\(\mathcal{I}_2\)-Cauchy double sequences in 2-normed spaces
DOI:
https://doi.org/10.14419/gjma.v3i1.4020Keywords:
Ideal, Double Sequences, \(\mathcal{I}_2\)-Convergence, \(\mathcal{I}_2\)-Cauchy, 2-normed spaces.Abstract
The concept \(\mathcal{I}\)-Cauchy and \(\mathcal{I}^{*}\)-Cauchy sequences were studied by Gudal and Acik in on \(\mathcal{I}\)-Cauchy sequences in 2-normed spaces, Math. Inequal. Appl. 11 (2) (2008), 349--354. In this paper, we introduce the
notions of \(\mathcal{I}_2\)-Cauchy and \(\mathcal{I}_2^{*}\)-Cauchy double sequences, and study their some properties with (AP2) in 2-normed spaces.
References
B. Altay, F. BaÅŸar, Some new spaces of double sequences, J. Math. Anal. Appl. 309 (1) (2005), 70-90.
P. Das, P. Kostyrko, W. Wilczynski, P. Malik, I and I-convergence of double sequences, Math. Slovaca 58 (5) (2008), 605-620.
P. Das, P. Malik, On extremal I-limit points of double sequences, Tatra Mt. Math. Publ. 40 (2008), 91-102.
E. Dündar, B. Altay, On some properties of I2-convergence and I2-Cauchy of double sequences, Gen. Math. Notes 7(1) (2011), 1-12.
H. Fast, Sur la convergence statistique, Colloq. Math. 2 (1951), 241-244.
J.A. Fridy, On statistical convergence, Analysis 5 (1985), 301-313.
S. Gähler, 2-metrische Räume und ihre topologische struktur, Math. Nachr. 26 (1963), 115-148.
S. Gähler, 2-normed spaces, Math. Nachr. 28 (1964), 1-43.
H. Gunawan, M. Mashadi, On n-normed spaces, Int. J. Math. Math. Sci. 27 (10) (2001), 631-639.
H. Gunawan, M. Mashadi, On _nite dimensional 2-normed spaces, Soochow J. Math. 27 (3) (2001), 321-329.
M. Gürdal, S. Pehlivan, The statistical convergence in 2-Banach spaces, Thai J. Math. 2 (1) (2004), 107-113.
M. Gürdal, I. Açık, On I-Cauchy sequences in 2-normed spaces, Math. Inequal. Appl. 11 (2) (2008), 349-354.
M. Gürdal, On ideal convergent sequences in 2-normed spaces, Thai J. Math. 4 (1) (2006), 85-91.
P. Kostyrko, T. Salát, W. Wilczynski, I-convergence, Real Anal. Exchange 26 (2) (2000), 669-686.
V. Kumar, On I and I-convergence of double sequences, Math. Commun. 12 (2007), 171-181.
Mursaleen, O.H.H. Edely, Statistical convergence of double sequences, J. Math. Anal. Appl. 288 (2003), 223-231.
M. Mursaleen, S.A. Mohiuddine, On ideal convergence of double sequences in probabilistic normed spaces, Math. Reports 12 (62) (4) (2010), 359-371.
M. Mursaleen, S.A. Mohiuddine, On ideal convergence in probabilistic normed spaces, Math. Slovaca 62 (2012), 49-62.
M. Mursaleen, A. Alotaibi, On I-convergence in random 2-normed spaces, Math. Slovaca 61 (6) (2011), 933-940.
F. Nuray, W.H. Ruckle, Generalized statistical convergence and convergence free spaces, J. Math. Anal. Appl. 245 (2000), 513-527.
A. Pringsheim, Zur theorie der zweifach unendlichen Zahlenfolgen, Math. Ann. 53 (1900), 289-321.
S. Sarabadan, S. Talebi, On I-convergence of double sequences in 2-normed spaces, Int. J. Comtemp. Math. Sci. 7 (14) (2012), 673-684.
S. Sarabadan, F.A. Arani, S. Khalehoghli, A condition for the equivalence of I and I-convergence in 2-normed spaces, Int. J. Comtemp. Math. Sci. 6 (43) (2011), 2147-2159.
S. Sarabadan, S. Talebi, Statistical convergence and ideal convergence of sequences of functions in 2-normed spaces, Internat. J. Math. Math. Sci. 2011 (2011), 10 pages.
I.J. Schoenberg, The integrability of certain functions and related summability methods, Amer. Math. Monthly 66 (1959), 361-375.
A. Şahiner, M. Gürdal, S. Saltan, H. Gunawan, Ideal convergence in 2-normed spaces, Taiwanese J. Math. 11 (2007), 1477-1484.
B. Tripathy, B.C. Tripathy, On I-convergent double sequences, Soochow J. Math. 31 (2005), 549-560.
B.C. Tripathy, M. Sen, S. Nath, I-convergence in probabilistic n-normed space, Soft Comput. 16 (2012), 1021-1027.
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