i-manager Publications

On I2-Cauchy Double Sequences in Fuzzy N-Normed Spaces

Muhammed Recai Turkmen *

Department of Mathematics, Faculty of Education, Afyon Kocatepe University, Afyonkarahisar, Turkey.

Periodicity:January - June'2020
DOI : https://doi.org/10.26634/jmat.9.1.17735

Abstract

In this paper, the concepts of I₂ -Cauchy, I₂* -Cauchy double sequences in fuzzy n-normed spaces were introduced and some properties and relations of them were studied. We show that if a double sequence (x_mr) in X is an I₂* -double Cauchy sequence, then it is I₂ -double Cauchy sequence, where I₂ denotes the ideal of subsets of N×N.

Keywords

Ideal, Double Sequences, Cauchy Sequence, I2 -Convergence.

How to Cite this Article?

Turkmen, M. R. (2020). On I2-Cauchy Double Sequences in Fuzzy N-Normed Spaces. i-manager's Journal on Mathematics, 9(1), 18-27. https://doi.org/10.26634/jmat.9.1.17735

References

[1]. Altay, B., & Başar, F. (2005). Some new spaces of double sequences. Journal of Mathematical Analysis and Applications, 309(1), 70-90. https://doi.org/10.1016/j.jmaa.2004.12.020

[2]. Altinok, H., Altin, Y., & Et, M. (2004). Lacunary almost statistical convergence of fuzzy numbers. Thai Journal of Mathematics, 2(2), 265-274.

[3]. Altinok, H., & Kasap, M. (2017). f-Statistical convergence of order β for sequences of fuzzy Numbers. Journal of Intelligent & Fuzzy Systems, 33(2), 705-712. https://doi.org/10.3233/JIFS-161654

[4]. Bag, T., & Samanta, S. K. (2008). Fixed point theorems in Felbin's type fuzzy normed linear spaces. Journal of Fuzzy Mathematics, 16(1), 243–260.

[5]. Çakan, C., & Altay, B. (2006). Statistically boundedness and statistical core of double sequences. Journal of Mathematical Analysis and Applications, 317(2), 690-697. https://doi.org/10.1016/j.jmaa.2005.06.006

[6]. Das, P., Kostyrko, P., Wilczyński, W., & Malik, P. (2008). I and I*-convergence of double sequences. Mathematica Slovaca, 58(5), 605-620. https://doi.org/10.2478/s12175-008-0096-x

[7]. Das, P., & Malik, P. (2008). On extremal I-limit points of double sequences. Tatra Mountains Mathematical Publications, 40, 91-102.

[8]. Diamond, P., & Kloeden, P. E. (1994). Metric spaces of fuzzy sets: Theory and applications. World Scientific.

[9]. Dündar, E., Ulusu, U., & Pancaroğlu, N. (2016). Strongly I₂ -lacunary convergence and I₂ -lacunary cauchy double sequences of sets. The Aligarh Bulletin of Mathematics, 35 (1-2), 1–15.

[10]. Dündar, E., & Talo, Ö. (2013). I₂ -convergence of double sequences of fuzzy numbers. Iranian Journal of Fuzzy Systems,10(3), 37-50.

[11]. Dündar, E., & Talo, Ö. (2013). I₂ -Cauchy double sequences of fuzzy numbers. Gen, 16(2), 103-114.

[12]. Fang, J. X., & Huang, H. (2004). On the level convergence of a sequence of fuzzy numbers. Fuzzy Sets and Systems, 147(3), 417-435. https://doi.org/10.1016/j.fss.2003.08.001

[13]. Fast, H. (1951). Sur la convergence statistique. In Colloquium Mathematicae (Vol. 2, No. 3-4, pp. 241-244).

[14]. Felbin, C. (1992). Finite dimensional fuzzy normed linear space. Fuzzy Sets and Systems, 48(2), 239-248. https://doi.org/10.1016/0165-0114(92)90338-5

[15]. Fridy, J. A. (1985). On statistical convergence. Analysis, 5(4), 301-314. https://doi.org/10.1524/anly.1985.5.4.301

[16]. Fridy, J. A. (1993). Statistical limit points. In Proceedings of the American Mathematical Society, 118(4), 1187-1192. https://doi.org/10.1090/S0002-9939-1993-1181163-6

[17]. Fridy, J. A., & Orhan, C. (1993). Lacunary statistical convergence. Pacific Journal of Mathematics, 160(1), 43-51.

[18]. Fridy, J., & Orhan, C. (1997). Statistical limit superior and limit inferior. In Proceedings of the American Mathematical Society, 125(12), 3625-3631. https://doi.org/10.1090/S0002-9939-97-04000-8

[19]. Hazarika, B. (2014). On ideal convergent sequences in fuzzy normed linear spaces. Afrika Matematika, 25(4), 987-999. https://doi.org/10.1007/s13370-013-0168-0

[20]. Kostyrko, P., Šalát,T., & Wilczyński, W. (2000). I-convergence, Real Anal. Exchange, 26(2), 669-686.

[21]. Kostyrko, P., Máčaj, M., Šalát, T., & Sleziak, M. (2005). I-convergence and extremal I-limit points. Mathematica Slovaca, 55(4), 443-464.

[22]. Kumar, V. (2007). On I and I*− convergence of double sequences. Mathematical Communications, 12(2), 171-181.

[23]. Kumar, V., & Kumar, K. (2008). On the ideal convergence of sequences of fuzzy numbers. Information Sciences, 178(24), 4670-4678. https://doi.org/10.1016/j.ins.2008.08.013

[24]. Matloka, M. (1986). Sequences of fuzzy numbers. Busefal, 28(1), 28-37.

