On Δᵐ-statistical convergence double sequences in intuitionistic fuzzy normed spaces

Authors

  • Reena Antal Chandigarh University.
  • Meenakshi Chawla Chandigarh University.
  • Vijay Kumar Chandigarh University.
  • Bipan Hazarika Gauhati University.

DOI:

https://doi.org/10.22199/issn.0717-6279-4633

Keywords:

statistical convergence, Δᵐ-statistical convergence, double sequence, Intuitionistic fuzzy normed spaces

Abstract

In the present paper, the basic objective of our work is to define Δ-statistical convergence in the setup of intuitionistic fuzzy normed spaces for double sequences. We have proved some examples which shows this method of convergence is more generalized. Further, we defined the Δ-statistical Cauchy sequences in these spaces and given the Cauchy convergence criterion for this new notion of convergence.

Author Biographies

Reena Antal, Chandigarh University.

Department of Mathematics.

Meenakshi Chawla, Chandigarh University.

Department of Mathematics.

Vijay Kumar, Chandigarh University.

Department of Mathematics.

Bipan Hazarika, Gauhati University.

Department of Mathematics.

References

R. Antal, M. Chawla, V. Kumar, “Statistical Λ-convergence in intuitionistic fuzzy normed spaces”, Buletinul Academiei de Stiinte a Republicii Moldova. Matematica, no. 3, pp. 22-33, 2019. [Online] Available: https://bit.ly/3Q0uQHd

R. Antal, M. Chawla, V. Kumar, “Generalized statistical convergence of order α in random n-normed space”, Advances and Applications in Mathematical Sciences, vol. 18, no. 8, pp. 715-729, 2019.

M. Chawla, M. S. Saroa, and V. Kumar, “On Δᵐ-statistical convergence of order ∝ in random 2-normed space”, Miskolc Mathematical Notes, vol. 16, no. 2, pp. 1003–1015, 2015.https://doi.org/10.18514/mmn.2015.821

M. Chawla and Palak, “Lacunary statistical convergence of double sequences of order α in probabilistic normed spaces”, Advances in Mathematics: Scientific Journal, vol. 9, no. 9, pp. 7257-7268, 2020. https://doi.org/10.37418/amsj.9.9.75

A. Esi, “On some new generalized difference double sequence spaces defined by Orlicz functions”, Matematika, vol. 27, pp. 31-40, 2011.

A. Esi and M. Kemal Ozdemir, “Generalized Δᵐ-Statistical convergence in probabilistic normed space”, Journal of Computational Analysis and Applications, vol. 13, no. 5, pp. 923-932, 2011.

M. Et and R. Çolak, “On some generalized difference sequence spaces”, Soochow Journal of Mathematical, vol. 21, no. 4, pp. 377-386, 1995

M. Et, F. Nuray, “Δᵐ-Statistical convergence”, Indian Journal of Pure and Applied Mathematics, vol. 32, no. 6, pp. 961-969, 2001.

H. Fast, “Sur La Convergence Statistique”, Colloquium Mathematicum, vol. 2, no. 3-4, pp. 241–244, 1951. https://doi.org/10.4064/cm-2-3-4-241-244

J. A. Fridy, “On statistical convergence”, Analysis, vol. 5, no. 4, pp. 301–313, 1985. https://doi.org/10.1524/anly.1985.5.4.301

B. Hazarika, “Lacunary generalized difference statistical convergence in random 2-normed spaces”, Proyecciones (Antofagasta), vol. 31, no. 4, pp. 373–390, 2012. https://doi.org/10.4067/s0716-09172012000400006

B. Hazarika, A. Alotaibi, and S. A. Mohiuddine, “Statistical convergence in measure for double sequences of fuzzy-valued functions”, Soft Computing, vol. 24, no. 9, pp. 6613–6622, 2020. https://doi.org/10.1007/s00500-020-04805-y

S. Karakus, “Statistical convergence on probabilistic normed space”, Mathematical Communications, vol. 12, pp. 11-23, 2007. [Online] Available: https://bit.ly/3ahCZq4

S. Karakus, K. Demirci, and O. Duman, “Statistical convergence on intuitionistic fuzzy normed spaces”, Chaos, Solitons and Fractals, vol. 35, no. 4, pp. 763–769, 2008. https://doi.org/10.1016/j.chaos.2006.05.046

H. Kizmaz, “On certain sequence spaces”, Canadian Mathematical Bulletin, vol. 24, no. 2, pp. 169–176, 1981. https://doi.org/10.4153/cmb-1981-027-5

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. https://doi.org/10.1017/s0305004100065312

