Previous |  Up |  Next

Article

Keywords:
double sequences; $\mu $-statistical convergence; divergence and Cauchy criteria; convergence; divergence and Cauchy criteria in $\mu $-density; condition (APO$_2)$
Summary:
In this paper, following the methods of Connor \cite {connor}, we extend the idea of statistical convergence of a double sequence (studied by Muresaleen and Edely \cite {moe}) to $\mu $-statistical convergence and convergence in $\mu $-density using a two valued measure $\mu $. We also apply the same methods to extend the ideas of divergence and Cauchy criteria for double sequences. We then introduce a property of the measure $\mu $ called the (APO$_2$) condition, inspired by the (APO) condition of Connor \cite {jc}. We mainly investigate the interrelationships between the two types of convergence, divergence and Cauchy criteria and ultimately show that they become equivalent if and only if the measure $\mu $ has the condition (APO$_2$).
References:
[1] Balcerzak, M., Dems, K.: Some types of convergence and related Baire systems. Real Anal. Exchange 30 (2004), 267-276. MR 2127531
[2] Connor, J.: Two valued measure and summability. Analysis 10 (1990), 373-385. DOI 10.1524/anly.1990.10.4.373 | MR 1085803
[3] Connor, J.: $R$-type summability methods, Cauchy criterion, $P$-sets and statistical convergence. Proc. Amer. Math. Soc. 115 (1992), 319-327. MR 1095221
[4] Connor, J., Fridy, J. A., Orhan, C.: Core equality results for sequences. J. Math. Anal. Appl. 321 (2006), 515-523. DOI 10.1016/j.jmaa.2005.07.067 | MR 2241135 | Zbl 1092.40001
[5] Das, P., Malik, P.: On the statistical and $I$ variation of double sequences. Real Anal. Exchange 33 (2008), 351-364. MR 2458252
[6] Das, P., Kostyrko, P., Wilczyński, W., Malik, P.: $I$ and $I^{*}$-convergence of double sequences. Math. Slovaca 58 (2008), 605-620. DOI 10.2478/s12175-008-0096-x | MR 2434680 | Zbl 1199.40026
[7] Dems, K.: On $I$-Cauchy sequences. Real Anal. Exchange 30 (2004), 123-128. MR 2126799
[8] Fast, H.: Sur la convergence statistique. Colloq. Math. 2 (1951), 241-244. MR 0048548 | Zbl 0044.33605
[9] Fridy, J. A.: On statistical convergence. Analysis 5 (1985), 301-313. DOI 10.1524/anly.1985.5.4.301 | MR 0816582 | Zbl 0588.40001
[10] Kostyrko, P., Šalát, T., Wilczyński, W.: $I$-Convergence. Real Anal. Exchange 26 (2000/2001), 669-686. MR 1844385
[11] Móricz, F.: Statistical convergence of multiple sequences. Arch. Math. 81 (2003), 82-89. DOI 10.1007/s00013-003-0506-9 | MR 2002719
[12] Muresaleen, Edely, Osama H. H.: Statistical convergence of double sequences. J. Math. Anal. Appl. 288 (2003), 223-231. DOI 10.1016/j.jmaa.2003.08.004 | MR 2019757
[13] Nuray, F., Ruckle, W. H.: Generalized statistical convergence and convergence free spaces. J. Math. Anal. Appl. 245 (2000), 513-527. DOI 10.1006/jmaa.2000.6778 | MR 1758553 | Zbl 0955.40001
[14] Pringsheim, A.: Zur Theorie der zweifach unendlichen Zahlenfolgen. Math. Ann. 53 (1900), 289-321. DOI 10.1007/BF01448977 | MR 1511092
[15] Savas, E., Muresaleen: {On statistically convergent double sequences of fuzzy numbers}. Information Sciences 162 (2004), 183-192. DOI 10.1016/j.ins.2003.09.005 | MR 2076238
[16] Šalát, T.: On statistically convergent sequences of real numbers. Math. Slovaca 30 (1980), 139-150. MR 0587239
[17] Schoenberg, I. J.: The integrability of certain functions and related summability methods. Amer. Math. Monthly. 66 (1959), 361-375. DOI 10.2307/2308747 | MR 0104946 | Zbl 0089.04002
Partner of
EuDML logo