Title: Lacunary I_2-Invariant Convergence and Some Properties
Author(s): Ugur Ulusu, Erdinc Dundar, Fatih Nuray
Pages: 317-327
Cite as:
Ugur Ulusu, Erdinc Dundar, Fatih Nuray, Lacunary I_2-Invariant Convergence and Some Properties, Int. J. Anal. Appl., 16 (3) (2018), 317-327.


In this paper, the concept of lacunary invariant uniform density of any subset $A$ of the set $\mathbb{N}\times\mathbb{N}$ is defined. Associate with this, the concept of lacunary $\mathcal{I}_2$-invariant convergence for double sequences is given. Also, we examine relationships between this new type convergence concept and the concepts of lacunary invariant convergence and $p$-strongly lacunary invariant convergence of double sequences. Finally, introducing lacunary $\mathcal{I}_2^*$-invariant convergence concept and lacunary $\mathcal{I}_2$-invariant Cauchy concepts, we give the relationships among these concepts and relationships with lacunary $\mathcal{I}_2$-invariant convergence concept.

Full Text: PDF



  1. P. Das, P. Kostyrko, W. Wilczy´ nski and P. Malik, I and I∗-convergence of double sequences, Math. Slovaca, 58(5) (2008), 605-620. Google Scholar

  2. E. Dündar and B. Altay, I2-convergence and I2-Cauchy of double sequences, Acta Math. Sci., 34B(2) (2014), 343-353. Google Scholar

  3. E. Dündar, U. Ulusu and F. Nuray, On ideal invariant convergence of double sequences and some properties, Creat. Math. Inf., 27(2) (2018), (in press). Google Scholar

  4. J. A. Fridy and C. Orhan, Lacunary statistical convergence, Pacific J. Math., 160(1) (1993), 43–51. Google Scholar

  5. P. Kostyrko, T.ˇSalát and W. Wilczy´ nski, I-Convergence, Real Anal. Exchange, 26(2) (2000), 669–686. Google Scholar

  6. V. Kumar, On I and I∗-convergence of double sequences, Math. Commun. 12 (2007), 171–181. Google Scholar

  7. S. A. Mohiuddine and E. Sava¸s, Lacunary statistically convergent double sequences in probabilistic normed spaces, Ann Univ. Ferrara, 58 (2012), 331–339. Google Scholar

  8. M. Mursaleen, Matrix transformation between some new sequence spaces, Houston J. Math., 9 (1983), 505–509. Google Scholar

  9. M. Mursaleen, On finite matrices and invariant means, Indian J. Pure Appl. Math., 10 (1979), 457–460. Google Scholar

  10. M. Mursaleen and O. H. H. Edely, On the invariant mean and statistical convergence, Appl. Math. Lett., 22(11) (2009), 1700–1704. Google Scholar

  11. A. Nabiev, S. Pehlivan and M. Gürdal, On I-Cauchy sequences, Taiwanese J. Math., 11(2) (2007), 569–576. Google Scholar

  12. F. Nuray, H. Gök and U. Ulusu, Iσ-convergence, Math. Commun. 16 (2011) 531–538. Google Scholar

  13. N. Pancaroˇglu and F. Nuray, Statistical lacunary invariant summability, Theor. Math. Appl., 3(2) (2013), 71–78. Google Scholar

  14. A. Pringsheim, Zur theorie der zweifach unendlichen Zahlenfolgen, Math. Ann., 53 (1900), 289?21. Google Scholar

  15. R. A. Raimi, Invariant means and invariant matrix methods of summability, Duke Math. J., 30(1) (1963), 81–94. Google Scholar

  16. E. Sava¸s, Some sequence spaces involving invariant means, Indian J. Math., 31 (1989), 1–8. Google Scholar

  17. E. Sava¸s, Strongly σ-convergent sequences, Bull. Calcutta Math., 81 (1989), 295–300. Google Scholar

  18. E. Sava¸s and R. Patterson, Double σ-convergence lacunary statistical sequences, J. Comput. Anal. Appl., 11(4) (2009). Google Scholar

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

  20. U. Ulusu and F. Nuray, Lacunary Iσ-convergence, (under review). Google Scholar


Copyright © 2021 IJAA, unless otherwise stated.