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Abstract and Applied Analysis
Volume 2013, Article ID 215612, 5 pages
http://dx.doi.org/10.1155/2013/215612
Research Article

Statistical Summability of Double Sequences through de la Vallée-Poussin Mean in Probabilistic Normed Spaces

Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia

Received 21 September 2013; Accepted 26 October 2013

Academic Editor: Mohammad Mursaleen

Copyright © 2013 S. A. Mohiuddine and Abdullah Alotaibi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. A. Pringsheim, “Zur theorie der zweifach unendlichen zahlenfolgen,” Mathematische Annalen, vol. 53, no. 3, pp. 289–321, 1900. View at Publisher · View at Google Scholar · View at MathSciNet
  2. H. Fast, “Sur la convergence statistique,” Colloquium Mathematicum, vol. 2, pp. 241–244, 1951. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. H. Steinhaus, “Sur la convergence ordinaire et la convergence asymptotique,” Colloquium Mathematicum, vol. 2, pp. 73–74, 1951. View at Google Scholar
  4. I. J. Schoenberg, “The integrability of certain function and related summability methods,” The American Mathematical Monthly, vol. 66, pp. 361–375, 1959. View at Publisher · View at Google Scholar
  5. T. Šalát, “On statistically convergent sequences of real numbers,” Mathematica Slovaca, vol. 30, no. 2, pp. 139–150, 1980. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. J. A. Fridy, “On statistical convergence,” Analysis, vol. 5, no. 4, pp. 301–313, 1985. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. H. Çakalli and M. K. Khan, “Summability in topological spaces,” Applied Mathematics Letters, vol. 24, no. 3, pp. 348–352, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. B. Hazarika, “On generalized statistical convergence in random 2-normed spaces,” Scientia Magna, vol. 8, no. 1, pp. 58–67, 2012. View at Google Scholar
  9. S. A. Mohiuddine and M. A. Alghamdi, “Statistical summability through a lacunary sequence in locally solid Riesz spaces,” Journal of Inequalities and Applications, vol. 2012, article 225, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  10. S. A. Mohiuddine and M. Aiyub, “Lacunary statistical convergence in random 2-normed spaces,” Applied Mathematics & Information Sciences, vol. 6, no. 3, pp. 581–585, 2012. View at Google Scholar · View at MathSciNet
  11. S. A. Mohiuddine, H. Şevli, and M. Cancan, “Statistical convergence in fuzzy 2-normed space,” Journal of Computational Analysis and Applications, vol. 12, no. 4, pp. 787–798, 2010. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. M. Mursaleen, “On statistical convergence in random 2-normed spaces,” Acta Scientiarum Mathematicarum, vol. 76, no. 1-2, pp. 101–109, 2010. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. M. Mursaleen and O. H. H. Edely, “Generalized statistical convergence,” Information Sciences, vol. 162, no. 3-4, pp. 287–294, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. M. Mursaleen, V. Karakaya, M. Ertürk, and F. Gürsoy, “Weighted statistical convergence and its application to Korovkin type approximation theorem,” Applied Mathematics and Computation, vol. 218, no. 18, pp. 9132–9137, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. A. Şahiner, M. Gürdal, S. Saltan, and H. Gunawan, “Ideal convergence in 2-normed spaces,” Taiwanese Journal of Mathematics, vol. 11, no. 5, pp. 1477–1484, 2007. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. U. Yamancı and M. Gürdal, “On lacunary ideal convergence in random 2-normed space,” Journal of Mathematics, vol. 2013, Article ID 868457, 8 pages, 2013. View at Publisher · View at Google Scholar
  17. M. Mursaleen and O. H. H. Edely, “Statistical convergence of double sequences,” Journal of Mathematical Analysis and Applications, vol. 288, no. 1, pp. 223–231, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. M. Mursaleen and S. A. Mohiuddine, “Statistical convergence of double sequences in intuitionistic fuzzy normed spaces,” Chaos, Solitons & Fractals, vol. 41, no. 5, pp. 2414–2421, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. S. A. Mohiuddine, H. Şevli, and M. Cancan, “Statistical convergence of double sequences in fuzzy normed spaces,” Filomat, vol. 26, no. 4, pp. 673–681, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. R. Çolak and Y. Altin, “Statistical convergence of double sequences of order α̃,” Journal of Function Spaces and Applications, vol. 2013, Article ID 682823, 5 pages, 2013. View at Publisher · View at Google Scholar
