Wijsman and Wijsman Randomly Triple Ideal Convergence Sequences of Sets in Probabilistic Metric Spaces

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DOI: https://doi.org/10.28924/2291-8639-22-2024-154
Published: Sep 16, 2024
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M.H.M. Rashid, Rabaa Al-Maita, Wijsman and Wijsman Randomly Triple Ideal Convergence Sequences of Sets in Probabilistic Metric Spaces, Int. J. Anal. Appl., 22 (2024), 154.

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M.H.M. Rashid, Rabaa Al-Maita

Abstract

Many authors have extended the concept of convergence from number sequences to sequences of sets. In this paper, we focus on two notable adaptations: Wijsman convergence and randomly ideal convergence. We introduce and analyze several new types of convergence for sequences of sets: IψW3-convergence, I∗,ψW3-convergence, IψW3-Cauchy, I∗,ψW3-Cauchy, (IψW3, IψW)-convergence, and (I∗,ψW3, I∗,ψ)-convergence. These new concepts expand the framework of convergence and provide a deeper understanding of the behavior of set sequences under various conditions. Through rigorous analysis, we demonstrate the relationships between these new forms of convergence and their classical counterparts, highlighting their theoretical significance and potential applications in mathematical analysis and related fields. Our findings offer a comprehensive exploration of these advanced convergence concepts, paving the way for further research and development in this area.

