A variation on strongly ideal lacunary ward continuity
DOI:
https://doi.org/10.5269/bspm.v38i7.46136Abstract
The main purpose of this paper is to introduce the concept of strongly ideal lacunary quasi-Cauchyness of order (alpha,beta) of sequences of real numbers. Strongly ideal lacunary ward continuity of order (alpha,beta) is also investigated. Interesting results are obtained.
References
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31. 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
32. T. Salat, B.C. Tripathy and M. Ziman, On some properties of I-convergence. Tatra Mt. Math. Publ. 28(2) (2004) 279-286.
33. T. Salat, B.C. Tripathy and M. Ziman, On I-convergence field. Ital. J. Pure Appl. Math. 17 (2005) 45-54.
34. T. Salat, On statistically convergent sequences of real numbers. Math. Slovaca 30(2) (1980) 139-150.
35. E. Savas and P. Das, A generalized statistical convergence via ideals. Appl. Math. Lett. 24(6) (2011) 826-830.
https://doi.org/10.1016/j.aml.2010.12.022
36. H. Sengul, H. Cakallı and M. Et, N ( , I)− ward continuity. AIP Conference Proceedings 2086, 030038 (2019);
https://doi.org/10.1063/1.5095123
37. H. Sengul and M. Et, On ( , I)-statistical convergence of order of sequences of function. Proc. Nat. Acad. Sci. India Sect. A 88(2) (2018) 181-186. https://doi.org/10.1007/s40010-017-0414-1
38. H. Sengul and M. Et, On I-lacunary statistical convergence of order of sequences of sets. Filomat 31(8) (2017) 2403-2412. https://doi.org/10.2298/FIL1708403S
39. I. Taylan, Abel statistical delta quasi Cauchy sequences of real numbers. Maltepe Journal of Mathematics, 1(1) (2019) 18-23. https://doi.org/10.1063/1.5095128
40. B.C. Tripathy, B. Hazarika and B. Choudhary, Lacunary I-Convergent Sequences Kyungpook Math. J. 52(4) (2012) 473-482. https://doi.org/10.5666/KMJ.2012.52.4.473
41. R.W. Vallin, Creating slowly oscillating sequences and slowly oscillating continuous functions. With an appendix by Vallin and H. Cakalli, Acta Math. Univ. Comenianae 80(1) (2011) 71-78.
42. S. Yildiz, Lacunary statistical p-quasi Cauchy sequences. Maltepe Journal of Mathematics 1(1) (2019) 9-17.
https://doi.org/10.1063/1.5095130
43. S. Yildiz, Variations on lacunary statistical quasi Cauchy sequences. AIP Conference Proceedings 2086, 030045 (2019); https://doi.org/10.1063/1.5095130
2. N.L. Braha and H. Cakalli, A new type continuity for real functions. J. Math. Anal. 7(6) (2016) 54-62.
3. D. Burton and J. Coleman, Quasi-Cauchy Sequences. Amer. Math. Monthly 117(4) (2010) 328-333.
https://doi.org/10.4169/000298910x480793
4. A. Caserta, G. Di Maio and L.D.R. Kocinac, Statistical convergence in function spaces. Abstr. Appl. Anal. 2011 (2011) Art. ID 420419, 11 pp. https://doi.org/10.1155/2011/420419
5. J. Connor and K.-G.Grosse-Erdmann, Sequential definitions of continuity for real functions. Rocky Mountain J. Math. 33(1) (2003) 93-121. https://doi.org/10.1216/rmjm/1181069988
6. J. S. Connor, The Statistical and strong p−Cesaro convergence of sequences. Analysis 8(1-2) (1988) 47-63.
https://doi.org/10.1524/anly.1988.8.12.47
7. H. Cakalli, Statistical quasi-Cauchy sequences. Math. Comput. Modelling 54(5-6) (2011) 1620-1624.
https://doi.org/10.1016/j.mcm.2011.04.037
8. H. Cakalli, Statistical ward continuity. Appl. Math. Lett. 24(10) (2011) 1724-1728. https://doi.org/10.1016/j.aml.2011.04.029
9. 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
10. H. Cakalli and B. Hazarika, Ideal quasi-Cauchy sequences. J. Inequal. Appl. 2012(234) (2012) 11pp. https://doi.org/10.1186/1029-242X-2012-234
11. H. Cakalli, A variation on ward continuity. Filomat 27(8) (2013) 1545-1549.
https://doi.org/10.2298/FIL1308545C
12. H. Cakalli, Slowly oscillating continuity. Abstr. Appl. Anal., Hindawi Publ. Corp., New York, 2008 Article ID 485706, (2008). https://doi.org/10.1155/2008/485706
13. H. Cakalli, On -quasi-slowly oscillating sequences. Comput. Math. Appl. 62(9) (2011) 3567-3574.
https://doi.org/10.1016/j.camwa.2011.09.004
14. H. Cakalli, Forward continuity. J. Comput. Anal. Appl. 13(2) (2011) 225-230.
15. H. Cakalli, -quasi-Cauchy sequences. Math. Comput. Modelling 53(1-2) (2011) 397-401.
https://doi.org/10.1016/j.mcm.2010.09.010
16. H. Cakalli, A. Sonmez and C¸ .G. Aras, -statistically ward continuity. An. Stiint. Univ. Al. I. Cuza Iasi. Mat. (N.S.) 63(2) (2017) 313-321.