[25]. Mizmoto, M., & Tanaka, K. (1979). Some properties of fuzzy numbers. In Advances in Fuzzy Sets Theory and Applications, North Holland, Amsterdam (pp. 153–164).

[26]. Mohiuddine, S. A., Şevli, H., & Cancan, M. (2012). Statistical convergence of double sequences in fuzzy normed spaces. Filomat, 26(4), 673-681.

[27]. Edely, O. H. (2003). Statistical convergence of double sequences. Journal of Mathematical Analysis and Applications, 288(1), 223-231. https://doi.org/10.1016/j.jmaa.2003.08.004

[28]. Nanda, S. (1989). On sequences of fuzzy numbers. Fuzzy Sets and systems, 33(1), 123-126.

[29]. Nuray, F., & Ruckle, W. H. (2000). Generalized statistical convergence and convergence free spaces. Journal of Mathematical Analysis and Applications, 245(2), 513-527. https://doi.org/10.1006/jmaa.2000.6778

[30]. Nuray, F. (1998). Lacunary statistical convergence of sequences of fuzzy numbers. Fuzzy Sets and Systems, 99(3), 353- 355. https://doi.org/10.1016/S0165-0114(98)00031-1

[31]. Nuray, F., & Savaş, E. (1995). Statistical convergence of sequences of fuzzy numbers. Mathematica Slovaca, 45(3), 269-273.

[32]. Nuray, F., Ulusu, U., & Dündar, E. (2016). Lacunary statistical convergence of double sequences of sets. Soft Computing, 20(7), 2883-2888. https://doi.org/10.1007/s00500-015-1691-8

[33]. Pringsheim, A. (1900). Zur theorie der zweifach unendlichen Zahlenfolgen. Mathematische Annalen, 53(3), 289-321. https://doi.org/10.1007/BF01448977

[34]. Rath, D., & Tripathy, B. C. (1994). On statistically convergent and statistically Cauchy sequences. Indian Journal of Pure and Applied Mathematics, 25, 381-381.

[35]. Saadati, R. (2008). On the I-fuzzy topological spaces. Chaos, Solitons and Fractals, 37(5), 1419–1426. https://doi.org/10.1016/j.chaos.2006.10.033

[36]. Šalát, T. (1980). On statistically convergent sequences of real numbers. Mathematica Slovaca, 30(2), 139-150.

[37]. Šalát, T., Tripathy, B. C., & Ziman, M. (2005). On I-convergence field. Ital. J. Pure Appl. Math, 17(5), 1-8.

[38]. Savaş, E. (2001). On statistically convergent sequences of fuzzy numbers. Information Sciences, 137(1-4), 277-282. https://doi.org/10.1016/S0020-0255(01)00110-4

[39]. Savaş, E. (2004). On statistically convergent double sequences of fuzzy numbers. Information Sciences, 162(3-4), 183- 192. https://doi.org/10.1016/j.ins.2003.09.005

[40]. Savaş, E. (2017). On I-lacunary statistical convergence of weight g of fuzzy numbers. Journal of Intelligent & Fuzzy Systems, 32(1), 1111-1117. https://doi.org/10.3233/JIFS-16292

[41]. Savaş, E. (2018). On lacunary p-summable convergence of weight g for fuzzy numbers via ideal. Journal of Intelligent & Fuzzy Systems, 34(4), 2121-2127. https://doi.org/10.3233/JIFS-17048

[42]. Şençimen, C., & Pehlivan, S. (2008). Statistical convergence in fuzzy normed linear spaces. Fuzzy Sets and Systems, 159(3), 361-370. https://doi.org/10.1016/j.fss.2007.06.008

[43]. Schoenberg, I. J. (1959). The integrability of certain functions and related summability methods. The American Mathematical Monthly 66(5), 361-775. https://doi.org/10.1080/00029890.1959.11989303

[44]. Talo, Ö., & Başar, F. (2009). Determination of the duals of classical sets of sequences of fuzzy numbers and related matrix transformations. Computers & Mathematics with Applications, 58(4), 717-733. https://doi.org/10.1016/j.camwa. 2009.05.002

[45]. Tripathy, B. & Tripathy, B. C. (2005). On I-convergent double sequences. Soochow Journal of Mathematıcs, 31(4), 549- 560.

[46]. Türkmen, M. R., & Çınar, M. (2017). Lacunary statistical convergence in fuzzy normed linear spaces. Applied and Computational Mathematics, 6(5), 233-237. https://doi.org/10.11648/j.acm.20170605.13

[47]. Türkmen, M. R., & Çınar, M. (2018). l-statistical convergence in fuzzy normed linear spaces. Journal of Intelligent & Fuzzy Systems, 34(6), 4023-4030. https://doi.org/10.3233/JIFS-171147

[48]. Türkmen, M. R., & Efe, H. (2013). On some properties of closability of farthest point maps in fuzzy n-Normed spaces. imanager's Journal on Mathematics, 2(4), 33-38. https://doi.org/10.26634/jmat.2.4.2613

[49]. Türkmen, M. R., & Dündar, E. (2019). On lacunary statistical convergence of double sequences and some properties in fuzzy normed spaces. Journal of Intelligent & Fuzzy Systems, 36(2), 1683-1690. https://doi.org/10.3233/JIFS-18841

[50]. L. A. Zadeh. (1965). Fuzzy sets. Information and Control, 8, 338–353.

If you have access to this article please login to view the article or kindly login to purchase the article

Single Article

	North Americas,UK, Middle East,Europe		India	Rest of world
	USD	EUR	INR	USD-ROW
Online	15	15

ADD TO CART

Options for accessing this content:

If you would like institutional access to this content, please recommend the title to your librarian.
If you already have i-manager's user account: Login above and proceed to purchase the article.
New Users: Please register, then proceed to purchase the article.