Invariant and Lacunary Invariant Statistical Convergence of Order $\eta$ for Double Set Sequences

Abstract

In this study, for double set sequences, we introduced the notions of invariant and lacunary invariant statistical convergence of order $\eta$ ($0<\eta\leq 1$) in the Wijsman sense. Also, we investigated the inclusion relations between them.

Keywords

• Double lacunary sequence
• invariant statistical convergence
• order $\eta$
• double set sequences
• convergence in the Wijsman sense

References

1. M. Baronti and P. Papini, Convergence of sequences of sets, In: Methods of Functional Analysis in Approximation Theory (pp.133-155), Birkhäuser, Basel, (1986).
2. G. Beer, Wijsman convergence: A survey, Set-Valued Anal., 2(1) (1994), 77-94.
3. R. Çolak, Statistical convergence of order α, In: Modern Methods in Analysis and Its Applications (pp.121-129), Anamaya Publishers, New Delhi, (2010).
4. E. Gülle and U. Ulusu, Double Wijsman lacunary statistical convergence of order α, J. Appl. Math. Inform., 39(3-4) (2021), 303-319.
5. M. Mursaleen and O.H.H. Edely, Statistical convergence of double sequences, J. Math. Anal. Appl., 288(1) (2003), 223-231.
6. F. Nuray and B.E. Rhoades, Statistical convergence of sequences of sets, Fasc. Math., 49 (2012), 87-99.
7. F. Nuray, U. Ulusu and E. Dündar, Lacunary statistical convergence of double sequences of sets, Soft Comput., 20(7) (2016), 2883-2888.
8. F. Nuray and U. Ulusu, Lacunary invariant statistical convergence of double sequences of sets, Creat. Math. Inform., 28(2) (2019), 143-150.

9. F. Nuray, E. Dündar and U. Ulusu, Wijsman statistical convergence of double sequences of sets, Iran. J. Math. Sci. Inform., 16(1) (2021), 55-64.
10. N. Pancarog ̆lu and F. Nuray, On invariant statistically convergence and lacunary invariant statistical convergence of sequences of sets, Progress Appl. Math., 5(2) (2013), 23-29.
11. R.F. Patterson and E. Savaş, Lacunary statistical convergence of double sequences, Math. Commun., 10(1) (2005), 55-61.
12. A. Pringsheim, Zur theorie der zweifach unendlichen Zahlenfolgen, Math. Ann., 53(3) (1900), 289-321.
13. E. Savaş and R.F. Patterson, Double σ-convergence lacunary statistical sequences, J. Comput. Anal. Appl., 11(4) (2009), 610-615.
14. E. Savaş, Double almost statistical convergence of order α, Adv. Difference Equ., 2013(62) (2013), 9 pages.
15. E. Savaş, Double almost lacunary statistical convergence of order α, Adv. Difference Equ., 2013(254) (2013), 10 pages.
16. E. Savaş, On I-lacunary statistical convergence of order α for sequences of sets, Filomat, 29(6) (2015), 1223-1229.
17. H. Şengül and M. Et, On lacunary statistical convergence of order α, Acta Math. Sci. Ser. B, 34(2) (2014), 473-482.
18. H. Şengül and M. Et, On I-lacunary statistical convergence of order α of sequences of sets, Filomat, 31(8) (2017), 2403-2412.
19. U. Ulusu and F. Nuray, Lacunary statistical convergence of sequences of sets, Progress Appl. Math., 4(2) (2012), 99-109.
20. U. Ulusu and E. Gülle, Some statistical convergence types of order α for double set sequences, Facta Univ. Ser. Math. Inform., 35(3) (2020), 595-603.