On (f, I) - Lacunary statistical convergence of Order α of sequences of sets
Abstract
In this paper we introduce the concepts of Wijsman $% \left( f,I\right) -$lacunary statistical{\Large \ }convergence of order $% \alpha $ and Wijsman strongly $\left( f,I\right) -$lacunary statistical% {\Large \ }convergence of order $\alpha ,$ and investigated between their relationship.
Downloads
References
A. Aizpuru, M. C. Listan-Garcıa and F. Rambla-Barreno, Density by moduli and statistical convergence. Quaest. Math. 37(4) (2014) 525-530. https://doi.org/10.2989/16073606.2014.981683
H. Altınok, M. Et, R. Colak, Some remarks on generalized sequence space of bounded variation of sequences of fuzzy numbers. Iran. J. Fuzzy Syst. 11(5) (2014) 39-46.
Y. Altın, H. Altınok and R. Colak, Statistical convergence of order for difference sequences. Quaest. Math. 38(4) (2015) 505-514. https://doi.org/10.2989/16073606.2014.981685
V. K. Bhardwaj and S. Dhawan, f−statistical convergence of order and strong Cesaro summability of order with respect to a modulus. J. Inequal. Appl. 2015(332) (2015) 14 pp. https://doi.org/10.1186/s13660-015-0850-x
A. Caserta, G. Di Maio and L. D. R. Koˇcinac, Statistical convergence in function spaces. Abstr. Appl. Anal. 2011 Art. ID 420419, (2011) 11 pp. https://doi.org/10.1155/2011/420419
H. Cakalli, Lacunary statistical convergence in topological groups. Indian J. Pure Appl. Math. 26(2) (1995) 113-119.
H. Cakalli and B. Hazarika, Ideal quasi-Cauchy sequences. J. Inequal. Appl. 2012(234) (2012) 11 pp.
https://doi.org/10.1186/1029-242X-2012-234
H. Cakalli, A new approach to statistically quasi Cauchy sequences. Maltepe Journal of Mathematics 1(1) (2019) 1-8. https://doi.org/10.1063/1.5095095
M. Cinar, M. Karaka¸s and M. Et, On pointwise and uniform statistical convergence of order for sequences of functions. Fixed Point Theory Appl. 2013(33) (2013) 11 pp. https://doi.org/10.1186/1687-1812-2013-33
R. Colak, Statistical convergence of order . Modern Methods in Analysis and Its Applications, New Delhi, India: Anamaya Pub. 2010 (2010) 121-129.
R. Colak, On -Statisitical Convergence. Conference on Summability and Applications 2011 Istanbul Commerce Univ. May 12-13, (2011), ˙Istanbul.
J. S. Connor, The Statistical and strong p−Cesaro convergence of sequences. Analysis 8 (1988) 47-63.
https://doi.org/10.1524/anly.1988.8.12.47
M. Et, A. Alotaibi and S. A. Mohiuddine, On ( m, I)−statistical convergence of order . The Scientific World Journal, 2014 Article ID 535419 (2014) 5 pages. https://doi.org/10.1155/2014/535419
M. Et and H. Sengul, Some Cesaro-type summability spaces of order and lacunary statistical convergence of order . Filomat 28(8), (2014), 1593-1602. https://doi.org/10.2298/FIL1408593E
M. Et, B. C. Tripathy and A. J. Dutta, On pointwise statistical convergence of order of sequences of fuzzy mappings. Kuwait J. Sci. 41(3) (2014) 17-30.
M. Et, R. Colak and Y. Altın, Strongly almost summable sequences of order . Kuwait J. Sci. 41(2) (2014) 35-47.
H. Fast, Sur la convergence statistique. Colloq. Math. 2 (1951) 241-244. https://doi.org/10.4064/cm-2-3-4-241-244
A. R. Freedman, J. J. Sember and M. Raphael, Some Cesaro-type summability spaces. Proc. London Math. Soc. 37(3) (1978) 508-520. https://doi.org/10.1112/plms/s3-37.3.508
J. A. Fridy, On statistical convergence. Analysis5 (1985) 301-313. https://doi.org/10.1524/anly.1985.5.4.301
J. A. Fridy and C. Orhan, Lacunary statistical convergence. Pacific J. Math. 160(1) (1993) 43-51.
https://doi.org/10.2140/pjm.1993.160.43
A. D. Gadjiev and C. Orhan, Some approximation theorems via statistical convergence. Rocky Mountain J. Math. 32(1) (2002) 129-138. https://doi.org/10.1216/rmjm/1030539612
M. Isık and K. E. Akbas, On −statistical convergence of order in probability. J. Inequal. Spec. Funct. 8(4) (2017) 57-64.
