Keywords:-
Article Content:-
Abstract
In this paper, we extend the concept of f-density to f-double natural density of a subset of N N where f is an unbounded modulus function. We explore the notion of f-statistical convergence, and strongly f-Cesaro summability defned by modulus f by f-double natural density. In this article, we also give some inclusion relations between f-statistical convergence and strongly f-Cesaro summability with respect to modulus f for double sequence of sets in the Wijsman sense.
References:-
References
Aizpuru A. , Listan-Garcia M. C. and Rambla-Barreno F., Density by moduli and statistical convergence, Quaestiones Mathematicae 37(4) (2014), 525 - 530.
Pringsheim A. , Zur theorie der zweifach unendlichen Zahlenfolgen, Math. Ann., 53 (1900), 289-321.
Dafadar A., Ganguly D.K., Wijsman invariant statistically convergence of double sequence of sets with respect to modulus function, Bull. Cal. Math. Soc., Vol. 112, No. 6, pp.495-510, 2020.
Dafadar A., Ganguly D.K.,, Wijsman Quasi-Invariant Convergence of Double Sequence of Sets, Bull. Cal. Math. Soc., 115, (6) 761-776 (2023).
Inan B. and Kucukaslan M., Ideal convergence of sequence of sets, Contem. Anal. Appl. Math. 3(2) 184 - 212, (2015),.
Leon-Saavedra F. , Listan-Garcia M.C. , Perez Fernandez F.J. , Romero de la Rosa M.P., On statistical convergence and strong Ces_x0012_aro convergence by moduli, J. Inequal. Appl. 12, 298 (2019).
Nuray F. and Rhoades B. E. , Statistical convergence of sequences of sets, Fasc. Math. 49 , 87- 99, (2012).
Nuray F. , Ulusu U. and Dundar E. , Lacunary statistical convergence of double sequences of sets, Soft Comput. 20(7), 2883 - 2888, (2016).
Nuray F. , Ulusu U. , and Erdinc D., Cesaaro summability of double sequences of sets, Gen. Math. Notes 25 , 8-18, (2014).
Nuray F. , Dundar E., Ulusu U., Wijsman Statistical Convergence of Double Sequences of Set, Iran. J. Math. Sci. and Inform., 16(1),, 55-64, (2021).
Nuray F., Ulusu U. and Dundar E., Lacunary statistical convergence of double sequences of sets, Soft Comput. 20(7) , 2883- 2888,(2016). DOI: 10.1007/s00500-015-1691-8.
Nakano H. , Modulared sequence spaces, Proc. Japan Acad. 27, 508-512, (1951).
Nakano H., Concave modulars, J. Math. Soc. Japan 5, 29-49. (1953).
Sengul H , Et M., and Altin Y. , f-lacunary statistical convergence and strong f-lacunary summability of order
of double sequences, F. U. Math. Inform. 35, no. 2, 495-506, (2020).
Maddox I. J., Sequence spaces defined by a modulus, Math. Proc. Camb. Philos. Soc, ,100:161-166, (1986).
Niven I. , Zuckerman H. S. and Montgomery H. L., An introduction to the theory of numbers, Fifth Ed., John Willey and Sons, New York., 1991.
Ibrahim I. Colak S., R., On The Sets of f-Strongly Cesaro Summable Sequences, KYUNGPOOK Math. J. 64, 235-244,(2024).
Christopher J., The asymptotic density of some k-dimensional sets, Amer. Math. Monthly 63 399-401, (1956).
Connor J.S., The statistical and strong p-Cesaro convergence of sequences, Analysis 8(1-2), 47-63 (1988).
Khan M.Z., Orhan C., Characterizations of strong and statistical convergences, Publ. Math. (Debr.) 76(1-2), 77-88 (2010).
Mursaleen M., Edely O.H., Statistical convergence of double sequences, J. Math. Anal. Appl. 288(1), 223-231 (2003).
Mursaleen M. and Edely Osama H.H., Statistical convergence of double sequences, J. Math. Anal. Appl. 288 223-231, (2003).
Pancaroglu N. and Nuray F., Invariant statistical convergence of sequences of sets with respect to a modulus function, Abstr. Appl. Anal. 2014 (2014).
Talo O., Sever Y. and Basar F., On statistically convergent sequences of closed sets, Filomat 30(6), 1497 -1509, (2016).
Wijsman R. A. , Convergence of sequences of convex sets, cones and functions, Bull. Am. Math. Soc. 70, 186- 188, (1964).
Wijsman R. A. , Convergence of sequences of convex sets, cones and functions II, Trans. Am. Math. Soc. 123(1) , 32 -45, (1966).
Colak R. and Altin Y., Statistical convergence of double sequences of order, J. Funct. Spaces Appl. 2013 , 682-823,(2013).
Bilgin T. , (σ, f )-Asymptotically lacunary equivalent sequences, Int. J. Anal.,(2014), Article ID 945902, 8 pages.
Ulusu U. and Nuray F., On strongly lacunary summability of sequences of sets, J. Appl. Math. Bioinform. 3, 75-88, (2013).
Bhardwaj V. K. and Dhawan S., f -statistical convergence of order and strong Cesaro summability of order with respect to a modulus, J. Ineq. Appl. (332) (2015), 14 pages.
Bhardwaj V. K. , Dhawan S. and Gupta S. , Density by moduli and statistical boundedness, Abst. Appl. Analysis (2016), Article ID 2143018, 6 pages.
Bhardwaj V. K. and Dhawan S., Density by moduli and lacunary statistical convergence, Abst. Appl. Analysis (2016), Article ID 9365037, 11 pages.
Bhardwaj V. K. and Dhawan S., Density by moduli and Wijsman lacunary statistical convergence of sequences of sets, J. Ineq. Appl. (25) (2017).
Bhardwaj V. K., Dhawan S. and Dovgoshey O. A., Density by moduli and Wijsman statistical convergence, Bull. Belg. Math. Soc. 24 , 393- 415,(2017).
Bhardwaj V. K. and Dhawan S., f-Statistical convergenceoforder and strong Ces`aro summability of order with respect to a modulus, J. Inequal. Appl., 67(4), 1-14, (2015).
Ruckle W. H., FK spaces in which the sequence of coordinate vectors is bounded, Can. J. Math. 25, 973- 978, (1973).