Asymptotically Lacunary I-Invariant Statistical Equivalence of Sequences of Sets Defined By A Modulus Function
Year 2018,
, 1857 - 1862, 01.12.2018
Erdinç Dundar
,
Nimet P. Akın
,
Uğur Ulusu
Abstract
In
this paper, we introduce the concepts of strongly asymptotically lacunary I-invariant equivalence, f-asymptotically
lacunary I-invariant equivalence, strongly f-asymptotically lacunary I-invariant equivalence and asymptotically lacunary I-invariant statistical equivalence for sequences of
sets. Also, we investigate some relationships among these concepts.
References
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- [8] Ö. Ki▁si, H. Gümü▁s, F. Nuray I-Asymptotically lacunary equivalent set sequences defined by modulus function, Acta Universitatis Apulensis, 41, 141-151,2015.
- [9] P. Kostyrko, T. Šalát, W. Wilczyński, I-Convergence, Real Anal. Exchange, 26(2), 669–686, 2000.
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- [15] M. Mursaleen, Matrix transformation between some new sequence spaces, Houston J. Math. 9 , 505–509, 1983.
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- [21] N. Pancarog ̆lu, F. Nuray, Statistical lacunary invariant summability, Theoretical Mathematics and Applications, 3(2) , 71–78, 2013.
- [22] N. Pancaroğlu, F. Nuray, On Invariant Statistically Convergence and Lacunary Invariant Statistically Convergence of Sequences of Sets, Progress in Applied Mathematics, 5(2), 23–29, 2013.
- [23] N. Pancarog ̆lu, F. Nuray and E. Savaş, On asymptotically lacunary invariant statistical equivalent set sequences, AIP Conf. Proc. 1558(780), http://dx.doi.org/10.1063/1.4825609, 2013.
- [24] N. Pancaroğlu and F. Nuray, Invariant Statistical Convergence of Sequences of Sets with respect to a Modulus Function, Abstract and Applied Analysis, Article ID 818020, 5 pages, 2014.
- [25] N. Pancaroğlu and F. Nuray, Lacunary Invariant Statistical Convergence of Sequences of Sets with respect to a Modulus Function, Journal of Mathematics and System Science, 5, 122–126, 2015.
- [26] N. Pancaroğlu, E. Dündar and U. Ulusu, Asymptotically I-Invariant Statistical Equivalence of Sequences of Set Defined By A Modulus Function, (Under review).
- [27] R. F. Patterson, On asymptotically statistically equivalent sequences, Demostratio Mathematica, 36(1), 149–153, 2003.
- [28] R. F. Patterson and E. Savaş, On asymptotically lacunary statistically equivalent sequences, Thai J. Math. 4(2), 267–272, 2006.
- [29] S. Pehlivan, B. Fisher, Some sequences spaces defined by a modulus, Mathematica Slovaca, 45, 275-280, 1995.
- [30] R. A. Raimi, Invariant means and invariant matrix methods of summability, Duke Math. J., 30 , 81–94, 1963.
- [31] E. Savaş, Some sequence spaces involving invariant means, Indian J. Math. 31, 1–8, 1989.
- [32] E. Savaş, Strong σ-convergent sequences, Bull. Calcutta Math. 81, 295–300, 1989.
- [33] E. Savaş, F. Nuray, On σ-statistically convergence and lacunary σ-statistically convergence, Math. Slovaca, 43(3), 309–315, 1993.
- [34] E. Savaş, On I-asymptotically lacunary statistical equivalent sequences, Adv. Differ. Equ. (111) , doi:10.1186/1687-1847-2013-111, 2013.
- [35] P. Schaefer, Infinite matrices and invariant means, Proc. Amer. Math. Soc. 36 , 104–110, 1972.
- [36] I. J. Schoenberg, The integrability of certain functions and related summability methods, Amer. Math. Monthly, 66 , 361–375, 1959.
