Research Article
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Asymptotically Lacunary I-Invariant Statistical Equivalence of Sequences of Sets Defined By A Modulus Function

Year 2018, Volume: 22 Issue: 6, 1857 - 1862, 01.12.2018

Erdinç Dundar Nimet P. Akın Uğur Ulusu

https://doi.org/10.16984/saufenbilder.445147
Cited By: 2

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.

Keywords

Asymptotic equivalence modulus function I-convergence

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.

Year 2018, Volume: 22 Issue: 6, 1857 - 1862, 01.12.2018

Erdinç Dundar Nimet P. Akın Uğur Ulusu

https://doi.org/10.16984/saufenbilder.445147
Cited By: 2

Abstract

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.
There are 43 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Research Articles
Authors

Erdinç Dundar 0000-0002-0545-7486

Nimet P. Akın 0000-0003-2886-3679

Uğur Ulusu 0000-0001-7658-6114

Publication Date December 1, 2018
Submission Date July 19, 2018
Acceptance Date August 16, 2018
Published in Issue Year 2018 Volume: 22 Issue: 6

Cite

APA Dundar, E., P. Akın, N., & Ulusu, U. (2018). Asymptotically Lacunary I-Invariant Statistical Equivalence of Sequences of Sets Defined By A Modulus Function. Sakarya University Journal of Science, 22(6), 1857-1862. https://doi.org/10.16984/saufenbilder.445147
AMA Dundar E, P. Akın N, Ulusu U. Asymptotically Lacunary I-Invariant Statistical Equivalence of Sequences of Sets Defined By A Modulus Function. SAUJS. December 2018;22(6):1857-1862. doi:10.16984/saufenbilder.445147
Chicago Dundar, Erdinç, Nimet P. Akın, and Uğur Ulusu. “Asymptotically Lacunary I-Invariant Statistical Equivalence of Sequences of Sets Defined By A Modulus Function”. Sakarya University Journal of Science 22, no. 6 (December 2018): 1857-62. https://doi.org/10.16984/saufenbilder.445147.
EndNote Dundar E, P. Akın N, Ulusu U (December 1, 2018) Asymptotically Lacunary I-Invariant Statistical Equivalence of Sequences of Sets Defined By A Modulus Function. Sakarya University Journal of Science 22 6 1857–1862.
IEEE E. Dundar, N. P. Akın, and U. Ulusu, “Asymptotically Lacunary I-Invariant Statistical Equivalence of Sequences of Sets Defined By A Modulus Function”, SAUJS, vol. 22, no. 6, pp. 1857–1862, 2018, doi: 10.16984/saufenbilder.445147.
ISNAD Dundar, Erdinç et al. “Asymptotically Lacunary I-Invariant Statistical Equivalence of Sequences of Sets Defined By A Modulus Function”. Sakarya University Journal of Science 22/6 (December 2018), 1857-1862. https://doi.org/10.16984/saufenbilder.445147.
JAMA Dundar E, P. Akın N, Ulusu U. Asymptotically Lacunary I-Invariant Statistical Equivalence of Sequences of Sets Defined By A Modulus Function. SAUJS. 2018;22:1857–1862.
MLA Dundar, Erdinç et al. “Asymptotically Lacunary I-Invariant Statistical Equivalence of Sequences of Sets Defined By A Modulus Function”. Sakarya University Journal of Science, vol. 22, no. 6, 2018, pp. 1857-62, doi:10.16984/saufenbilder.445147.
Vancouver Dundar E, P. Akın N, Ulusu U. Asymptotically Lacunary I-Invariant Statistical Equivalence of Sequences of Sets Defined By A Modulus Function. SAUJS. 2018;22(6):1857-62.

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