  21. S. A. Mohiuddine, A. Alotaibi, and M. Mursaleen, “Statistical convergence of double sequences in locally solid Riesz spaces,” Abstract and Applied Analysis, vol. 2012, Article ID 719729, 9 pages, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  22. S. A. Mohiuddine, B. Hazarika, and A. Alotaibi, “Double lacunary density and some inclusion results in locally solid Riesz spaces,” Abstract and Applied Analysis, vol. 2013, Article ID 507962, 8 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  23. H. Dutta, “A characterization of the class of statistically pre-Cauchy double sequences of fuzzy numbers,” Applied Mathematics & Information Sciences, vol. 7, no. 4, pp. 1437–1440, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  24. M. Mursaleen, “λ-statistical convergence,” Mathematica Slovaca, vol. 50, no. 1, pp. 111–115, 2000. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  25. C. Belen and S. A. Mohiuddine, “Generalized weighted statistical convergence and application,” Applied Mathematics and Computation, vol. 219, no. 18, pp. 9821–9826, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  26. R. Çolak and C. A. Bektaş, “λ-statistical convergence of order α,” Acta Mathematica Scientia B, vol. 31, no. 3, pp. 953–959, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  27. H. Dutta and T. Bilgin, “Strongly (Vλ,A,Δvmn,p)-summable sequence spaces defined by an Orlicz function,” Applied Mathematics Letters, vol. 24, no. 7, pp. 1057–1062, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  28. O. H. H. Edely, S. A. Mohiuddine, and A. K. Noman, “Korovkin type approximation theorems obtained through generalized statistical convergence,” Applied Mathematics Letters, vol. 23, no. 11, pp. 1382–1387, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  29. M. Et, M. Çınar, and M. Karakaş, “On λ-statistical convergence of order α of sequences of function,” Journal of Inequalities and Applications, vol. 2013, article 204, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  30. S. A. Mohiuddine and Q. M. D. Lohani, “On generalized statistical convergence in intuitionistic fuzzy normed space,” Chaos, Solitons & Fractals, vol. 42, no. 3, pp. 1731–1737, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  31. S. A. Mohiuddine, A. Alotaibi, and M. Mursaleen, “A new variant of statistical convergence,” Journal of Inequalities and Applications, vol. 2013, article 309, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  32. M. Mursaleen, C. Çakan, S. A. Mohiuddine, and E. Savaş, “Generalized statistical convergence and statistical core of double sequences,” Acta Mathematica Sinica, vol. 26, no. 11, pp. 2131–2144, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  33. E. Savaş and S. A. Mohiuddine, “λ̅-statistically convergent double sequences in probabilistic normed spaces,” Mathematica Slovaca, vol. 62, no. 1, pp. 99–108, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  34. E. P. Klement, R. Mesiar, and E. Pap, Triangular Norms, vol. 8 of Trends in Logic, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2000. View at MathSciNet
  35. A. N. Šerstnev, “Random normed spaces: problems of completeness,” Kazanskij Gosudarstvennyj Universitet Imeni V.I. Ul'janova-Lenina. Učenye Zapiski, vol. 122, no. 4, pp. 3–20, 1962. View at Google Scholar · View at MathSciNet
  36. K. Menger, “Statistical metrics,” Proceedings of the National Academy of Sciences of the United States of America, vol. 28, pp. 535–537, 1942. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  37. C. Alsina, B. Schweizer, and A. Sklar, “On the definition of a probabilistic normed space,” Aequationes Mathematicae, vol. 46, no. 1-2, pp. 91–98, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  38. B. Schweizer and A. Sklar, “Statistical metric spaces,” Pacific Journal of Mathematics, vol. 10, pp. 313–334, 1960. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  39. B. Schweizer and A. Sklar, Probabilistic Metric Spaces, Elsevier, New York, NY, USA, 1983.
  40. B. Schweizer and A. Sklar, Probabilistic Metric Spaces, Dover Publication, Mineola, NY, USA, 2nd edition, 2005.