A New Type of Ideal Convergence of Difference Sequence in Probabilistic Normed Space

Vakeel A. Khan, Henna Altaf, Mohammad Faisal Khan


The idea of difference sequence sets \(X(\Delta)=\{x=(x_{k}):\Delta x\in X\}\) with \(X=l_{\infty}\), \(c\) and \(c_{0}\) was introduced by Kizmaz [10]. Mursaleen and Mohiuddine [13] defined the idea of probabilistic normed space(PNS) and the ideal convergence in PNS. Motivated by the above two concepts, we in this paper introduce the notion of   difference \(I\)-convergent  sequence  in PNS and study the elementary properties of this convergence.


Triangular norm; Probabilistic normed space; \(\Delta I\)-convergence; \(\Delta I^{*}\)-convergence; \(\Delta I\)-limit points; \(\Delta I\)-cluster points

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C. Alsina, B. Schweizer and A. Sklar, Continuity properties of probabilistic norms, Journal of Mathematical Analysis and Applications, 208 (1997), 446 – 452, https://core.ac.uk/download/pdf/82030803.pdf.

M. Balcerzak, K. Dems and A. Komisarski, Statistical convergence and ideal convergence for sequences of functions, Journal of Mathematical Analysis and Applications 328 (2007), 715 – 729, DOI: 10.1016/j.jmaa.2006.05.040.

Mursaleen and O.H.H. Edely, Statistical convergence of double sequences, Journal of Mathematical Analysis and Applications 288 (2003), 223 – 231, DOI: 10.1016/j.jmaa.2003.08.004.

H. Fast, Sur la convergence statistique, Colloquium Mathematicae 2 (1951), 241 – 244, http: //matwbn.icm.edu.pl/ksiazki/cm/cm2/cm2137.pdf.

M.J. Frank, Probabilistic topological spaces, Journal of Mathematical Analysis and Applications, 34 (1971), 67 – 81, https://core.ac.uk/download/pdf/82210079.pdf.

J.A. Fridy, On statistical convergence. Analysis 5 (1985), 301 – 314, DOI: 10.1524/anly.1985.5.4.301.

H.G. Gumus and F. Nuray, ±{m}-ideal convergence, Selcuk Journal of Applied Mathematics 12 (2011), 101 –110, http://sjam.selcuk.edu.tr/sjam/article/view/308.

V.A. Khan and N. Khan, On zweier i-convergent double sequence spaces, Filomat, 30 (2016), 3361 – 3369, DOI: 10.2298/FIL1612361K.

V.A. Khan, Y. Khan, H. Altaf, A. Esi and A. Ahamd, On paranorm intuitionistic fuzzy i-convergent sequence spaces defined by compact operator, International Journal of Advanced and Applied Sciences 4 (2017), 138 – 143, 10.21833/ijaas.2017.05.024.

H Kizmaz, Certain sequence spaces, Can. Math. Bull. 24(1981), 169 – 176, DOI: 10.4153/CMB-1981-027-5.

P. Kostyrko, W. Wilczy ´ nski and T. Šalát, I-convergence, Real Analysis Exchange 26 (2000), 669 – 686, https://projecteuclid.org/download/pdf_1/euclid.rae/1214571359.

K. Menger, Statistical metrics, Proceedings of the National Academy of Sciences 28 (1942), 535 – 537, https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1078534/pdf/pnas01647-0029.pdf.

M. Mursaleen and S. Mohiuddine, On ideal convergence in probabilistic normed spaces, Mathematica Slovaca 62 (2012), 49 – 62, DOI: 10.2478/s12175-011-0071-9.

M. Mursaleen and S.A. Mohiuddine, On ideal convergence of double sequences in probabilistic normed spaces, Mathematical Reports 12 (2010), 359 – 371, https://www.researchgate.net/profile/Mohammad_Mursaleen/publication/265369529.

T. Šalát, B.C. Tripathy and M. Ziman, On some properties of i-convergence, Tatra Mt. Math. Publ. 28 (2004), 274 – 286, https://www.researchgate.net/profile/Binod_Tripathy/publication/228432524.

B. Schweizer and A. Sklar, Statistical metric spaces, Pacific Journal of Mathematics 10 (1960), 313 – 334, https://www.researchgate.net/profile/Binod_Tripathy/publication/228432524.

H. Steinhaus, Sur la convergence ordinaire et la convergence asymptotique, Colloq. Math. 2 (1951), 73 – 74, https://www.impan.pl/en/publishing-house/journals-and-series/colloquium-mathematicum.

B.C. Tripathy and R. Goswami, On triple difference sequences of real numbers in probabilistic normed spaces, Proyecciones (Antofagasta) 33 (2014), 157 – 174, DOI: 10.4067/S0716-09172014000200003.

DOI: http://dx.doi.org/10.26713%2Fcma.v9i4.1197


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