Asymptotically Lacunary $J_\delta$-Equivalence of Sequences of Sets

In this study, we introduce the notions of Wijsman asymptotically strongly p -lacunary invariant equivalence (${\left[{W^L}_{N\delta\theta}\right]}_p$), Wijsman asymptotically lacunary J -invariant equivalence (${\left[{W^L}_{N\delta\theta}\right]}_p$) and Wijsman asymptotically lacunary J* -invariant equivalence ($W_{J^\ast\delta\vartheta}^L$) for sequences of sets. Also, the relationships among the notions of Wijsman asymptotically lacunary invariant equivalence, Wijsman asymptotically lacunary invariant statistical equivalence, ${\left[{W^L}_{N\delta\theta}\right]}_p$, ${\left[{W^L}_{N\delta\theta}\right]}_p$ and ($W_{J^\ast\delta\vartheta}^L$) are investigated.

Küme Dizilerinin Asimptotik Lacunary $J_\delta$ -Denkliği

Bu çalışmada, küme dizileri için Wijsman asimptotik kuvvetli p -lacunary invaryant denklik (${\left[{W^L}_{N\delta\theta}\right]}_p$), Wijsman asimptotik lacunary J -invaryant denklik (${\left[{W^L}_{N\delta\theta}\right]}_p$) ve Wijsman asimptotik lacunary J* -invaryant denklik ($W_{J^\ast\delta\vartheta}^L$) kavramları tanıtıldı. Ayrıca, Wijsman asimptotik lacunary invaryant denklik, Wijsman asimptotik lacunary invaryant istatistiksel denklik, ${\left[{W^L}_{N\delta\theta}\right]}_p$, ${\left[{W^L}_{N\delta\theta}\right]}_p$ ve ($W_{J^\ast\delta\vartheta}^L$) kavramları arasındaki ilişkiler araştırıldı.

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Baronti, M., Papini, P. 1986. Convergence of sequences of sets, In: Methods of Functional Analysis in Approximation Theory. Birkhäuser, Basel, pp. 133–155.

Beer, G. 1985. On convergence of closed sets in a metric space and distance functions. Bull. Aust. Math. Soc., 31(3): 421–432.

Beer, G. 1994. Wijsman convergence: A survey. Set-Valued Anal., 2(1-2): 77–94.

Fast, H. 1951. Sur la convergence statistique. Colloq. Math., 2(3- 4): 241–244.

Fridy, JA. 1985. On statistical convergence. Analysis, 5(4): 301– 314.

Fridy, JA., Orhan, C. 1993. Lacunary statistical convergence. Pacific J. Math., 160(1): 43-51.

Hazarika, B. 2015. On asymptotically ideal equivalent sequences. Journal of the Egyptian Mathematical Society, 23(1): 67–72.

Kişi, Ö., Nuray, F. 2013. New convergence definitions for sequences of sets. Abstract and Applied Analysis, 2013(Article ID 852796): 6 pages.

Kostyrko, P., Wilczyński, W., Šalát, T. 2000. J -Convergence. Real Anal. Exchange, 26(2): 669–686.

Marouf, M. 1993. Asymptotic equivalence and summability. Int. J. Math. Math. Sci., 16(4): 755–762.

Mursaleen, M. 1979. Invariant means and some matrix transformations. Tamkang J. Math., 10(2): 183–188.

Mursaleen M., Edely, OHH. 2009. On the invariant mean and statistical convergence. Appl. Math. Lett., 22(11): 1700–1704.

Nuray, F., Gök, H., Ulusu, U. 2011. $J_\delta$ -convergence. Math. Commun., 16: 531–538.

Nuray, F., Rhoades, BE. 2012. Statistical convergence of sequences of sets. Fasc. Math., 49: 87–99.

Pancaroğlu, N., Nuray, F. 2013. On invariant statistically convergence and lacunary invariant statistically convergence of sequences of sets. Progress in Applied Mathematics, 5(2): 23–29.

Pancaroğlu, N., Nuray, F. 2013. Statistical lacunary invariant summability. Theoretical Mathematics and Applications, 3(2): 71–78.

Pancaroğlu, N., Nuray, F., Savaş, E. 2013. On asymptotically lacunary invariant statistical equivalent set sequences. AIP Conf. Proc., 1558(1): 780–781.

Pancaroğlu Akın, N., Dündar, E., Ulusu, U. 2019. Wijsman lacunary J -invariant convergence of sequences of sets. Proc. Nat. Acad. Sci. India Sect. A. (in press).

Patterson, R.F., Savaş, E. 2006. On asymptotically lacunary statistically equivalent sequences. Thai J. Math., 4(2): 267–272. Raimi, R.A. 1963. Invariant means and invariant matrix methods of summability. Duke Math. J., 30(1): 81–94.

Šalát, T. 1980. On statistically convergent sequences of real numbers. Math. Slovaca, 30(2): 139–150.

Savaş, E. 1989. Some sequence spaces involving invariant means. Indian J. Math., 31: 1–8.

Savaş, E. 1990. On lacunary strong v -convergence. Indian J. Pure Appl. Math., 21(4): 359–365.

Savaş, E. 2013. On J -asymptotically lacunary statistical equivalent sequences. Adv. Difference Equ., 2013(111): 7 pages.

Savaş, E., Nuray, F. 1993. On $\delta$ -statistically convergence and lacunary v -statistically convergence. Math. Slovaca, 43(3): 309–315.

Savaş, E., Patterson, RF. 2006. $\delta$-asymptotically lacunary statistical equivalent sequences. Cent. Eur. J. Math., 4(4): 648– 655.

Schaefer, P. 1972. Infinite matrices and invariant means. Proc. Amer. Math. Soc., 36(1): 104–110.

Sever, Y., Ulusu, U., Dündar, E. 2014. On strongly J and J* -lacunary convergence of sequences of sets. AIP Conf. Proc., 1611(1): 357–362.

Ulusu, U., Dündar, E. 2014. J -lacunary statistical convergence of sequences of sets. Filomat, 28(8): 1567–1574.

Ulusu, U., Dündar, E. 2018. Asymptotically J -Cesàro equivalence of sequences of sets. Universal Journal of Mathematics and Applications, 1(2): 101-105.

Ulusu, U., Nuray, F. 2012. Lacunary statistical convergence of sequences of sets. Progress in Applied Mathematics, 4(2): 99– 109.

Ulusu, U., Nuray, F. 2013. On asymptotically lacunary statistical equivalent set sequences. Journal of Mathematics, 2013(Article ID 310438): 5 pages.

Ulusu, U., Nuray, F. 2016. Lacunary c -convergence. 2nd International Conference on Analysis and Its Applications, pp: 321, Kırşehir.

Wijsman, RA. 1964. Convergence of sequences of convex sets, cones and functions. Bull. Amer. Math. Soc., 70(1): 186–188.