ON f−LACUNARY STATISTICAL CONVERGENCE OF ORDER β OF DOUBLE SEQUENCES FOR DIFFERENCE SEQUENCES OF FRACTIONAL ORDER

Nazlım Deniz Aral, Hacer Şengül Kandemir

DOI Number
https://doi.org/10.22190/FUMI211007023A
First page
329
Last page
343

Abstract


In this study, by using definition of lacunary statistical convergence we introduce the concepts of f− lacunary statistical convergence of order β and strongly f−lacunary summability of order β of double sequences for different sequences of fractional order spaces. Also, we establish some inclusion relations between these concepts.


Keywords

Difference sequences, Lacunary statistical convergence, Modulus function.

Full Text:

PDF

References


A. Aizpuru, M. C. Lista ́n-Garc ́ıa and F. Rambla-Barreno: Density by moduli and statistical convergence. Quaest. Math., 37(4) (2014), 525–530.

Y. Altın and M. Et: Generalized difference sequence spaces defined by a modulus function in a locally convex space. Soochow J. Math. 31(2) (2005), 233–243.

H. Altınok, M. Et and R. C ̧ olak: Some remarks on generalized sequence space of bounded variation of sequences of fuzzy numbers. Iran. J. Fuzzy Syst. 11(5) (2014), 39–46, 109.

N. D. Aral and M. Et: On lacunary statistical convergence of order β of dif- ference sequences of fractional order. International Conference of Mathematical Sciences, (ICMS 2019), Maltepe University, Istanbul, Turkey.

N. D. Aral and M. Et: Generalized difference sequence spaces of fractional order defined by Orlicz functions. Commun. Fac. Sci. Univ. Ank. Ser. A1. Math. Stat. 69(1) (2020), 941–951.

N. D. Aral and H. S ̧engu ̈l: I−Lacunary Statistical Convergence of Order β of Difference Sequences of Fractional Order. Facta Universitatis, Series: Mathematics and Informatics, 36(1) (2021), 43–55.

N. D. Aral: Statistical Convergence Dened by a Modulus Function of order. Maltepe Journal of Mathematics, 4(1) (2022), 15-23.

P. Baliarsingh: Some new difference sequence spaces of fractional order and their dual spaces. Appl. Math. Comput. 219(18) (2013), 9737–9742.

P. Baliarsingh, U. Kadak and M. Mursaleen: On statistical convergence of difference sequences of fractional order and related Korovkin type approximation theorems. Quaest. Math. 41(8) (2018), 1117–1133.

C. Belen and S. A. Mohiuddine: Generalized weighted statistical convergence and application. Applied Mathematics and Computation 219 (2013), 9821–9826.

V. K. Bhardwaj and S. Dhawan: Density by moduli and lacunary statistical convergence. Abstr. Appl. Anal. 2016 (2016), Art. ID 9365037, 11 pp.

A. Caserta, G. Di Maio and L. D. R. Kocˇinac: Statistical convergence in function spaces. Abstr. Appl. Anal. 2011 (2011), Art. ID 420419, 11 pp.

J. S. Connor: The statistical and strong p−Cesaro convergence of sequences. Analysis 8 (1988), 47–63.

H. C ̧ akallı: Lacunary statistical convergence in topological groups. Indian J. Pure Appl. Math. 26(2) (1995) 113–119.

H. C ̧akallı and H. Kaplan: A variation on lacunary statistical quasi Cauchy sequences. Commun. Fac. Sci. Univ. Ank. S ́er. A1 Math. Stat. 66(2) (2017), 71–79.

H. C ̧akallı and H. Kaplan: A study on Nθ-quasi-Cauchy sequences. Abstr. Appl. Anal. 2013 (2013), Article ID 836970, 4 pages.

H. C ̧akallı: A study on statistical convergence. Funct. Anal. Approx. Comput. 1(2) (2009), 19–24.

H. Cakalli: A New Approach to Statistically Quasi Cauchy Sequences. Maltepe Journal of Mathematics, 1(1) (2019), 1-8.

M. C ̧ ınar, M. Karaka ̧s and M. Et: On pointwise and uniform statistical conver- gence of order α for sequences of functions. Fixed Point Theory And Applications, 2013(33) (2013), 11 pages.

R. C ̧olak: Statistical convergence of order α. Modern Methods in Analysis and Its Applications, New Delhi, India: Anamaya Pub. 2010 (2010), 121–129.

G. Di Maio and L. D. R. Kocˇinac: Statistical convergence in topology. Topology Appl. 156 (2008), 28–45.

M. Et and R. C ̧olak: On some generalized difference sequence spaces. Soochow J. Math. 21 (4) (1995), 377-386 .

M. Et, Y. Altın and H. Altınok: On some generalized difference sequence spaces defined by a modulus function. Filomat 17 (2003), 23–33.

M. Et, B. C. Tripathy and A. J. Dutta: On pointwise statistical convergence of order α of sequences of fuzzy mappings. Kuwait J. Sci. 41(3) (2014), 17–30.

M. Et, M. Mursaleen and M. I ̧sık: On a class of fuzzy sets defined by Orlicz functions. Filomat 27(5) (2013), 789–796.

M. Et and H. S ̧engu ̈l: Some Cesaro-type summability spaces of order α and lacunary statistical convergence of order α. Filomat 28(8) (2014), 1593–1602.

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

A. R. Freedman, J. J. Sember and M. Raphael: Some Ces`aro-type summability spaces. Proc. London Math. Soc. (3) 37(3) (1978), 508–520.

