On (λ,A)-statistical convergence of order α

Year 2019, Volume: 68 Issue: 2, 2094 - 2103, 01.08.2019

Hacer Şengül , Özlem Koyun

https://doi.org/10.31801/cfsuasmas.512628

https://izlik.org/JA59JF79AD

Abstract

In the paper [B. de Malafosse and V. Rakočević, Linear Algebra Appl. 420, no. 2-3, (2007), 377--387], authors defined the concept of (λ,A)-statistical convergence. In this paper, the concept of (λ,A)-statistical convergence is generalized to (λ,A)-statistical convergence of order α. Also, we introduce the concept of strong (V,λ,A)-convergence of order α and give some inclusion relations between the concepts of (λ,A)-statistical convergence of order α and strong (V,λ,A)-convergence of order α.

Keywords

λ-statistical convergence , statistical convergence , strong (V lamda A)-convergence

References

• Atlihan, Ö. G., Ünver, M. and Duman, O., Korovkin theorems on weighted spaces: revisited, Period. Math. Hungar., 75(2) (2017) 201--209.
• Bhardwaj, V. K. and Dhawan, S., Density by moduli and lacunary statistical convergence, Abstr. Appl. Anal., 2016, Art. ID 9365037, 11 pp.
• Bilgin, T., Lacunary strong A-convergence with respect to a modulus, Studia Univ. Babeş-Bolyai Math., 46(4) (2001) 39--46.
• Bilgin, T., Lacunary strong A-convergence with respect to a sequence of modulus functions, Appl. Math. Comput,. 151(3) (2004) 595--600.
• Caserta, A., Di Maio, G. and Kočinac, L. D. R., Statistical convergence in function spaces, Abstr. Appl. Anal., 2011, Art. ID 420419, 11 pp.
• Connor, J. S., The statistical and strong p-Cesaro convergence of sequences, Analysis, 8 (1988) 47--63.
• Çakallı, H., Lacunary statistical convergence in topological groups, Indian J. Pure Appl. Math., 26(2) (1995) 113--119.
• Çakallı, H., A study on statistical convergence, Funct. Anal. Approx. Comput., 1(2) (2009) 19--24.
• Çınar, M., Karakaş, M. and Et, M., On pointwise and uniform statistical convergence of order α for sequences of functions, Fixed Point Theory and Applications, 2013(33) (2013) 11 pp.
• Çolak, R., Statistical convergence of order α, Modern Methods in Analysis and Its Applications, New Delhi, India: Anamaya Pub., 2010 (2010) 121--129.
• Çolak, R., On λ-Statistical Convergence, Conference on Summability and Aplications, Commerce University, May 12-13 (2011), Istanbul, Turkey.
• Çolak, R. and Bektaş, Ç. A., On λ-Statistical Convergence of Order α, Acta Math. Sci., 31(3) (2011) 953--959.
• Çolak, R., Altın, Y. and Et, M., λ-almost statistical convergence of order α. An. Ştiinţ. Univ. Al. I. Cuza Iaşi. Mat. (N.S.), 60(2) (2014) 437--448.
• De Malafosse, B. and Rakočević, V., Matrix transformation and statistical convergence, Linear Algebra Appl., 420(2-3) (2007) 377--387.
• Demirci, K., Strong A-summability and A-statistical convergence, Indian J. Pure Appl. Math., 27(6) (1996) 589--593.
• Di Maio, G. and Kočinac, L. D. R., Statistical convergence in topology, Topology Appl., 156 (2008) 28--45.
• Doğru, O. and Duman, O., Statistical approximation of Meyer-König and Zeller operators based on q-integers, Publ. Math. Debrecen, 68(1-2) (2006) 199--214.
• Et, M., Çınar, M. and Karakaş, M., On λ-statistical convergence of order α of sequences of functions, J. Inequal. Appl., 2013(204) (2013) 8 pp.
• Et, M., Alotaibi, A. and Mohiuddine, S. A., On (Δ^{m},I) statistical convergence of order α, Scientific World Journal, 2014 (2014) Article Number: 535419.
• Et, M. and Şengül, H., Some Cesaro-type summability spaces of order α and lacunary statistical convergence of order α, Filomat, 28(8) (2014) 1593--1602.
• Et, M., On pointwise λ-statistical convergence of order α of sequences of fuzzy mappings, Filomat, 28(6) (2014) 1271--1279.
• Fast, H., Sur la convergence statistique,Colloq. Math., 2 (1951) 241--244.
• Fridy, J., On statistical convergence, Analysis 5 (1985), 301--313.
• Gadjiev, A. D. and Orhan, C., Some approximation theorems via statistical convergence, Rocky Mountain J. Math., 32(1) (2002) 129--138.
• Işık, M. and Akbaş, K. E., On λ-statistical convergence of order α in probability, J. Inequal. Spec. Funct., 8(4) (2017) 57--64.
• Işık, M., Generalized vector-valued sequence spaces defined by modulus functions, J. Inequal, Appl., 2010 (2010) Art. ID 457892, 7 pp.
• Işık, M. and Et, M., Almost λ_{r}-statistical and strongly almost λ_{r}-convergence of order β of sequences of fuzzy numbers, J. Funct. Spaces, 2015 (2015) Art. ID 451050, 6 pp.
• Işık, M. and Altın, Y., f_{(λ,μ)}-statistical convergence of order α for double sequences, J. Inequal. Appl., 246 (2017) 8 pp.
• Malkowsky, E., The continuous duals of the spaces c₀(Λ) and c(Λ) for exponentially bounded sequences Λ, Acta Sci. Math. (Szeged), 61(1-4) (1995) 241--250.
• Sağıroğlu, S. and Ünver, M., Statistical convergence of sequences of sets in hyperspaces, Hacet. J. Math. Stat., 47(4) (2018) 889--896.
• Salat, T., On statistically convergent sequences of real numbers, Math. Slovaca, 30 (1980) 139--150.
• Schoenberg, I. J., The integrability of certain functions and related summability methods, Amer. Math. Monthly, 66 (1959) 361--375.
• Söylemez, D. and Ünver, M., Korovkin type theorems for Cheney-Sharma operators via summability methods, Results Math., 72(3) (2017) 1601--1612.
• Steinhaus, H., Sur la convergence ordinaire et la convergence asymptotique, Colloq. Math., 2 (1951) 73--74.
• Şengül, H. and Et, M., On lacunary statistical convergence of order α, Acta Math. Sci. Ser. B Engl. Ed., 34 (2) (2014) 473--482.
• Şengül, H. and Et, M., On (λ,I)-statistical convergence of order α of sequences of function, Proc. Nat. Acad. Sci. India Sect. A, 88(2) (2018) 181--186.
• Şengül, H. and Arıca, Z., Lacunary A-statistical convergence and lacunary strong A-convergence of order α with respect to a modulus, AIP Conference Proceedings 2086, 030037 (2019), doi: https://doi.org/10.1063/1.5095122.
• Turan, C. and Duman, O., Statistical convergence on timescales and its characterizations, Advances in applied mathematics and approximation theory, Springer, New York, (2013), pp. 57--71.