17. H. Cakalli and M. Albayrak, New type continuities via Abel convergence. The Scientific World Journal 2014 (2014) Article ID 398379, 6 pages. https://doi.org/10.1155/2014/398379
18. H. Cakalli and A. Sonmez, Slowly oscillating continuity in abstract metric spaces. Filomat 27(5) (2013) 925-930.
https://doi.org/10.2298/FIL1305925C
19. H. Cakalli, N -ward continuity. Abstr. Appl. Anal., Hindawi Publ. Corp., New York, 2012 Article ID 680456, (2012) 8 pp. https://doi.org/10.1155/2012/680456
20. H. Cakalli, Sequential definitions of compactness. Appl. Math. Lett. 21(6) (2008) 594-598. https://doi.org/10.1016/j.aml.2007.07.011
21. H. Cakalli, On G-continuity. Comput. Math. Appl. 61(2) (2011) 313-318. https://doi.org/10.1016/j.camwa.2010.11.006
22. H. Cakalli, M. Et and H. Sengul, A variation on N ward continuity. Georgian Mathematical Journal. https://doi.org/10.1515/gmj-2018-0037
23. I. Canak and M. Dik, New Types of Continuities. Abstr. Appl. Anal., Hindawi Publ. Corp., New York, 2010 Article ID 258980, (2010). https://doi.org/10.1155/2010/258980
24. P. Das, E. Savas and S. Kr. Ghosal, On generalizations of certain summability methods using ideals. Appl. Math. Lett. 24(9) (2011) 1509-1514. https://doi.org/10.1016/j.aml.2011.03.036
25. H. Fast, Sur la convergence statistique. Colloq. Math. 2 (1952) 241-244. https://doi.org/10.4064/cm-2-3-4-241-244
26. J. Fridy, On statistical convergence. Analysis 5 (1985) 301-313. https://doi.org/10.1524/anly.1985.5.4.301
27. 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
28. 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
29. P. Kostyrko, T. Salat and W. Wilczynski, I−convergence. Real Anal. Exchange 26(2) (2000/2001) 669-686.
30. P. Kostyrko, M. Macaj, M. Sleziak and T. Salat, I-convergence and extremal I-limit points. Math. Slovaca 55(4) (2005) 443-464.
31. 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
32. T. Salat, B.C. Tripathy and M. Ziman, On some properties of I-convergence. Tatra Mt. Math. Publ. 28(2) (2004) 279-286.
33. T. Salat, B.C. Tripathy and M. Ziman, On I-convergence field. Ital. J. Pure Appl. Math. 17 (2005) 45-54.
34. T. Salat, On statistically convergent sequences of real numbers. Math. Slovaca 30(2) (1980) 139-150.
35. E. Savas and P. Das, A generalized statistical convergence via ideals. Appl. Math. Lett. 24(6) (2011) 826-830.
https://doi.org/10.1016/j.aml.2010.12.022
36. H. Sengul, H. Cakallı and M. Et, N ( , I)− ward continuity. AIP Conference Proceedings 2086, 030038 (2019);
https://doi.org/10.1063/1.5095123
37. H. Sengul and M. Et, On ( , I)-statistical convergence of order of sequences of function. Proc. Nat. Acad. Sci. India Sect. A 88(2) (2018) 181-186. https://doi.org/10.1007/s40010-017-0414-1
38. H. Sengul and M. Et, On I-lacunary statistical convergence of order of sequences of sets. Filomat 31(8) (2017) 2403-2412. https://doi.org/10.2298/FIL1708403S
39. I. Taylan, Abel statistical delta quasi Cauchy sequences of real numbers. Maltepe Journal of Mathematics, 1(1) (2019) 18-23. https://doi.org/10.1063/1.5095128
40. B.C. Tripathy, B. Hazarika and B. Choudhary, Lacunary I-Convergent Sequences Kyungpook Math. J. 52(4) (2012) 473-482. https://doi.org/10.5666/KMJ.2012.52.4.473
41. R.W. Vallin, Creating slowly oscillating sequences and slowly oscillating continuous functions. With an appendix by Vallin and H. Cakalli, Acta Math. Univ. Comenianae 80(1) (2011) 71-78.
42. S. Yildiz, Lacunary statistical p-quasi Cauchy sequences. Maltepe Journal of Mathematics 1(1) (2019) 9-17.
https://doi.org/10.1063/1.5095130
43. S. Yildiz, Variations on lacunary statistical quasi Cauchy sequences. AIP Conference Proceedings 2086, 030045 (2019); https://doi.org/10.1063/1.5095130
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