Rough Statistical Convergence of Double Sequences in Probabilistic Normed Spaces

Authors

DOI:

https://doi.org/10.26713/cma.v14i5.2239

Keywords:

Probabilistic normed space, Rough statistical convergence of double sequences, Rough statistical cluster points of double sequences

Abstract

In this paper, we have defined rough convergence and rough statistical convergence of double sequences in probabilistic normed spaces which is more generalized version than the rough statistical convergence of double sequences in normed linear spaces. Also, we have defined rough statistical cluster points of double sequences and then, investigated some important results associated with the set of rough statistical limits of double sequences in these spaces. Moreover, in the same spaces, we have proved an important relation between the set of all rough statistical cluster points and rough statistical limits under certain condition.

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References

C. Alsina, B. Schweizer and A. Sklar, On the definition of a probabilistic normed space, Aequationes Mathematicae 46 (1993), 91 – 98, DOI: 10.1007/BF01834000.

A. Aghajani and K. Nourouzi, Convex sets in probabilistic normed spaces, Chaos, Solitons & Fractals 36(2) (2008), 322 – 328, DOI: 10.1016/j.chaos.2006.06.051.

S. Aytar, Rough statistical convergence, Numerical Functional Analysis and Optimization 29(3-4) (2008), 291 – 303, DOI: 10.1080/01630560802001064.

R. Antal, M. Chawla and V. Kumar, Rough statistical convergence in intuitionistic fuzzy normed spaces, Filomat 35(13) (2021), 4405 – 4416, URL: https://journal.pmf.ni.ac.rs/filomat/index.php/filomat/article/view/14757.

R. Antal, M. Chawla and V. Kumar, Rough statistical convergence in probabilistic normed spaces, Thai Journal of Mathematics 20(4) (2022), 1707 – 1719, URL: https://thaijmath2.in.cmu.ac.th/index.php/thaijmath/article/view/1433.

J. S. Connor, The statistical and strong p-Cesaro convergence of sequences, Analysis 8(1-2) (1988), 47 – 64, DOI: 10.1524/anly.1988.8.12.47.

H. Çakallı and E. Sava¸s, Statistical convergence of double sequences in topological groups, Journal of Computational Analysis and Applications 12(2) (2010), 421 – 426.

F. Nuray, E. Dündar and U. Ulusu, Wijsman statistical convergence of double sequences of set, Iranian Journal of Mathematical Sciences and Informatics 16(1) (2021), 55 – 64, DOI: DOI:10.29252/ijmsi.16.1.55.

H. Fast, Sur la convergence statistique, Colloquium Mathematicae 2(3-4) (1951), 241 – 244, URL: https://eudml.org/doc/209960.

M. J. Frank, Probabilistic topological spaces, Journal of Mathematical Analysis and Applications 34(1) (1971), 67 – 81, DOI: 10.1016/0022-247X(71)90158-2.

J. A. Fridy, On statistical convergence, Analysis 5(4) (1985), 301 – 313, DOI: 10.1524/anly.1985.5.4.301.

J. A. Fridy and C. Orhan, Lacunary statistical convergence, Pacific Journal of Mathematics 160(1) (1993), 43 – 51, URL: https://projecteuclid.org/journals/pacific-journal-of-mathematics/volume-160/issue-1/Lacunary-statistical-convergence/pjm/1102624563.full.

S. Ghosal and M. Banerjee, Effects on rough I-lacunary statistical convergence to induce the weighted sequence, Filomat 32(10) (2018), 3557 – 3568, DOI: DOI:10.2298/FIL1810557G.

N. Hossain and A. K. Banerjee, Rough I-convergence in intuitionistic fuzzy normed spaces, Bulletin of Mathematical Analysis and Applications 14(4) (2022), 1 – 10, URL: https://emis.de/journals/BMAA/repository/docs/BMAA14-4-1.pdf.

E. P. Klement, R. Mesiar and E. Pap, Triangular norms. Position paper I: basic analytical and algebraic properties, Fuzzy Sets and Systems 143(1) (2004), 5 – 26, DOI: 10.1016/j.fss.2003.06.007.

