Invariant and Lacunary Invariant Statistical Equivalence of Order β for Double Set Sequences

Year 2021, Volume: 11 Issue: 2, 421 - 430, 31.12.2021

Uğur Ulusu Erdinç Dündar Fatih Nuray

https://doi.org/10.37094/adyujsci.956986

Cited By: 1

Abstract

In this study, as a new approach to the concept of asymptotical equivalence in the Wijsman sense for double set sequences, the new concepts which are called asymptotical invariant statistical equivalence of order β and asymptotical lacunary invariant statistical equivalence of order β (0<β≤1) in the Wijsman sense for double set sequences are introduced and explained with examples. In addition, the existence of some relations between these concepts and furthermore, the relationships between these concepts and previously studied asymptotical equivalence concepts in the Wijsman sense for double set sequences are investigated.

Keywords

Asymptotical equivalence, Convergence in the Wijsman sense, Double lacunary sequence, Invariant statistical convergence, Order β, Sequences of sets, Sequences of sets

Çift Küme Dizileri için β ıncı Mertebeden İnvaryant ve Lacunary İnvaryant İstatistiksel Denklik

Abstract

Bu çalışmada, çift küme dizileri için Wijsman anlamında asimptotik denklik kavramına yeni bir yaklaşım olarak, çift küme dizileri için Wijsman anlamında β (0<β≤1) yıncı mertebeden asimptotik invaryant istatistiksel denklik ve asimptotik lacunary invaryant istatistiksel denklik olarak adlandırılan yeni kavramlar tanıtıldı ve örneklerle açıklandı. Ayrıca, bu kavramlar arasında bazı ilişkilerin varlığı ve dahası bu kavramlar ve daha önceden çift küme dizileri için Wijsman anlamında çalışılmış asimptotik denklik kavramları arasındaki ilişkiler incelendi.   

Keywords

Asimptotik denklik; Wijsman anlamında yakınsaklık; Çift lacunary dizi; İnvaryant istatistiksel yakınsaklık; β ıncı mertebe; Küme dizisi.

References

• [1] Pringsheim, A., Zur theorie der zweifach unendlichen Zahlenfolgen, Mathematische Annalen, 53, 289-321, 1900.
• [2] Mursaleen, M., Edely, O.H.H., Statistical convergence of double sequences, Journal of Mathematical Analysis and Applications, 288, 223-231, 2003.
• [3] Patterson, R.F., Savaş, E., Lacunary statistical convergence of double sequences, Mathematical Communications, 10, 55-61, 2005.
• [4] Savaş, E., Patterson, R.F., Double σ-convergence lacunary statistical sequences, Journal of Computational Analysis and Applications, 11, 610-615, 2009.
• [5] Savaş, E., Double almost statistical convergence of order α, Advances in Difference Equations, 62, 1-9, 2013.
• [6] Savaş, E., Double almost lacunary statistical convergence of order α, Advances in Difference Equations, 254, 1-10, 2013.
• [7] Patterson, R.F., Rates of convergence for double sequences, Southeast Asian Bulletin of Mathematics, 26, 469-478, 2003.
• [8] Baronti, M., Papini, P., Convergence of sequences of sets, ın: methods of functional analysis in approximation theory, Birkhäuser, Basel, 133-155, 1986.
• [9] Beer, G., Wijsman convergence: A survey, Set-Valued Analysis, 2, 77-94, 1994.
• [10] Nuray, F., Ulusu, U., Dündar, E., Lacunary statistical convergence of double sequences of sets, Soft Computing, 20, 2883-2888, 2016.
• [11] Nuray, F., Ulusu, U., Lacunary invariant statistical convergence of double sequences of sets, Creative Mathematics and Informatics, 28, 143-150, 2019.
• [12] Nuray, F., Dündar, E., Ulusu, U., Wijsman statistical convergence of double sequences of sets, Iranian Journal of Mathematical Sciences and Informatics, 16, 55-64, 2021.
• [13] Nuray, F., Patterson, R.F., Dündar, E., Asymptotically lacunary statistical equivalence of double sequences of sets, Demonstratio Mathematica, 49, 183-196, 2016.
• [14] Çolak, R., Statistical convergence of order α, In: Modern Methods in Analysis and Its Applications, Anamaya Publishers, New Delhi, 121-129, 2010.
• [15] Gülle, E., Double Wijsman asymptotically statistical equivalence of order α, Journal of Intelligent & Fuzzy Systems, 38, 2081-2087, 2020.
• [16] Pancaroğlu, N., Nuray, F., On invariant statistically convergence and lacunary invariant statistical convergence of sequences of sets, Progress in Applied Mathematics, 5, 23-29, 2013.
• [17] Pancaroğlu, N., Nuray, F., Savaş, E., On asymptotically lacunary invariant statistical equivalent set sequences, AIP Conference Proceedings, 1558, 780-781, 2013.
• [18] Savaş, E., Nuray, F., On σ-statistically convergence and lacunary σ-statistically convergence, Mathematica Slovaca, 43, 309-315, 1993.
• [19] Savaş, E., Asymptotically I-lacunary statistical equivalent of order α for sequences of sets, Journal of Nonlinear Science and its Applications, 10, 2860-2867, 2017.
• [20] Şengül, H., Et, M., On lacunary statistical convergence of order α, Acta Mathematica Scientia. Series B, 34, 473-482, 2014.
• [21] Şengül, H., On Wijsman I-lacunary statistical equivalence of order (η,μ), Journal of Inequalities and Special Functions, 9, 92-101, 2018.
• [22] Ulusu, U., Nuray, F., On asymptotically lacunary statistical equivalent set sequences, Journal of Mathematics, 310438, 1-5, 2013.
• [23] Ulusu, U., Gülle, E., Some statistical convergence types of order α for double set sequences, Facta Universitatis, Series: Mathematics and Informatics, 35, 595-603, 2020.
• [24] Ulusu, U., Nuray, F., Lacunary I-invariant convergence, Cumhuriyet Science Journal, 41, 617-624, 2020.
• [25] Ulusu, U., Dündar, E., Pancaroğlu Akın, N., Lacunary invariant statistical equivalence for double set sequences, Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, (Accepted).