- [37] U. Ulusu and F. Nuray, Lacunary statistical convergence of sequence of sets, Progress in Applied Mathematics 4(2), 99–109, 2012.
- [38] U. Ulusu, F. Nuray, On asymptotically lacunary statistical equivalent set sequences, Journal of Mathematics, Article ID 310438, 5 pages, 2013.
- [39] U. Ulusu and E. Dündar, I-Lacunary Statistical Convergence of Sequences of Sets. Filomat 28(8), 1567–1574, 2013.
- [40] U. Ulusu, F. Nuray, Lacunary I_σ-convergence, (Under review).
- [41] U. Ulusu, E. Gülle, Asymptotically I_σ-equivalence of sequences of sets,(Under review).
- [42] R. A. Wijsman, Convergence of sequences of convex sets, cones and functions, Bull. Amer. Math. Soc. 70, 186–188, 1964.
- [43] R. A. Wijsman, Convergence of Sequences of Convex sets, Cones and Functions II, Trans. Amer. Math. Soc., 123(1), 32–45, 1966.
Year 2018,
, 1857 - 1862, 01.12.2018
Erdinç Dundar
,
Nimet P. Akın
,
Uğur Ulusu
References
- [1] M. Baronti, and P. Papini, Convergence of sequences of sets, In Methods of functional analysis in approximation theory, ISNM 76, Birkhäuser, Basel, 133-155,1986.
- [2] G. Beer, On convergence of closed sets in a metric space and distance functions, Bull. Aust. Math. Soc. 31, 421–432, 1985.
- [3] G. Beer, Wijsman convergence: A survey, Set-Valued Anal. 2 ,77–94, 1994.
- [4] H. Fast, Sur la convergence statistique, Colloq. Math. 2, 241–244, 1951.
- [5] J. A. Fridy, On statistical convergence, Analysis, 5, 301–313,1985.
- [6] J. A. Fridy and C. Orhan, Lacunary statistical convergence, Pacific J. Math. 160(1), 43–51,1993.
- [7] Ö. Kişi and Nuray, F. A new convergence for sequences of sets, Abstract and Applied Analysis, Article ID 852796, 6 pages, 2013.
- [8] Ö. Ki▁si, H. Gümü▁s, F. Nuray I-Asymptotically lacunary equivalent set sequences defined by modulus function, Acta Universitatis Apulensis, 41, 141-151,2015.
- [9] P. Kostyrko, T. Šalát, W. Wilczyński, I-Convergence, Real Anal. Exchange, 26(2), 669–686, 2000.
- [10] V. Kumar, A. Sharma, Asymptotically lacunary equivalent sequences defined by ideals and modulus function, Mathematical Sciences. 6(23), 5 pages, 2012.
- [11] G. Lorentz, A contribution to the theory of divergent sequences, Acta Math. 80, 167–190, 1948.
- [12] I. J. Maddox, Sequence spaces defined by a modulus, Math. Proc. Camb. Phil. Soc. 100, 161–166, 1986.
- [13] M. Marouf, Asymptotic equivalence and summability, Int. J. Math. Math. Sci. 16(4), 755-762, 1993.
- [14] M. Mursaleen, O. H. H. Edely, On the invariant mean and statistical convergence, Appl. Math. Lett., 22, 1700–1704, 2009.
- [15] M. Mursaleen, Matrix transformation between some new sequence spaces, Houston J. Math. 9 , 505–509, 1983.
- [16] M. Mursaleen, On finite matrices and invariant means, Indian J. Pure and Appl. Math. 10, 457–460, 1979.
- [17] H. Nakano, Concave modulars, J. Math. Soc. Japan, 5, 29-49, 1953.
- [18] F. Nuray, B. E. Rhoades, Statistical convergence of sequences of sets, Fasc. Math. 49, 87–99, 2012.
- [19] F. Nuray, E. Savaş, Invariant statistical convergence and A-invariant statistical convergence, Indian J. Pure Appl. Math. 10, 267–274, 1994.