E. Kayan, R. Colak and Y. Altın, d-statistical convergence of order and d-statistical boundedness of order in metric spaces. Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys. 80(4) (2018) 229-238.
P. Kostyrko, T. Salat and W. Wilczynski, I−convergence. Real Anal. Exchange 26 (2000/2001) 669-686.
P. Kostyrko, M. Macaj, M. Sleziak and T. Salat, I−convergence and extremal I−limit points. Math. Slovaca 55(4) (2005) 443-464.
H. Nakano, Modulared sequence spaces. Proc. Japan Acad. 27 (1951) 508-512. https://doi.org/10.3792/pja/1195571225
F. Nuray and W. H. Ruckle, Generalized statistical convergence and convergence free spaces. J. Math. Anal. Appl. 245(2) (2000) 513-527. https://doi.org/10.1006/jmaa.2000.6778
F. Nuray and B. E. Rhoades, Statistical convergence of sequences of sets. Fasc. Math. 49 (2012) 87-99.
T. Salat, On statistically convergent sequences of real numbers. Math. Slovaca 30 (1980) 139-150.
T. Salat, B. C. Tripathy and M. Ziman, On I−convergence field. Ital. J. Pure Appl. Math. 17 (2005) 45-54.
E. Savas and M. Et, On ( m , I)− statistical convergence of order . Period. Math. Hungar. 71(2) (2015) 135-145.
https://doi.org/10.1007/s10998-015-0087-y
I. J. Schoenberg, The integrability of certain functions and related summability methods. Amer. Math. Monthly 66 (1959) 361-375. https://doi.org/10.2307/2308747
H. Sengul and M. Et, On lacunary statistical convergence of order . Acta Math. Sci. Ser. B Engl. Ed. 34(2) (2014) 473-482. https://doi.org/10.1016/S0252-9602(14)60021-7
H. Sengul, Some Cesaro-type summability spaces defined by a modulus function of order ( , ). Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 66(2) (2017) 80-90. https://doi.org/10.1501/Commua1_0000000803
H. Sengul, M. Et and H. C¸ akallı, On Wijsman (f, I)−lacunary statistical convergence of order . AIP Conference Proceedings 2086, 030040 (2019); https://doi.org/10.1063/1.5095125
H. M. Srivastava and M. Et, Lacunary statistical convergence and strongly lacunary summable functions of order . Filomat 31(6) (2017) 1573-1582. https://doi.org/10.2298/FIL1706573S
H. Steinhaus, Sur la convergence ordinaire et la convergence asymptotique. Colloq. Math. 2 (1951) 73-74.
I. Taylan, Abel statistical delta quasi Cauchy sequences of real numbers., Maltepe Journal of Mathematics, 1, 1, 18-23, (2019). https://doi.org/10.1063/1.5095128
U. Ulusu and F. Nuray, Lacunary statistical convergence of sequence of sets. Prog. Appl. Math. 4(2) (2012) 99-109.
U. Ulusu and E. Dundar, I−lacunary statistical convergence of sequences of sets. Filomat 28(8) (2014) 1567-1574. https://doi.org/10.2298/FIL1408567U
S. Yildiz, Lacunary statistical p-quasi Cauchy sequences., Maltepe Journal of Mathematics, 1, 1, 9-17, (2019).
Copyright (c) 2019 Boletim da Sociedade Paranaense de Matemática
This work is licensed under a Creative Commons Attribution 4.0 International License.
When the manuscript is accepted for publication, the authors agree automatically to transfer the copyright to the (SPM).
The journal utilize the Creative Common Attribution (CC-BY 4.0).