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References

  1. N.H. Altaweel, M.H.M. Rashid, O. Albalawi, M.G. Alshehri, N.H.E. Eljaneid, R. Albalawi, On the Ideal Convergent Sequences in Fuzzy Normed Space, Symmetry 15 (2023), 936. https://doi.org/10.3390/sym15040936.
  2. N.H. Altaweel, M.H.M. Rashid, M.G. Alshehri, R. Albalawi, Wijsman Lacunary Statistical Convergence of Triple Sequences with Order α, Missouri J. Math. Sci. In Press.
  3. M. Balcerzak, K. Dems, A. Komisarski, Statistical Convergence and Ideal Convergence for Sequences of Functions, J. Math. Anal. Appl. 328 (2007), 715–729. https://doi.org/10.1016/j.jmaa.2006.05.040.
  4. F. Bani-Ahmad, M.H.M. Rashid, Regarding the Ideal Convergence of Triple Sequences in Random 2-Normed Spaces, Symmetry 15 (2023), 1983. https://doi.org/10.3390/sym15111983.
  5. C. Granados, B.O. Osu, Wijsman and Wijsman Regularly Triple Ideal Convergence Sequences of Sets, Sci. Afr. 15 (2022), e01101. https://doi.org/10.1016/j.sciaf.2022.e01101.
  6. A. Caserta, G. Di Maio, L.D.R. Koˇcinac, Statistical Convergence in Function Spaces, Abstr. Appl. Anal. 2011 (2011), 420419. https://doi.org/10.1155/2011/420419.
  7. E. Dündar, N.P. Akın, Wijsman Regularly Ideal Convergence of Double Sequences of Sets, J. Intell. Fuzzy Syst. 37 (2019), 8159–8166. https://doi.org/10.3233/jifs-190626.
  8. P. Erdös, G. Tenenbaum, Sur les densities de certaines suites d’entiers, Proc. London. Math. Soc. 3 (1989), 417–438.
  9. A.D. Gadjiev, C. Orhan, Some Approximation Theorems via Statistical Convergence, Rocky Mt. J. Math. 32 (2002), 129–138. https://www.jstor.org/stable/44238888.
  10. S. Ghosal, Statistical Convergence of a Sequence of Random Variables and Limit Theorems, Appl. Math. 58 (2013), 423–437. https://doi.org/10.1007/s10492-013-0021-7.
  11. H. Fast, Sur la Convergence Statistique, Colloq. Math. 2 (1951), 241–244.
  12. J.A. Fridy, On Statistical Convergence, Analysis 5 (1985), 301–313.
  13. J.A. Fridy, M.K. Khan, Tauberian Theorems via Statistical Convergence, J. Math. Anal. Appl. 228 (1998), 73–95. https://doi.org/10.1006/jmaa.1998.6118.
  14. P. Kostyrko, W. Wilczy ´nski, T. Šalát, I-Convergence, Real Anal. Exchange 26 (2000/2001), 669–686.
  15. E.P. Klement, R. Mesiar, E. Pap, Triangular Norms: Basic Notions and Properties, in: Logical, Algebraic, Analytic and Probabilistic Aspects of Triangular Norms, Elsevier, 2005: pp. 17–60. https://doi.org/10.1016/B978-044451814-9/50002-1.
  16. O. Kaleva, S. Seikkala, On Fuzzy Metric Spaces, Fuzzy Sets Syst. 12 (1984), 215–229. https://doi.org/10.1016/0165-0114(84)90069-1.
  17. Ö. Ki¸si, F. Nuray, New Convergence Definitions for Sequences of Sets, Abstr. Appl. Anal. 2013 (2013), 852796. https://doi.org/10.1155/2013/852796.
  18. P. Kostyrko, M. Macaj, T. Salat, M. Sleziak, I-Convergence and Extremal I-Limit Points, Math. Slovaca 550 (2005), 443–464. http://dml.cz/dmlcz/132906.
  19. I. Kramosil, J. Michálek, Fuzzy Metrics and Statistical Metric Spaces, Kybernetika, 11 (1975), 336–344. http://dml.cz/dmlcz/125556.
  20. M. Maˇcaj, T. Šalát, Statistical Convergence of Subsequences of a Given Sequence, Math. Bohem. 126 (2001), 191–208. https://doi.org/10.21136/mb.2001.133923.
  21. V. Giménez-Martínez, G. Sánchez-Torrubia, C. Torres-Blanc, A Statistical Convergence Application for the Hopfield Networks, Inf. Theor. Appl. 15 (2008), 84–88.
  22. F. Nuray, B.E. Rhoades, Statistical Convergence of Sequences of Sets, Fasc. Math. 49 (2012), 87–99.
  23. F. Nuray, U. Ulusu, E. Dündar, Lacunary Statistical Convergence of Double Sequences of Sets, Soft Comp. 20 (2016), 2883–2888. https://doi.org/10.1007/s00500-015-1691-8.
  24. S. Pehlivan, M.A. Mamedov, Statistical Cluster Points and Turnpike, Optimization 48 (2000), 91–106. https://doi.org/10.1080/02331930008844495.
  25. M.H.M. Rashid, S.A. Al-Subh, Statistical Convergence with Rough I3-Lacunary and Wijsman Rough I3-Statistical Convergence in 2-Normed Spaces, Int. J. Anal. Appl. 22 (2024), 115. https://doi.org/10.28924/2291-8639-22-2024-115.
  26. H. Steinhaus, Sur la Convergence Ordinaire et la Convergence Asymptotique, Colloq. Math. 2 (1951), 73–34.
  27. B. Schweizer, A. Sklar, Probabilistic Metric Spaces, Dover Publications, New York, 2005.
  28. N. Subramanian, A. Esi, Wijsman Rough Lacunary Statistical Convergence on I Cesaro Triple Sequences, Int. J. Anal. Appl. 16 (2018), 643–653. https://doi.org/10.28924/2291-8639-16-2018-643.
  29. E. Sava¸s, On I-Lacunary Statistical Convergence of Weight g of Sequences of Sets, Filomat 31 (2017), 5315–5322. https://doi.org/10.2298/fil1716315s.
  30. S. Tortop, E. Dündar, Wijsman I2-Invariant Convergence of Double Sequences of Sets, J. Ineq. Spec. Funct. 9 (2018), 90–100.
  31. U. Ulusu, E. Dündar, I-Lacunary Statistical Convergence of Sequences of Sets, Filomat 28 (2014), 1567–1574. https://www.jstor.org/stable/24897654.
  32. U. Ulusu, E. Dündar, Asymptotically I2-Lacunary Statistical Equivalence of Double Sequences of Sets, J. Ineq. Spec. Funct. 7 (2016), 44–56.
  33. U. Ulusu, F. Nuray, Lacunary Statistical Convergence of Sequence of Sets, Progr. Appl. Math. 4 (2012), 99–109.
  34. U. Ulusu, F. Nuray, On Strongly Lacunary Summability of Sequences of Sets, J. Appl. Math. Bioinf. 3 (2013), 75–88.
  35. R.A. Wijsman, Convergence of sequences of convex sets, cones and functions, Bull. Amer. Math. Soc. 70 (1964), 186–188.
  36. R.A. Wijsman, Convergence of Sequences of Convex sets, Cones and Functions II, Trans. Amer. Math. Soc. 123 (1966), 32–45. https://www.jstor.org/stable/1994611.