About this Journal Submit a Manuscript Table of Contents
Abstract and Applied Analysis
Volume 2012 (2012), Article ID 719729, 9 pages
http://dx.doi.org/10.1155/2012/719729
Research Article

Statistical Convergence of Double Sequences in Locally Solid Riesz Spaces

1Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia
2Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India

Received 26 May 2012; Accepted 13 August 2012

Academic Editor: Ziemowit Popowicz

Copyright © 2012 S. A. Mohiuddine et al. 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. H. Fast, “Sur la convergence statistique,” Colloquium Mathematicum, vol. 2, pp. 241–244, 1951. View at Zentralblatt MATH
  2. H. Steinhaus, “Sur la convergence ordinaire et la convergence asymptotique,” Colloquium Mathematicum, vol. 2, pp. 73–34, 1951.
  3. A. Zygmund, Trigonometric Series, Cambridge University Press, New York, NY, USA, 1st edition, 1959.
  4. G. G. Lorentz, “A contribution to the theory of divergent sequences,” Acta Mathematica, vol. 80, pp. 167–190, 1948. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  5. J. A. Fridy, “On statistical convergence,” Analysis, vol. 5, no. 4, pp. 301–313, 1985. View at Zentralblatt MATH
  6. R. Çolak and Ç. A. Bektaş, “λ-statistical convergence of order α,” Acta Mathematica Scientia Series B, vol. 31, no. 3, pp. 953–959, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  7. V. Kumar and M. Mursaleen, “On (λ,μ)-statistical convergence of double sequences on intuitionistic fuzzy normed spaces,” Filomat, vol. 25, no. 2, pp. 109–120, 2011. View at Publisher · View at Google Scholar
  8. 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
  9. 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.
  10. M. Mursaleen and A. Alotaibi, “On I-convergence in random 2-normed spaces,” Mathematica Slovaca, vol. 61, no. 6, pp. 933–940, 2011. View at Publisher · View at Google Scholar
  11. S. A. Mohuiddine, A. Alotaibi, and S. M. Alsulami, “Ideal convergence of double sequences in 2 random 2-normed spaces,” Advances in Difference Equations, vol. 2012, article 149, 2012.
  12. M. Mursaleen and S. A. Mohiuddine, “On lacunary statistical convergence with respect to the intuitionistic fuzzy normed space,” Journal of Computational and Applied Mathematics, vol. 233, no. 2, pp. 142–149, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  13. M. Mursaleen and S. A. Mohiuddine, “On ideal convergence of double sequences in probabilistic normed spaces,” Mathematical Reports, vol. 12, no. 4, pp. 359–371, 2010. View at Zentralblatt MATH
  14. M. Mursaleen and S. A. Mohiuddine, “On ideal convergence in probabilistic normed spaces,” Mathematica Slovaca, vol. 62, no. 1, pp. 49–62, 2012. View at Publisher · View at Google Scholar
  15. M. Mursaleen, S. A. Mohiuddine, and O. H. H. Edely, “On the ideal convergence of double sequences in intuitionistic fuzzy normed spaces,” Computers & Mathematics with Applications, vol. 59, no. 2, pp. 603–611, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  16. E. Savaş and S. A. Mohiuddine, “λ¯-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
  17. I. J. Maddox, “Statistical convergence in a locally convex space,” Mathematical Proceedings of the Cambridge Philosophical Society, vol. 104, no. 1, pp. 141–145, 1988. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  18. H. Cakalli, “On statistical convergence in topological groups,” Pure and Applied Mathematika Sciences, vol. 43, no. 1-2, pp. 27–31, 1996. View at Zentralblatt MATH
  19. H. Çakalli and E. Savaş, “Statistical convergence of double sequences in topological groups,” Journal of Computational Analysis and Applications, vol. 12, no. 2, pp. 421–426, 2010. View at Zentralblatt MATH
  20. S. A. Mohiuddine and E. Savaş, “Lacunary statistically convergent double sequences in probabilistic normed spaces,” Annali dell'Universita di Ferrara. In press. View at Publisher · View at Google Scholar
  21. 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
  22. 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 Zentralblatt MATH
  23. M. Mursaleen, “On statistical convergence in random 2-normed spaces,” Acta Scientiarum Mathematicarum, vol. 76, no. 1-2, pp. 101–109, 2010.
  24. H. Albayrak and S. Pehlivan, “Statistical convergence and statistical continuity on locally solid Riesz spaces,” Topology and its Applications, vol. 159, no. 7, pp. 1887–1893, 2012. View at Publisher · View at Google Scholar
  25. A. C. Zaanen, Introduction to Operator Theory in Riesz Spaces, Springer, Berlin, Germany, 1997. View at Publisher · View at Google Scholar
  26. G. T. Roberts, “Topologies in vector lattices,” Mathematical Proceedings of the Cambridge Philosophical Society, vol. 48, pp. 533–546, 1952. View at Zentralblatt MATH
  27. 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
  28. 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
  29. 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
  30. 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