f-Asymptotically I_2^σ-Equivalence of Double Sequences of Sets

Year 2019, Volume: 19 Issue: 1, 79 - 86, 28.05.2019

Erdinç Dündar , Nimet Akın

https://doi.org/10.35414/akufemubid.479439

Cited By: 1

Abstract

In this study, first, we present the
concepts of strongly asymptotically -equivalence, -asymptotically -equivalence, strongly -asymptotically -equivalence for double sequences of
sets. Then, we investigated some properties and relationships among this new
concepts. After, we present asymptotically -statistical equivalence for double
sequences of sets. Also we investigate relationships between asymptotically -statistical equivalence and strongly
-asymptotically -equivalence.

Keywords

Asymptotic Equivalence, I_2-Convergence, Invariant Convergence, Wijsman Convergence, Modulus Function

References

• Akın, N. P. and Dündar, E., 2018a. ”Asymptotically I-invariant statistical equivalence of sequences of set defined by a modulus function”, AKU J. Sci. Eng. 18(2), 477–485.
• Akın, N. P., Dündar, E., and Ulusu, U., 2018b. “Asymptotically I_σθ-statistical Equivalence of Sequences of Set Defined By A Modulus Function”, Sakarya University Journal of Science, 22(6) , 10.16984/saufenbilder.Akın, N. P., “Wijsman lacunary I_2-invariant convergence of double sequences of sets”, (In review).
• Baronti M., and Papini P., 1986. Convergence of sequences of sets, In: Methods of functional analysis in approximation theory (pp. 133-155), ISNM 76, Birkhäuser, Basel.
• Beer G., 1985. On convergence of closed sets in a metric space and distance functions. Bulletin of the Australian Mathematical Society, 31, 421-432.Beer G., 1994. Wijsman convergence: A survey. Set-Valued Analysis, 2 , 77-94.
• Das, P., Kostyrko, P., Wilczyński, W. and Malik, P., 2008.”I and I^*-convergence of double sequences”, Math. Slovaca, 58 (5) , 605–620.
• Dündar, E., Ulusu, U. and Nuray, F., “On asymptotically ideal invariant equivalence of double sequences”, (In review).
• Fast, H., 1951. Sur la convergence statistique. Colloquium Mathematicum, 2, 241-244.
• Khan, V. A. and Khan, N., 2013. “On Some I-Convergent Double Sequence Spaces Defined by a Modulus Function”, Engineering, 5, 35–40.
• Kılınç, G. and Solak, İ., 2014. “Some Double Sequence Spaces Defined by a Modulus Function”, Gen. Math. Notes, 25(2), 19–30.
• Kişi, Ö. and Nuray, F., 2013. A new convergence for sequences of sets. Abstract and Applied Analysis, Article ID 852796.
• Kişi Ö., Gümüş H. and Nuray F., 2015. I-Asymptotically lacunary equivalent set sequences defined by modulus function. Acta Universitatis Apulensis, 41, 141-151.
• Kostyrko P., Šalát T. and Wilczyński W., 2000. I-Convergence. Real Analysis Exchange, 26(2), 669-686.
• Kumar V. and Sharma A., 2012. Asymptotically lacunary equivalent sequences defined by ideals and modulus function. Mathematical Sciences, 6(23), 5 pages. Lorentz G., 1948. A contribution to the theory of divergent sequences. Acta Mathematica, 80, 167-190.
• Maddox J., 1986. Sequence spaces defined by a modulus. Mathematical Proceedings of the Cambridge Philosophical Society, 100, 161-166.
• Marouf, M., 1993. Asymptotic equivalence and summability. Int. J. Math. Math. Sci., 16(4), 755-762.
• Mursaleen, M. and Edely, O. H. H., 2009. On the invariant mean and statistical convergence. Applied Mathematics Letters, 22(11), 1700-1704.
• Mursaleen, M., 1983. Matrix transformation between some new sequence spaces. Houston Journal of Mathematics, 9, 505-509.
• Mursaleen, M., 1979. On finite matrices and invariant means. Indian Journal of Pure and Applied Mathematics, 10, 457-460.
• Nakano H., 1953. Concave modulars. Journal of the Mathematical Society Japan, 5 ,29-49.
• Nuray F. and Rhoades B. E., 2012. Statistical convergence of sequences of sets. Fasiciculi Mathematici, 49 , 87-99.
• Nuray, F. and Savaş, E., 1994. Invariant statistical convergence and A-invariant statistical convergence. Indian Journal of Pure and Applied Mathematics, 25(3), 267-274.
• Nuray, F., Gök, H. and Ulusu, U., 2011. I_σ-convergence. Mathematical Communications, 16, 531-538.
• Pancaroğlu, N. and Nuray, F., 2013a. Statistical lacunary invariant summability. Theoretical Mathematics and Applications, 3(2), 71-78.
• Pancarog ̆lu N. and Nuray F., 2013b. On Invariant Statistically Convergence and Lacunary Invariant Statistically Convergence of Sequences of Sets. Progress in Applied Mathematics, 5(2), 23-29. Pancarog ̆lu N. and Nuray F. and Savaş E., 2013. On asymptotically lacunary invariant statistical equivalent set sequences. AIP Conf. Proc. 1558(780) http://dx.doi.org/10.1063/1.4825609
• Pancarog ̆lu N. and Nuray F., 2014. Invariant Sta- tistical Convergence of Sequences of Sets with respect to a Modulus Function. Abstract and Applied Analysis, Article ID 818020, 5 pages. Patterson, R. F., 2003. On asymptotically statistically equivalent sequences. Demostratio Mathematica, 36(1), 149-153.
• Pehlivan S., and Fisher B., 1995. Some sequences spaces defined by a modulus. Mathematica Slovaca, 45, 275-280.
• Raimi, R. A., 1963. Invariant means and invariant matrix methods of summability. Duke Mathematical Journal, 30(1), 81-94.
• Savaş, E., 1989a. Some sequence spaces involving invariant means. Indian Journal of Mathematics, 31, 1-8.
• Savaş, E., 1989b. Strongly σ-convergent sequences. Bulletin of Calcutta Mathematical Society, 81, 295-300.
• Savaş, E., 2013. On I-asymptotically lacunary statistical equivalent sequences. Advances in Difference Equations, 111(2013), 7 pages. doi:10.1186/1687-1847-2013-111.
• Savaş, E. and Nuray, F., 1993. On σ-statistically convergence and lacunary σ-statistically convergence. Mathematica Slovaca, 43(3), 309-315.
• Schaefer, P., 1972. Infinite matrices and invariant means. Proceedings of the American Mathe-matical Society, 36, 104-110.
• Schoenberg I. J., 1959. The integrability of certain functions and related summability methods. American Mathematical Monthly, 66, 361-375.
• Tortop, Ş. and Dündar, E., “Wijsman I2-invariant convergence of double sequences of sets”, Journal of Inequalities and Special Functions, (In press).
• Ulusu U. and Nuray F., 2013. On asymptotically lacunary statistical equivalent set sequences. Journal of Mathematics, Article ID 310438, 5 pages.
• Ulusu U. and Gülle E., Asymptotically I_σ-equiva-lence of sequences of sets. (In press).
• Wijsman R. A., 1964. Convergence of sequences of convex sets, cones and functions. Bulletin American Mathematical Society, 70, 186-188.
• Wijsman R. A., 1966. Convergence of Sequences of Convex sets, Cones and Functions II. Transactions of the American Mathematical Society, 123(1) , 32-45.