S. A. Mohiuddine and Q. M. Danish Lohani, “On generalized statistical convergence in intuitionistic fuzzy normed space”, Chaos, Solitons and Fractals, vol. 42, no. 3, pp. 1731–1737, 2009. https://doi.org/10.1016/j.chaos.2009.03.086

S. A. Mohiuddine, B. Hazarika, and A. Alotaibi, “On statistical convergence of double sequences of fuzzy valued functions”, Journal of Intelligent and Fuzzy Systems, vol. 32, no. 6, pp. 4331–4342, 2017.https://doi.org/10.3233/jifs-16974

F. Móricz, “Statistical convergence of multiple sequences”, Archiv der Mathematik, vol. 81, no. 1, pp. 82–89, 2003. https://doi.org/10.1007/s00013-003-0506-9

M. Mursaleen, A. K. Noman, “On the spaces of λ-convergent and bounded sequences”, Thai Journal of Mathematics, vol. 8, no. 2, pp. 311-329, 2010.

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. https://doi.org/10.1016/j.cam.2009.07.005

M. Mursaleen and S. A. Mohiuddine, “Statistical convergence of double sequences in intuitionistic fuzzy normed spaces”, Chaos, Solitons and Fractals, vol. 41, no. 5, pp. 2414–2421, 2009. https://doi.org/10.1016/j.chaos.2008.09.018

M. Mursaleen, O. H. H. Edely, “Statistical convergence of double sequences”, Journal of Mathematical Analysis and Applications, vol. 288, no. 1, pp. 223-231, 2003.

J. H. Park, “Intuitionistic fuzzy metric spaces”, Chaos, Solitons and Fractals, vol. 22, no. 5, pp. 1039–1046, 2004. https://doi.org/10.1016/j.chaos.2004.02.051

A. Pringsheim, “Zur Theorie der Zweifach Unendlichen zahlenfolgen”, Mathematische Annalen, vol. 53, no. 3, pp. 289–321, 1900. https://doi.org/10.1007/bf01448977

R. Saadati and J. H. Park, “On the intuitionistic fuzzy topological spaces”, Chaos, Solitons and Fractals, vol. 27, no. 2, pp. 331–344, 2006. https://doi.org/10.1016/j.chaos.2005.03.019

B. Schweizer and A. Sklar, “Statistical Metric Spaces”, Pacific Journal of Mathematics, vol. 10, no. 1, pp. 313–334, 1960. https://doi.org/10.2140/pjm.1960.10.313

M. Sen and M. Et, “Lacunary statistical and lacunary strongly convergence of generalized difference sequences in intuitionistic fuzzy normed linear spaces”, Boletim da Sociedade Paranaense de Matemática, vol. 38, no. 1, pp. 117–129, 2018. https://doi.org/10.5269/bspm.v38i1.34814

B. C. Tripathy, B. Sarma, “On some classes of difference double sequence spaces”, Fasciculi Mathematici, vol. 41, pp. 135-142, 2009.

B. C. Tripathy and B. Sarma, “Statistically convergent difference double sequence spaces”, Acta Mathematica Sinica, English Series, vol. 24, no. 5, pp. 737–742, 2008. https://doi.org/10.1007/s10114-007-6648-0

B. C. Tripathy and S. Borgohain, “Statistically convergent difference sequence spaces of fuzzy real numbers defined by Orlicz function”, Thai Journal of Mathematics, vol. 11, no. 2, pp. 357-370, 2013.

B. C. Tripathy and M. Sen, “On lacunary strongly almost convergent double sequences of fuzzy numbers”, Annals of the University of Craiova - Mathematics and Computer Science series, vol. 42, no. 2, pp. 254-259, 2015

B. C. Tripathy and R. Goswami, “Statistically convergent multiple sequences in probabilistic normed spaces”, University Politehnica of Bucharest - Scientific Bulletin- Serie A, vol. 78, no. 4, pp. 83-94, 2016.

B. C. Tripathy, “Statistically convergent double sequences”, Tamkang Journal of Mathematics, vol. 34, no. 3, pp. 231–237, 2003. https://doi.org/10.5556/j.tkjm.34.2003.314

L. A. Zadeh, “Fuzzy sets”, Information and Control, vol. 8, no. 3, pp. 338–353, 1965. https://doi.org/10.1016/s0019-9958(65)90241-x

Published

2022-06-01

How to Cite

[1]
R. . Antal, M. . Chawla, V. . Kumar, and B. Hazarika, “On Δᵐ-statistical convergence double sequences in intuitionistic fuzzy normed spaces”, Proyecciones (Antofagasta, On line), vol. 41, no. 3, pp. 697-713, Jun. 2022.

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