On (f, I) - Lacunary statistical convergence of Order α of sequences of sets
DOI:
https://doi.org/10.5269/bspm.v38i7.46259Abstract
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.
References
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30. T. Salat, B. C. Tripathy and M. Ziman, On I−convergence field. Ital. J. Pure Appl. Math. 17 (2005) 45-54.
31. 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
32. 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
33. 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
34. 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
35. 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
36. 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
37. H. Steinhaus, Sur la convergence ordinaire et la convergence asymptotique. Colloq. Math. 2 (1951) 73-74.
38. 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
39. U. Ulusu and F. Nuray, Lacunary statistical convergence of sequence of sets. Prog. Appl. Math. 4(2) (2012) 99-109.
40. 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
41. S. Yildiz, Lacunary statistical p-quasi Cauchy sequences., Maltepe Journal of Mathematics, 1, 1, 9-17, (2019).
https://doi.org/10.1063/1.5095130
2. 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.
3. 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
4. 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
5. 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
6. H. Cakalli, Lacunary statistical convergence in topological groups. Indian J. Pure Appl. Math. 26(2) (1995) 113-119.
7. 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
8. 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
9. 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
10. R. Colak, Statistical convergence of order . Modern Methods in Analysis and Its Applications, New Delhi, India: Anamaya Pub. 2010 (2010) 121-129.
11. R. Colak, On -Statisitical Convergence. Conference on Summability and Applications 2011 Istanbul Commerce Univ. May 12-13, (2011), ˙Istanbul.
12. 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
13. 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
14. 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
15. 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.
16. M. Et, R. Colak and Y. Altın, Strongly almost summable sequences of order . Kuwait J. Sci. 41(2) (2014) 35-47.
17. H. Fast, Sur la convergence statistique. Colloq. Math. 2 (1951) 241-244. https://doi.org/10.4064/cm-2-3-4-241-244
18. 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
19. J. A. Fridy, On statistical convergence. Analysis5 (1985) 301-313. https://doi.org/10.1524/anly.1985.5.4.301
20. 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
21. 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
22. M. Isık and K. E. Akbas, On −statistical convergence of order in probability. J. Inequal. Spec. Funct. 8(4) (2017) 57-64.
23. 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.
24. P. Kostyrko, T. Salat and W. Wilczynski, I−convergence. Real Anal. Exchange 26 (2000/2001) 669-686.
25. P. Kostyrko, M. Macaj, M. Sleziak and T. Salat, I−convergence and extremal I−limit points. Math. Slovaca 55(4) (2005) 443-464.
26. H. Nakano, Modulared sequence spaces. Proc. Japan Acad. 27 (1951) 508-512. https://doi.org/10.3792/pja/1195571225
27. 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
28. F. Nuray and B. E. Rhoades, Statistical convergence of sequences of sets. Fasc. Math. 49 (2012) 87-99.
29. T. Salat, On statistically convergent sequences of real numbers. Math. Slovaca 30 (1980) 139-150.
30. T. Salat, B. C. Tripathy and M. Ziman, On I−convergence field. Ital. J. Pure Appl. Math. 17 (2005) 45-54.
31. 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
32. 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
33. 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
34. 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
35. 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
36. 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
37. H. Steinhaus, Sur la convergence ordinaire et la convergence asymptotique. Colloq. Math. 2 (1951) 73-74.
38. 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
39. U. Ulusu and F. Nuray, Lacunary statistical convergence of sequence of sets. Prog. Appl. Math. 4(2) (2012) 99-109.
40. 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
41. S. Yildiz, Lacunary statistical p-quasi Cauchy sequences., Maltepe Journal of Mathematics, 1, 1, 9-17, (2019).
https://doi.org/10.1063/1.5095130
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