$\mathcal{I}_2$-convergence of double sequences of\\ fuzzy numbers

Document Type: Research Paper


1 Department of Mathematics, Afyon Kocatepe University, 03200 Afyonkarahisarn,Turkey

2 Department of Mathematics, Celal Bayar University, 45040 Manisa, Turkey


In this paper, we introduce and study the concepts of $\mathcal{I}_2$-convergence, $\mathcal{I}_2^{*}$-convergence for double sequences of fuzzy real numbers, where $\mathcal{I}_2$ denotes the ideal of subsets of $\mathbb N \times \mathbb N$. Also, we study some properties and relations of them.


B. Altay and F. Bad{s}ar, emph{Some new spaces of double sequences}, J. Math. Anal. Appl., textbf{309(1)} (2005), 70--90.

H. Alt{i}nok, Y. Alt{i}n and M. Id{s}{i}k, emph{Statistical convergence and strong p-Ces'{a}ro summability of order $beta$ in sequences of fuzzy numbers}, Iranian Journal of Fuzzy
Systems, textbf{9(2)} (2012), 63--73.

bibitem{bede}B. Bede and S. G. Gal, textit{Almost periodic fuzzy-number-valued functions},
Fuzzy Sets and Systems, textbf{147} (2004), 385--403.

d{C}. CÇakan and B. Altay, emph{Statistically boundedness and statistical core
of double sequences}, J. Math. Anal. Appl., textbf{317} (2006), 690--697.

bibitem{das 1}
P. Das, P. Kostyrko, W. Wilczy'{n}ski and P. Malik, emph{I and
$I^{*}$-convergence of double sequences}, Math. Slovaca, textbf{58(5)} (2008), 605--620.

bibitem{das 2}
P. Das and P. Malik, emph{On extremal I-limit points of double
sequences}, Tatra Mt. Math. Publ., textbf{40} (2008), 91--102.

bibitem{edba FU}
E. D"{u}ndar and B. Altay emph{$mathcal{I}_2$-uniform convergence of double
sequences of functions}, (under communication).

bibitem{fang}J. X. Fang and H. Huang, textit{On the level convergence of a
sequence of fuzzy numbers}, Fuzzy Sets and Systems, textbf{147} (2004), 417-415.

H. Fast, emph {Sur la convergence statistique}, Colloq. Math.,
textbf{2} (1951), 241--244.

J. A. Fridy, emph{On statistical convergence}, Analysis,
textbf{5} (1985), 301--313.

bibitem{fr- c.o}
J. A. Fridy and C. Orhan, emph{Statistical limit superior and inferior}, Proc. Amer. Math. Soc., textbf{125} (1997), 3625--3631.

J. A. Fridy, emph{Statistical limit points}, Proc. Amer. Math. Soc.,
textbf{118} (1993), 1187--1192.

P. Kostyrko, T. u{S}al'{a}t and W. Wilczy'{n}ski, emph{I-convergence},
Real Anal. Exchange, textbf{26(2)} (2000), 669-686.

P. Kostyrko, M. Mav{c}aj, T. u{S}al'{a}t and M. Sleziak, emph{I-convergence
and extremal I-limit points}, Math. Slovaca, textbf{55} (2005), 443--464.

bibitem{kumar 1}
V. Kumar, emph{On I and $I^{*}$-convergence of double sequences},
Math. Commun., textbf {12} (2007), 171--181.

bibitem{kumar F}
V. Kumar and K. Kumar, emph{On the ideal convergence of sequences of fuzzy numbers}, Information Sciences, textbf{178} (2008), 4670--4678.

M. Matloka, emph{Sequences of fuzzy numbers}, Busefal, textbf{28} (1986), 28--37.

Mursaleen and O. H. H. Edely, emph{Statistical convergence of double
sequences}, J. Math. Anal. Appl., textbf{288} (2003), 223--231.

S. Nanda, emph{On sequences of fuzzy numbers}, Fuzzy Sets and Systems, textbf{33} (1989), 123--126.

A. Nabiev, S. Pehlivan and M. G"{u}rdal, emph{On I-Cauchy sequence},
Taiwanese J. Math., textbf {11(2)} (2007), 569--576.

F. Nuray and W. H. Ruckle, emph{Generalized statistical convergence and convergence free spaces}, J. Math. Anal. Appl., textbf{245} (2000), 513--527.

bibitem{nuray 2}
F. Nuray, emph{Lacunary statistical convergence of sequences of fuzzy numbers},
Fuzzy Sets and Systems, textbf{99} (1998), 353--355.

bibitem{nuray 3}
F. Nuray and E. Savad{s}, emph{Statistical convergence of sequences of fuzzy numbers}, Math. Slovaca, textbf{45(3)} (1995), 269--273.

A. Pringsheim, emph{Zur theorie der zweifach unendlichen Zahlenfolgen},
Math. Ann., textbf{53} (1900), 289--321.

D. Rath and B. C. Tripaty, emph{On statistically convergence and
statistically Cauchy sequences}, Indian J. Pure Appl. Math., textbf{25(4)} (1994), 381--386.

R. Saadati, emph{On the I-fuzzy topological spaces}, Chaos, Solitons and Fractals,
textbf{37} (2008), 1419--1426.

bibitem{salat st}
T. u{S}al'{a}t, emph {On statistically convergent sequences of
real numbers}, Math. Slovaca, textbf{30} (1980), 139--150.

T. u{S}al'{a}t, B. C. Tripaty and M. Ziman, emph{On I-convergence
field},  Ital. J. Pure Appl. Math., textbf {17} (2005), 45--54.

E. Savad{s}, emph{On statistical convergent sequences of fuzzy numbers},
Information Sciences, textbf{137} (2001), 277--282.

E. Savad{s} and Mursaleen, emph{On statistically convergent
double sequences of fuzzy numbers}, Information Sciences, textbf{162} (2004), 183--192.

E. Savad{s}, emph{A note on double sequences of fuzzy numbers}, Turk. Jour. Math., textbf{20(20)} (1996), 175--178.

E. Savad{s}, emph{$(A)_{Delta}$-double sequence spaces of fuzzy numbers via orlicz function}, Iranian Journal of Fuzzy
Systems, textbf{8(2)} (2011), 91--103.

I. J. Schoenberg, emph {The integrability of certain functions and
related summability methods}, Amer. Math. Monthly, textbf {66}
(1959), 361--375.

"{O}. Talo and F. Bad{s}ar, emph{Determination of the
duals of classical sets of sequences of fuzzy numbers and related
matrix transformations}, Comput. Math. Appl., textbf{58} (2009),

bibitem{tri 1}
B. Tripathy and B. C. Tripathy, emph{On I-convergent double
sequences}, Soochow J. Math., textbf {31} (2005), 549--560.

bibitem{tri 2}
B. C. Tripathy, emph{Statistically convergent double sequences}, Tamkang J. Math., textbf{34(3)} (2003), 231--237.

bibitem{tri 3}
B. C. Tripathy and B. Sarma, emph{Double sequence spaces of fuzzy numbers defined by Orlicz function}, Acta Math. Sci., textbf{31B(1)} (2011), 134--140.

L. A. Zadeh, textit{Fuzzy sets}, Information and Control, textbf{8}(1965), 338--353.