J. Fridy: On statistical convergence. Analysis 5 (1985), 301–313.

J. A. Fridy and C. Orhan: Lacunary statistical convergence. Pacific J. Math. 160 (1993), 43–51.

J. A. Fridy and C. Orhan: Lacunary statistical summability. J. Math. Anal. Appl. 173(2) (1993), 497–504.

A. K. Gaur and M. Mursaleen: Difference sequence spaces defined by a se- quence of moduli. Demonstratio Math. 31(2) (1998), 275–278.

M. I ̧sık and K. E. Et: On lacunary statistical convergence of order α in probability. AIP Conference Proceedings 1676 020045 (2015), doi: http://dx.doi.org/10.1063/1.4930471.

M. Isık and K. E. Akba ̧s: On λ−statistical convergence of order α in probability. J. Inequal. Spec. Funct. 8(4) (2017), 57–64.

M. I ̧sık: Generalized vector-valued sequence spaces defined by modulus functions. J. Inequal. Appl. 2010 Art. ID 457892 (2010), 7 pp.

U. Kadak: Generalized lacunary statistical difference sequence spaces of fractional order. Int. J. Math. Math. Sci. 2015, Art. ID 984283, 6 pp.

H. Kaplan and H. C ̧akallı: Variations on strong lacunary quasi-Cauchy se- quences. J. Nonlinear Sci. Appl. 9(6) (2016), 4371–4380.

M. Karaka ̧s, M. Et and V. Karakaya: Some geometric properties of a new difference sequence space involving lacunary sequences. Acta Math. Sci. Ser. B (Engl. Ed.) 33(6) (2013), 1711–1720.

H. Kızmaz: On certain sequence spaces. Canad. Math. Bull. 24(2) (1981), 169- 176.

I. J. Maddox: Sequence spaces defined by a modulus. Math. Proc. Camb. Philos. Soc. 100 (1986), 161–166.

S. A. Mohiuddine, A. Alotaibi and M. Mursaleen: Statistical convergence of double sequences in locally solid Riesz spaces. Abstr. Appl. Anal. 2012 Article ID 719729 (2012), 9 pages.

M. Mursaleen and O. H. H. Edely: Statistical convergence of double sequences. Journal of Mathematical Analysis and Applications 288(1) (2003), 223–231.

M. Mursaleen and S. A. Mohiuddine: On lacunary statistical convergence with respect to the intuitionistic fuzzy normed space. J. Comput. Appl. Math. 233(2) (2009), 142–149.

H. Nakano: Modulared sequence spaces. Proc. Japan Acad. 27 (1951), 508–512.

L. Nayak, M. Et and P. Baliarsingh: On certain generalized weighted mean fractional difference sequence spaces. Proc. Nat. Acad. Sci. India Sect. A 89(1) (2019), 163–170.

F. Nuray, E. Sava ̧s: Some new sequence spaces defined by a modulus function. Indian J. Pure Appl. Math. 24(11) (1993), 657-663.

R. F. Patterson and E. Sava ̧s: Lacunary statistical convergence of double se- quences. Math. Commun. 10(1) (2005), 55–61.

S. Pehlivan and B. Fisher: Lacunary strong convergence with respect to a se- quence of modulus functions. Comment. Math. Univ. Carolin. 36(1) (1995), 69–76.

S. Pehlivan and B. Fisher: Some sequence spaces defined by a modulus. Math. Slovaca 45(3) (1995), 275–280.

A. Pringsheim: Zur Theorie der zweifach unendlichen Zahlenfolgen. Mathema- tische Annalen 53(3) (1900), 289–321.

T. Salat: On statistically convergent sequences of real numbers. Math. Slovaca 30 (1980), 139–150.

E. Sava ̧s and M. Et: On (∆mλ ,I)−statistical convergence of order α. Period. Math. Hungar. 71(2) (2015), 135–145.

I. J. Schoenberg: The integrability of certain functions and related summability methods. Amer. Math. Monthly 66 (1959), 361–375.

H. M. Srivastava and M. Et: Lacunary statistical convergence and strongly lacunary summable functions of order α. Filomat 31(6) (2017), 1573–1582.

H. Steinhaus: Sur la convergence ordinaire et la convergence asymptotique. Col- loq. Math. 2 (1951), 73–74.

H. S ̧engu ̈l: Some Cesa`ro-type summability spaces defined by a modulus function of order (α,β). Commun. Fac. Sci. Univ. Ank. S ́er. A1 Math. Stat. 66(2) (2017), 80–90.

H. S ̧engu ̈l and M. Et: On lacunary statistical convergence of order α. Acta Math. Sci. Ser. B Engl. Ed. 34(2) (2014), 473–482.

H. Sengul Kandemir: On Statistical Convergence in Topological Groups. Maltepe Journal of Mathematics, 4(1) (2022), 9-14.

B. C. Tripathy and M. Et: On generalized difference lacunary statistical con- vergence. Studia Univ. Babe ̧s-Bolyai Math., 50(1) (2005), 119–130.

B. C. Tripathy, A. Esi and B. K. Tripathy: On a new type of generalized difference Cesaro Sequence spaces. Soochow J. Math. 31(3) (2005), 333-340.

T. Yaying, A. Das, B. Hazarika and P. Baliarsingh: Compactness of bino- mial difference operator of fractional order and sequence spaces. Rend. Circ. Mat. Palermo (2) 68(3) (2019), 459–476.

T. Yaying and B. Hazarika: On sequence spaces generated by binomial differ- ence operator of fractional order. Math. Slovaca 69(4) (2019), 901-918.




DOI: https://doi.org/10.22190/FUMI211007023A

Refbacks

  • There are currently no refbacks.




© University of Niš | Created on November, 2013
ISSN 0352-9665 (Print)
ISSN 2406-047X (Online)