- [20] F. Nuray, H. Gök, U. Ulusu, I_σ-convergence, Math. Commun. 16, 531–538, 2011.
- [21] N. Pancarog ̆lu, F. Nuray, Statistical lacunary invariant summability, Theoretical Mathematics and Applications, 3(2) , 71–78, 2013.
- [22] N. Pancaroğlu, F. Nuray, On Invariant Statistically Convergence and Lacunary Invariant Statistically Convergence of Sequences of Sets, Progress in Applied Mathematics, 5(2), 23–29, 2013.
- [23] N. Pancarog ̆lu, F. Nuray and E. Savaş, On asymptotically lacunary invariant statistical equivalent set sequences, AIP Conf. Proc. 1558(780), http://dx.doi.org/10.1063/1.4825609, 2013.
- [24] N. Pancaroğlu and F. Nuray, Invariant Statistical Convergence of Sequences of Sets with respect to a Modulus Function, Abstract and Applied Analysis, Article ID 818020, 5 pages, 2014.
- [25] N. Pancaroğlu and F. Nuray, Lacunary Invariant Statistical Convergence of Sequences of Sets with respect to a Modulus Function, Journal of Mathematics and System Science, 5, 122–126, 2015.
- [26] N. Pancaroğlu, E. Dündar and U. Ulusu, Asymptotically I-Invariant Statistical Equivalence of Sequences of Set Defined By A Modulus Function, (Under review).
- [27] R. F. Patterson, On asymptotically statistically equivalent sequences, Demostratio Mathematica, 36(1), 149–153, 2003.
- [28] R. F. Patterson and E. Savaş, On asymptotically lacunary statistically equivalent sequences, Thai J. Math. 4(2), 267–272, 2006.
- [29] S. Pehlivan, B. Fisher, Some sequences spaces defined by a modulus, Mathematica Slovaca, 45, 275-280, 1995.
- [30] R. A. Raimi, Invariant means and invariant matrix methods of summability, Duke Math. J., 30 , 81–94, 1963.
- [31] E. Savaş, Some sequence spaces involving invariant means, Indian J. Math. 31, 1–8, 1989.
- [32] E. Savaş, Strong σ-convergent sequences, Bull. Calcutta Math. 81, 295–300, 1989.
- [33] E. Savaş, F. Nuray, On σ-statistically convergence and lacunary σ-statistically convergence, Math. Slovaca, 43(3), 309–315, 1993.
- [34] E. Savaş, On I-asymptotically lacunary statistical equivalent sequences, Adv. Differ. Equ. (111) , doi:10.1186/1687-1847-2013-111, 2013.
- [35] P. Schaefer, Infinite matrices and invariant means, Proc. Amer. Math. Soc. 36 , 104–110, 1972.
- [36] I. J. Schoenberg, The integrability of certain functions and related summability methods, Amer. Math. Monthly, 66 , 361–375, 1959.
- [37] U. Ulusu and F. Nuray, Lacunary statistical convergence of sequence of sets, Progress in Applied Mathematics 4(2), 99–109, 2012.
- [38] U. Ulusu, F. Nuray, On asymptotically lacunary statistical equivalent set sequences, Journal of Mathematics, Article ID 310438, 5 pages, 2013.
- [39] U. Ulusu and E. Dündar, I-Lacunary Statistical Convergence of Sequences of Sets. Filomat 28(8), 1567–1574, 2013.
- [40] U. Ulusu, F. Nuray, Lacunary I_σ-convergence, (Under review).
- [41] U. Ulusu, E. Gülle, Asymptotically I_σ-equivalence of sequences of sets,(Under review).
- [42] R. A. Wijsman, Convergence of sequences of convex sets, cones and functions, Bull. Amer. Math. Soc. 70, 186–188, 1964.
- [43] R. A. Wijsman, Convergence of Sequences of Convex sets, Cones and Functions II, Trans. Amer. Math. Soc., 123(1), 32–45, 1966.