Hazarika, Some classes of ideal convergent sequences and generalized difference matrix operator, Filomat 31(6) (2017), 1827-1834
  23. S. A. Mohiuddine, B. Hazarika and A. Alotaibi, On statistical convergence of double sequences of fuzzy valued functions, J. Intell. Fuzzy Syst. 32 (2017), 4331-4342.
  24. M. Mursaleen, λ-Statistical convergence, Math. Slovaca 50 (2000), 111-115.
  25. M. Mursaleen, C. Cakan, S. A. Mohiuddine and E. Sava ¸s, Generalized statistical convergence and statistical core of double sequences, Acta Math. Sin. 26(11) (2010), 2131-2144.
  26. M. Mursaleen and O. H. H. Edely, Statistical convergence of double sequences, J. Math. Anal. Appl. 288 (2003), 223-231.
  27. M. Mursaleen and S. A. Mohiuddine, Statistical convergence of double sequences in intuitionistic fuzzy normed spaces, Chaos Solitons Fractals 41(5) (2009), 2414-2421.
  28. M. Mursaleen and S. A. Mohiuddine, On lacunary statistical convergence with respect to the intuitionistic fuzzy normed space, J. Comput. Appl. Math. 233 (2009), 142-149.
  29. A. Pringsheim, Zur Ttheorie der zweifach unendlichen Zahlenfolgen, Math. Ann. 53 (1900), 289-321.
  30. I. J. Schoenberg, The integrability of certain functions and related summability methods, Amer. Math. Monthly 66 (1959), 361-375.
  31. T. Salat, On statistically convergent sequences of real numbers, Math. Slovaca 30 (1980), 139-150.
  32. A. Zygmund, Trigonometrical Series, vol. 5(1935) of Monografyas de Matematicas, Warszawa-Lwow.