Asymptotically lacunary µ-statistical equivalence of generalized difference sequences in probabilistic normed spaces
DOI:
https://doi.org/10.5269/bspm.51915Resumen
The current article introduces the notion of asymptotically lacunary $(\Delta^n,\mu)$-statistical equivalent sequence in the settings of a probabilistic norm $N$. Furthermore, the article presents the concepts of asymptotically $(\Delta^n,\mu)$-strongly Ces\'{a}ro equivalent sequences and asymptotically $(\Delta^n,\mu)$-strongly Ces\'{a}ro Orlicz equivalent sequences in the theory of probabilistic normed spaces and also investigates their various properties including some inclusion relations as well as some equivalent conditions in this new settings.
Referencias
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27. Tripathy, B. C., Dutta, H., On some lacunary difference sequence spaces defined by a sequence of Orlicz functions and q-lacunary ∆nm-statistical convergence, An. S¸tiint¸. Univ. "Ovidius" Constant¸a Ser. Mat, 20(1), 417-430, (2012). https://doi.org/10.2478/v10309-012-0028-1
28. Tripathy, B. C., Goswami, R., On triple difference sequences of real numbers in probabilistic normed spaces, Proyecciones, 33(2), 157-174, (2014). https://doi.org/10.4067/S0716-09172014000200003
29. Tripathy, B. C., Goswami, R., Multiple sequences in probabilistic normed spaces, Afr. Mat., 26(5-6), 753-760, (2015). https://doi.org/10.1007/s13370-014-0243-1
30. Tripathy, B. C., Goswami, R., Fuzzy real valued p-absolutely summable multiple sequences in probabilistic normed spaces, Afr. Mat., 26(7-8), 1281-1289, (2015). https://doi.org/10.1007/s13370-014-0280-9
31. Tripathy, B. C., Goswami, R., Statistically convergent multiple sequences in probabilistic normed spaces, U.P.B. Sci. Bull., Ser. A, 78(4), 83-94, (2016).
32. Tripathy, B. C., Hazarika, B., Choudhary, B., Lacunary I-convergent sequences, Kyungpook Math. J., 52(4), 473-482, (2012). https://doi.org/10.5666/KMJ.2012.52.4.473
2. Altin, Y., Et, M., Colak, R., Lacunary statistical and lacunary strongly convergence of generalized difference sequences in fuzzy numbers, Comput. Math. Appl., 52, 1011-1020, (2016). https://doi.org/10.1016/j.camwa.2006.03.025
3. Braha, N. L., On asymptotically ∆m lacunary statistical equivalent sequences, Appl. Math. Comput., 219, 280-288, (2012). https://doi.org/10.1016/j.amc.2012.06.016
4. Connor, J., The statistical and strong p-Ces'aro convergence of sequences, Analysis, 8, 47-63, (1988). https://doi.org/10.1524/anly.1988.8.12.47
5. Connor, J., Two valued measure and summability, Analysis, 10, 373-385, (1990). https://doi.org/10.1524/anly.1990.10.4.373
6. Esi, A., Ozdemir, M. K., Generalized ∆m-statistical convergence in probabilistic normed space, J. Comput. Appl. Math., 13(5), 923-932, (2011).
7. Esi, A., On asymptotically lacunay statistical equivalent sequences in probabilistic normed space, J. Math. Stat., 9(2), 144-148, (2013). https://doi.org/10.3844/jmssp.2013.144.148
8. Et, M., Colak, R., On some generalized difference sequence spaces, Soochow J. Math., 21(4), 377-386, (1995).
9. Fast, H., Sur la convergence statistique, Colloq. Math., 2, 241-244, (1951). https://doi.org/10.4064/cm-2-3-4-241-244
10. Frechet, M., Sur quelques points du calcul functionnel, Rend. Circ. Mat. Palermo, 22, 1-74, (1906). https://doi.org/10.1007/BF03018603
11. Freedman, A. R., Sember, J. J., Raphael, M. Some Cesa'ro-type summability spaces, Proc. Lond. Math. Soc., 37(3), 508-520, (1978). https://doi.org/10.1112/plms/s3-37.3.508
12. Fridy, J. A., Orhan, C., Lacunary statistical convergence, Pacific J. Math., 160, 43-51, (1993). https://doi.org/10.2140/pjm.1993.160.43
13. Fridy, J. A., Orhan, C., Statistical limit superior and inferior, Proc. Amer. Math. Soc., 125, 3625-3631, (1997). https://doi.org/10.1090/S0002-9939-97-04000-8
14. Guillen, B. L., Lallena, J. A., Sempi, C., Some classes of probabilistic normed spaces, Rend. Math., 17(7), 237-252, (1997).
