Abstract and Applied Analysis

On Some Further Generalizations of Strong Convergence in Probabilistic Metric Spaces Using Ideals

Abstract

Following the line of (Das et al., 2011, Savas and Das, 2011), we make a new approach in this paper to extend the notion of strong convergence and more general strong statistical convergence (Şençimen and Pehlivan, 2008) using ideals and introduce the notion of strong $ℐ$- and ${ℐ}^{*}$-statistical convergence and two related concepts, namely, strong $ℐ$-lacunary statistical convergence and strong $ℐ$-$\lambda$-statistical convergence in a probabilistic metric space endowed with strong topology. We mainly investigate their interrelationship and study some of their important properties.

Article information

Source
Abstr. Appl. Anal., Volume 2013 (2013), Article ID 765060, 8 pages.

Dates
First available in Project Euclid: 27 February 2014

https://projecteuclid.org/euclid.aaa/1393512062

Digital Object Identifier
doi:10.1155/2013/765060

Mathematical Reviews number (MathSciNet)
MR3108647

Citation

Das, Pratulananda; Dutta, Kaustubh; Karakaya, Vatan; Ghosal, Sanjoy. On Some Further Generalizations of Strong Convergence in Probabilistic Metric Spaces Using Ideals. Abstr. Appl. Anal. 2013 (2013), Article ID 765060, 8 pages. doi:10.1155/2013/765060. https://projecteuclid.org/euclid.aaa/1393512062

References

• H. Fast, “Sur la convergence statistique,” vol. 2, pp. 241–244, 1951.
• J. S. Connor, “The statistical and strong $p$-Cesàro convergence of sequences,” Analysis, vol. 8, no. 1-2, pp. 47–63, 1988.
• J. A. Fridy, “On statistical convergence,” Analysis, vol. 5, no. 4, pp. 301–313, 1985.
• T. Šalát, “On statistically convergent sequences of real numbers,” Mathematica Slovaca, vol. 30, no. 2, pp. 139–150, 1980.
• J. A. Fridy and C. Orhan, “Lacunary statistical convergence,” Pacific Journal of Mathematics, vol. 160, no. 1, pp. 43–51, 1993.
• J. Li, “Lacunary statistical convergence and inclusion properties between lacunary methods,” International Journal of Mathematics and Mathematical Sciences, vol. 23, no. 3, pp. 175–180, 2000.
• Mursaleen, “$\lambda$-statistical convergence,” Mathematica Slovaca, vol. 50, no. 1, pp. 111–115, 2000.
• V. Karakaya, N. Şimşek, M. Ertürk, and F. Gürsoy, “Lacunary statistical convergence of sequences of functions in intuitionistic fuzzy normed space,” Journal of Intelligent & Fuzzy Systems. In press.
• V. Karakaya, N. Şimşek, M. Ertürk, and F. Gürsoy, “$\lambda$-statistical convergence of sequences of functions in intuitionistic fuzzy normed spaces,” Journal of Function Spaces and Applications, vol. 2012, Article ID 926193, 14 pages, 2012.
• V. Karakaya, N. Şimşek, M. Ertürk, and F. Gürsoy, “Lacunary statistical convergencečommentComment on ref. [16?]: Please update the information of this reference, if possible. of sequences of functions in intuitionistic fuzzy normed space,” Journal of Intelligent & Fuzzy Systems. In press.
• P. Kostyrko, T. Šalát, and W. Wilczyński, “$\mathcal{I}$-convergence,” Real Analysis Exchange, vol. 26, no. 2, pp. 669–685, 2000.
• P. Das, E. Savas, and S. Kr. Ghosal, “On generalizations of certain summability methods using ideals,” Applied Mathematics Letters, vol. 24, no. 9, pp. 1509–1514, 2011.
• E. Savas and P. Das, “A generalized statistical convergence via ideals,” Applied Mathematics Letters, vol. 24, no. 6, pp. 826–830, 2011.
• K. Dems, “On $\mathcal{I}$-Cauchy sequences,” Real Analysis Exchange, vol. 30, no. 1, pp. 123–128, 2004/05.
• A. Nabiev, S. Pehlivan, and M. Gürdal, “On $\mathcal{I}$-Cauchy sequences,” Taiwanese Journal of Mathematics, vol. 11, no. 2, pp. 569–576, 2007.
• P. Das and S. Kr. Ghosal, “Some further results on $\mathcal{I}$-Cauchy sequences and condition (AP),” Computers & Mathematics with Applications, vol. 59, no. 8, pp. 2597–2600, 2010.
• M. Mursaleen and S. A. Mohiuddine, “On ideal convergence in probabilistic normed spaces,” Mathematica Slovaca, vol. 62, no. 1, pp. 49–62, 2012.
• K. Menger, “Statistical metrics,” Proceedings of the National Academy of Sciences of the United States of America, vol. 28, pp. 535–537, 1942.
• B. Schweizer and A. Sklar, “Statistical metric spaces,” Pacific Journal of Mathematics, vol. 10, pp. 313–334, 1960.
• B. Schweizer, A. Sklar, and E. Thorp, “The metrization of statistical metric spaces,” Pacific Journal of Mathematics, vol. 10, pp. 673–675, 1960.
• B. Schweizer and A. Sklar, “Statistical metric spaces arising from sets of random variables in Euclidean $n$-space,” Akademija Nauk SSSR, vol. 7, pp. 456–465, 1962.
• B. Schweizer and A. Sklar, “Triangle inequalities in a class of statistical metric spaces,” Journal of the London Mathematical Society, vol. 38, pp. 401–406, 1963.
• R. M. Tardiff, “Topologies for probabilistic metric spaces,” Pacific Journal of Mathematics, vol. 65, no. 1, pp. 233–251, 1976.
• B. Schweizer and A. Sklar, Probabilistic Metric Spaces, Dover, Mineola, NY, USA, 2005.
• C. Şençimen and S. Pehlivan, “Strong statistical convergence in probabilistic metric spaces,” Stochastic Analysis and Applications, vol. 26, no. 3, pp. 651–664, 2008.
• C. Şençimen and S. Pehlivan, “Strong ideal convergence in probabilistic metric spaces,” Indian Academy of Sciences, vol. 119, no. 3, pp. 401–410, 2009.
• C. Şençimen and S. Pehlivan, “Statistical continuity in probabilistic normed spaces,” Applicable Analysis, vol. 87, no. 3, pp. 377–384, 2008.
• C. Şençimen and S. Pehlivan, “Statistically $D$-bounded sequences in probabilistic normed spaces,” Applicable Analysis, vol. 88, no. 8, pp. 1133–1142, 2009.
• D. A. Sibley, “A metric for weak convergence of distribution functions,” The Rocky Mountain Journal of Mathematics, vol. 1, no. 3, pp. 427–430, 1971.
• A. R. Freedman, J. J. Sember, and M. Raphael, “Some Cesàro-type summability spaces,” Proceedings of the London Mathematical Society. Third Series, vol. 37, no. 3, pp. 508–520, 1978.
• L. Leindler, “Über die verallgemeinerte de la Vallée-Poussinsche Summierbarkeit allgemeiner Orthogonalreihen,” Acta Mathematica Academiae Scientiarum Hungaricae, vol. 16, pp. 375–387, 1965.