Skip to main content
SpringerLink
Log in

Wijsman Lacunary \({\mathbf{\mathcal{I}}}\)-Invariant Convergence of Sequences of Sets

  • Research Article
  • Published:
Proceedings of the National Academy of Sciences, India Section A: Physical Sciences Aims and scope Submit manuscript

Abstract

In this paper, we study the concepts of Wijsman lacunary \({\mathcal{I}}\)-invariant convergence \(\left( {{\mathcal{I}}_{\sigma \theta }^{W} } \right),\) Wijsman lacunary \({\mathcal{I}}^{ *}\)-invariant convergence \(\left( {{\mathcal{I}}_{\sigma \theta }^{ *W} } \right),\) Wijsman \(p\)-strongly lacunary invariant convergence \(\left( {[WN_{\sigma \theta } ]_{p} } \right)\) of sequences of sets and investigate the relationships among Wijsman lacunary invariant convergence, \([WN_{\sigma \theta } ]_{p}\), \({\mathcal{I}}_{\sigma \theta }^{W}\) and \({\mathcal{I}}_{\sigma \theta }^{ *W}\). Also, we introduce the concepts of \({\mathcal{I}}_{\sigma \theta }^{W}\)-Cauchy sequence and \({\mathcal{I}}_{\sigma \theta }^{ *W}\)-Cauchy sequence of sets.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Fast H (1951) Sur la convergence statistique. Colloq Math 2:241–244

    Article  MathSciNet  MATH  Google Scholar 

  2. Schoenberg IJ (1959) The integrability of certain functions and related summability methods. Am Math Monthly 66:361–375

    Article  MathSciNet  MATH  Google Scholar 

  3. Nuray F, Ruckle WH (2000) Generalized statistical convergence and convergence free spaces. J Math Anal Appl 245:513–527

    Article  MathSciNet  MATH  Google Scholar 

  4. Kostyrko P, Šalát T, Wilczyński W (2000) \({\mathcal{I}}\)-Convergence. Real Anal Exchange 6(2):669–686

    Article  MATH  Google Scholar 

  5. Nuray F, Rhoades BE (2012) Statistical convergence of sequences of sets. Fasc Math 49:87–99

    MathSciNet  MATH  Google Scholar 

  6. Ulusu U, Nuray F (2012) Lacunary statistical convergence of sequence of sets. Progress Appl Math 4(2):99–109

    MATH  Google Scholar 

  7. Kişi Ö, Nuray F (2013) A new convergence for sequences of sets. Abstract Appl Anal. https://doi.org/10.1155/2013/852796

    Article  MathSciNet  MATH  Google Scholar 

  8. Baronti M, Papini P (1986) Convergence of sequences of sets. Methods of functional analysis in approximation theory. Birkhäuser, Basel, pp 133–155

    Google Scholar 

  9. Beer G (1985) On convergence of closed sets in a metric space and distance functions. Bull Aust Math Soc 31:421–432

    Article  MathSciNet  MATH  Google Scholar 

  10. Beer G (1994) Wijsman convergence: a survey. Set-Valued Anal 2:77–94

    Article  MathSciNet  MATH  Google Scholar 

  11. Sever Y, Ulusu U, Dündar E (2014) On strongly \({\mathcal{I}}\) and \({\mathcal{I}}^{ *}\)-lacunary convergence of sequences of sets. In: AIP conference proceedings, vol 1611, no 357, pp 7

  12. Talo Ö, Sever Y, Başar F (2016) On statistically convergent sequences of closed set. Filomat 30(6):1497–1509

    Article  MathSciNet  MATH  Google Scholar 

  13. Wijsman RA (1964) Convergence of sequences of convex sets, cones and functions. Bull Am Math Soc 70:186–188

    Article  MathSciNet  MATH  Google Scholar 

  14. Wijsman RA (1966) Convergence of sequences of convex sets, cones and functions II. Trans Am Math Soc 123(1):32–45

    Article  MathSciNet  MATH  Google Scholar 

  15. Ulusu U, Nuray F (2015) Lacunary statistical summability of sequences of sets. Konuralp J Math 3(2):176–184

    MathSciNet  MATH  Google Scholar 

  16. Ulusu U, Nuray F (2013) On strongly lacunary summability of sequences of sets. J Appl Math Bioinf 3(3):75–88

    MATH  Google Scholar 

  17. Ulusu U, Dündar E (2013) \({\mathcal{I}}\)-lacunary statistical convergence of sequences of sets. Filomat 28(8):1567–1574

    Article  MathSciNet  MATH  Google Scholar 

  18. Raimi RA (1963) Invariant means and invariant matrix methods of summability. Duke Math J 30:81–94

    Article  MathSciNet  MATH  Google Scholar 

  19. Schaefer P (1972) Infinite matrices and invariant means. Proc Am Math Soc 36:104–110

    Article  MathSciNet  MATH  Google Scholar 

  20. Mursaleen M (1979) On finite matrices and invariant means. Indian J Pure and Appl Math 10:457–460

    MathSciNet  MATH  Google Scholar 

  21. Pancaroǧlu N, Nuray F (2013) Statistical lacunary invariant summability. Theor Math Appl 3(2):71–78

    MATH  Google Scholar 

  22. Nuray F, Gök H, Ulusu U (2011) \({\mathcal{I}}_{\sigma }\)-convergence. Math Commun 16:531–538

    MathSciNet  MATH  Google Scholar 

  23. Mursaleen M (1983) Matrix transformation between some new sequence spaces. Houston J Math 9:505–509

    MathSciNet  MATH  Google Scholar 

  24. Savaş E, Nuray F (1993) On σ-statistically convergence and lacunary σ-statistically convergence. Math Slovaca 43(3):309–315