15. Kizmaz, H., On certain sequence spaces, Canad. Math. Bull., 24(2), 169-176, (1981). https://doi.org/10.4153/CMB-1981-027-5
16. Marouf, M., Asymptotic equivalence and summability, Int. J. Math. Sci., 16(4), 755-762, (1993). https://doi.org/10.1155/S0161171293000948
17. Menger, K., Statistical metrices, Proc. Nat. Acad. Sci. USA., 28, 535-537, (1942). https://doi.org/10.1073/pnas.28.12.535
18. Patterson, R. F., On asymptotically statistically equivalent sequences, Demonstr. Math., 36(1), 149-153, (2003). https://doi.org/10.1515/dema-2003-0116
19. Patterson, R. F., Savas, E., On asymptotically lacunary statistical equivalent sequences, Thai J. Math., 4(2), 267-272, (2006).
20. Salat, T., On statistically convergent sequences of real numbers, Math. Slovaca, 30, 139-150, (1980).
21. Schweizer, B., Sklar, A., Probabilistic Metric Spaces, Elsevier, New York, (1983).
22. Sen, M., Et, M. Lacunary statistical and lacunary strongly convergence of generalized difference sequences in intuitionistic fuzzy normed linear spaces, Bol. Soc. Paran. Mat., 38(1), 117-129, (2020). https://doi.org/10.5269/bspm.v38i1.34814
23. Sen, M., Nath, S., Tripathy, B. C., Best approximation in quotient probabilistic normed space, J. Appl. Anal., 23(1), 53-57, (2017). https://doi.org/10.1515/jaa-2017-0008
24. Serstnev, A. N., Random normed spaces, questions of completeness, Kazan Gos. Univ. Uchen. Zap, 122(4), 3-20, (1962).
25. Steinhaus, H., Sur la convergence ordinaire et la convergence asymptotique, Colloq. Math., 2, 73-74, (1951).
26. Tripathy, B. C., Baruah, A., Lacunary statistically convergent and lacunary strongly convergent generalized difference sequences of fuzzy real numbers, Kyungpook Math. J., 50(4), 565-574, (2010). https://doi.org/10.5666/KMJ.2010.50.4.565
27. Tripathy, B. C., Dutta, H., On some lacunary difference sequence spaces defined by a sequence of Orlicz functions and q-lacunary ∆nm-statistical convergence, An. S¸tiint¸. Univ. "Ovidius" Constant¸a Ser. Mat, 20(1), 417-430, (2012). https://doi.org/10.2478/v10309-012-0028-1
28. Tripathy, B. C., Goswami, R., On triple difference sequences of real numbers in probabilistic normed spaces, Proyecciones, 33(2), 157-174, (2014). https://doi.org/10.4067/S0716-09172014000200003
29. Tripathy, B. C., Goswami, R., Multiple sequences in probabilistic normed spaces, Afr. Mat., 26(5-6), 753-760, (2015). https://doi.org/10.1007/s13370-014-0243-1
30. Tripathy, B. C., Goswami, R., Fuzzy real valued p-absolutely summable multiple sequences in probabilistic normed spaces, Afr. Mat., 26(7-8), 1281-1289, (2015). https://doi.org/10.1007/s13370-014-0280-9
31. Tripathy, B. C., Goswami, R., Statistically convergent multiple sequences in probabilistic normed spaces, U.P.B. Sci. Bull., Ser. A, 78(4), 83-94, (2016).
32. Tripathy, B. C., Hazarika, B., Choudhary, B., Lacunary I-convergent sequences, Kyungpook Math. J., 52(4), 473-482, (2012). https://doi.org/10.5666/KMJ.2012.52.4.473
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2022-12-23
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