S. Karakus, Statistical convergence on probalistic normed spaces, Mathematical Communications 12(1) (2007), 11 – 23, URL:https://hrcak.srce.hr/file/19396.

S. Karakus and K. Demırcı, Statistical convergence of double sequences on probabilistic normed spaces, International Journal of Mathematics and Mathematical Sciences 2007 (2007), Article ID 014737, 11 pages, DOI: 10.1155/2007/14737.

Ö. Kisi and H. K. Ünal, Rough statistical convergence of difference double sequences in normed linear spaces, Honam Mathematical Journal 43(1) (2021), 47 – 58, DOI: 10.5831/HMJ.2021.43.1.47.

K. Menger, Statistical metrics, Proceedings of the National Academy of Sciences 28(12) (1942), 535 – 537, DOI: 10.1073/pnas.28.12.535.

M. Mursaleen, λ-statistical convergence, Mathematica Slovaca 50(1) (2000), 111 – 115, URL: https://dml.cz/handle/10338.dmlcz/136769.

M. Mursaleen and O. H. H. Edely, Statistical convergence of double sequences, Journal of Mathematical Analysis and Applications 288(1) (2003), 223 – 231, DOI: 10.1016/j.jmaa.2003.08.004.

M. Mursaleen and S. A. Mohiuddine, Statistical convergence of double sequences in intuitionistic fuzzy normed spaces, Chaos, Solitons & Fractals 41(5) (2009), 2414 – 2421, DOI: 10.1016/j.chaos.2008.09.018.

P. Malik and M. Maity, On rough convergence of double sequence in normed linear spaces, Bulletin of the Allahabad Mathematical Society 28(1) (2013), 89 – 99, DOI: ?????.

P. Malik and M. Maity, On rough statistical convergence of double sequences in normed linear spaces, Afrika Matematika 27(1-2) (2016), 141 – 148, DOI: 10.1007/s13370-015-0332-9.

F. Nuray and E. Sava¸s, Statistical convergence of sequences of fuzzy numbers, Mathematica Slovaca 45(3) (1995), 269 – 273, URL: https://dml.cz/handle/10338.dmlcz/129143.

A. Özcan and A. Or, Rough statistical convergence of double sequences in intuitionistic fuzzy normed spaces, Journal of New Results in Science 11(3) (2022), 233 – 246, DOI: 10.54187/jnrs.1198582.

H. X. Phu, Rough convergence in normed linear spaces, Numerical Functional Analysis and Optimization 22(1-2) (2001), 199 – 222, DOI: 10.1081/NFA-100103794.

H. X. Phu, Rough continuity of linear operators, Numerical Functional Analysis and Optimization 23(1-2) (2002), 139 – 146, DOI: 10.1081/NFA-120003675.

H. X. Phu, Rough convergence in infinite dimensional normed spaces, Numerical Functional Analysis and Optimization 24(3-4) (2003), 285 – 301, DOI: 10.1081/NFA-120022923.

H. Steinhaus, Sur la convergence ordinaire et la convergence asymptotique, Colloquium Mathematicum 2(1) (1951), 73 – 74.

A. N. Šerstnev, Random normed spaces: Problems of completeness, Uchenye Zapiski Kazanskogo Universiteta 122 (1962), 3 – 20.

T. Šalát, On statistically convergent sequences of real numbers, Mathematica Slovaca 30(2) (1980), 139 – 150, URL: http://eudml.org/doc/34081.

B. Schweizer and A. Sklar, Probabilistic Metric Spaces, North Holland, Amsterdam (1983).

S. Sarabadan and S. Talebi, Statistical convergence of double sequences in 2-normed spaces, International Journal of Contemporary Mathematical Sciences 6 (5-8) (2011), 373 – 380, URL: http://www.m-hikari.com/ijcms-2011/5-8-2011/talebiIJCMS5-8-2011.pdf.

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Published

31-12-2023
CITATION

How to Cite

Mondal, R., & Hossain, N. (2023). Rough Statistical Convergence of Double Sequences in Probabilistic Normed Spaces. Communications in Mathematics and Applications, 14(5), 1835–1846. https://doi.org/10.26713/cma.v14i5.2239

Issue

Vol. 14 No. 5 (2023)

Section

Research Article

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