    MathSciNet  MATH  Google Scholar 

  25. Savaş E (1989) Some sequence spaces involving invariant means. Indian J Math 31:1–8

    MathSciNet  MATH  Google Scholar 

  26. Ulusu U, Nuray F (in review) Lacunary \({\mathcal{I}}_{\sigma }\)-convergence

  27. Pancaroǧlu Akın N, Dündar E, Nuray F (in review) Wijsman \({\mathcal{I}}\)-invariant convergence of sequences of sets

  28. Mursaleen M, Edely OHH (2009) On the invariant mean and statistical convergence. Appl Math Lett 22:1700–1704

    Article  MathSciNet  MATH  Google Scholar 

  29. Nuray F, Savaş E (1994) Invariant statistical convergence and A-invariant statistical convergence. Indian J Pure Appl Math 10:267–274

    MathSciNet  MATH  Google Scholar 

  30. Pancaroǧlu N, Nuray F (2013) On invariant statistically convergence and lacunary invariant statistically convergence of sequences of sets. Progress Appl Math 5(2):23–29

    Google Scholar 

  31. Lorentz G (1948) A contribution to the theory of divergent sequences. Acta Math 80:167–190

    Article  MathSciNet  MATH  Google Scholar 

  32. Savaş E (1989) Strong σ-convergent sequences. Bull Calcutta Math 81:295–300

    MathSciNet  MATH  Google Scholar 

  33. Nath M, Nath B, Roy S (2017) On some new classes of ideal convergent triple sequences of fuzzy numbers associated with multiplier sequences. Adv Fuzzy Sets Syst 22(1):1–23

    Article  MathSciNet  MATH  Google Scholar 

  34. Saha S, Roy S (accepted) Classes of multiplier ideal convergent triple sequence spaces of fuzzy real numbers defined by Orlicz function. Kuwait J Sci

  35. Saha S, Roy S (2017) Some \({\mathcal{I}}\)-convergent triple sequence spaces of fuzzy numbers defined by Orlicz function. Int J Control Theory Appl 10(19):115

    Google Scholar 

  36. Saha S, Roy S (2016) On lacunary p-absolutely summable fuzzy real-valued triple sequence space. Int J Adv Inf Sci Technol 55(55):84

    Google Scholar 

  37. Mursaleen M, Mohiuddine SA, Edely OHH (2010) On the ideal convergence of double sequences in intuitionistic fuzzy normed spaces. Comput Math Appl 59(2):603–611

    Article  MathSciNet  MATH  Google Scholar 

  38. Mursaleen M, Mohiuddine SA (2010) On ideal convergence of double sequences in probabilistic normed spaces. Math Rep 12(62):359–371

    MathSciNet  MATH  Google Scholar 

  39. Mursaleen M, Alotaibi A (2011) On \({\mathcal{I}}\)-convergence in random 2-normed spaces. Mathematica Slovaca 61(6):933–940

    MathSciNet  MATH  Google Scholar 

  40. Mursaleen M, Mohiuddine S (2012) On ideal convergence in probabilistic normed spaces. Mathematica Slovaca 62(1):49–62

    Article  MathSciNet  MATH  Google Scholar 

  41. Dündar E, Altay B (2014) \({\mathcal{I}}_{2}\)-convergence and \({\mathcal{I}}_{2}\)-Cauchy double sequences. Acta Mathematica Scientia 34(2):343–353

    MathSciNet  MATH  Google Scholar 

  42. Ulusu U, Dündar E (2019) Asymptotically lacunary \({\mathcal{I}}_{2}\)-invariant equivalence. J Intell Fuzzy Syst 36(1):467–472

    Google Scholar 

  43. Dündar E, Talo O (2013) \({\mathcal{I}}_{2}\)-convergence of double sequences of fuzzy numbers. Iranian J Fuzzy Syst 10(3):37–50

    MathSciNet  MATH  Google Scholar 

  44. Dems K (2004/2005) On \({\mathcal{I}}\)-cauchy sequences. Real Anal Exchange 30:123–128

  45. Nabiev A, Pehlivan S, Gürdal M (2007) On \({\mathcal{I}}\)-cauchy sequences. Taiwanese J Math 11(2):569–5764

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

This study is supported by Afyon Kocatepe University Scientific Research Coordination Unit with the project number 17.KARİYER.20 conducted by Erdinç Dündar.

Author information

Authors and Affiliations

  1. Department of Mathematics, Afyon Kocatepe University, Afyonkarahisar, Turkey

    Erdinç Dündar, Nimet Pancaroğlu Akın & Uğur Ulusu

Authors
  1. Erdinç Dündar
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Nimet Pancaroğlu Akın
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Uğur Ulusu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Erdinç Dündar.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Dündar, E., Pancaroğlu Akın, N. & Ulusu, U. Wijsman Lacunary \({\mathbf{\mathcal{I}}}\)-Invariant Convergence of Sequences of Sets. Proc. Natl. Acad. Sci., India, Sect. A Phys. Sci. 91, 517–522 (2021). https://doi.org/10.1007/s40010-020-00694-w

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40010-020-00694-w

Keywords

2010 Mathematics Subject Classification

Search

Navigation