On $\mathcal{I}_2$-Cauchy Double Sequences in Fuzzy Normed Spaces

Year 2019, Volume: 2 Issue: 2, 154 - 160, 27.06.2019

Erdinç Dundar , Muhammed Recai Türkmen

Abstract

In this paper, we investigate relationship between $\mathcal{I}_2$-convergence and $\mathcal{I}_2$-Cauchy double sequences in fuzzy normed spaces. After, we introduce the concepts of $\mathcal{I}_2^{*}$-Cauchy double sequences and study relationships between $\mathcal{I}_2$-Cauchy and $\mathcal{I}_2^{*}$-Cauchy double sequences in fuzzy normed spaces.

Keywords

Double sequences, $\mathcal{I}_2$-convergence, $\mathcal{I}_2$-Cauchy, Fuzzy normed space, $\mathcal{I}_2$-convergence

References

• [1] H. Fast, Sur la convergence statistique, Colloq. Math., 2 (1951), 241–244.
• [2] I.J. Schoenberg, The integrability of certain functions and related summability methods, Amer. Math. Monthly 66 (1959), 361–375.
• [3] M. Mursaleen, O.H.H. Edely, Statistical convergence of double sequences, J. Math. Anal. Appl. 288 (2003), 223–231.
• [4] D. Rath, B.C. Tripaty, On statistically convergence and statistically Cauchy sequences, Indian J. Pure Appl. Math. 25(4) (1994), 381–386.
• [5] P. Kostyrko, T. Salat, W. Wilczynski, $\mathcal{I}$-convergence, Real Anal. Exchange, 26(2) (2000), 669–686.
• [6] P. Das, P. Kostyrko, W. Wilczynski, P. Malik, $\mathcal{I}$ and $\mathcal{I}^{*}$-convergence of double sequences, Math. Slovaca, 58(5) (2008), 605–620.
• [7] P. Das, P. Malik, On extremal $\mathcal{I}$-limit points of double sequences, Tatra Mt. Math. Publ., 40 (2008), 91–102.
• [8] V. Kumar, On $\mathcal{I}$ and $\mathcal{I}^{*}$-convergence of double sequences, Math. Commun. 12 (2007), 171–181.
• [9] T. Salat, B.C. Tripaty, M. Ziman, On $\mathcal{I}$-convergence field, Ital. J. Pure Appl. Math. 17 (2005), 45–54.
• [10] B. Tripathy, B.C. Tripathy, On $\mathcal{I}$-convergent double sequences, Soochow J. Math. 31 (2005), 549–560.
• [11] M. Matloka, Sequences of fuzzy numbers, Busefal, 28 (1986), 28–37.
• [12] S. Nanda, On sequences of fuzzy numbers, Fuzzy Sets Syst. 33 (1989), 123–126.
• [13] C.Sençimen, S. Pehlivan, Statistical convergence in fuzzy normed linear spaces, Fuzzy Sets and Systems, 159 (2008), 361–370.
• [14] B. Hazarika, On ideal convergent sequences in fuzzy normed linear spaces, Afrika Mat., 25(4) (2013), 987–999.
• [15] E. Dündar, Ö. Talo, $\mathcal{I}_2$-convergence of double sequences of fuzzy numbers, Iran. J. Fuzzy Syst., 10(3) (2013), 37–50
• [16] E. Dündar, Ö. Talo, Ö., $\mathcal{I}_2$-Cauchy Double Sequences of Fuzzy Numbers, Gen. Math. Notes, 16(2) (2013), 103–114.
• [17] B. Hazarika, V. Kumar, Fuzzy real valued $\mathcal{I}$-convergent double sequences in fuzzy normed spaces, J. Intell. Fuzzy Syst., 26 (2014), 2323–2332.
• [18] E. Dündar, M. R. Türkmen, On $\mathcal{I}_2$-Convergence and $\mathcal{I}_2^{*}$-Convergence of Double Sequences in Fuzzy Normed Spaces, Konuralp J. Math., (in press).
• [19] V. Kumar, K. Kumar, On the ideal convergence of sequences of fuzzy numbers, Inform. Sci. 178 (2008), 4670–4678.
• [20] S.A. Mohiuddine, H.Şevli, M. Cancan, Statistical convergence of double sequences in fuzzy normed spaces, Filomat, 26(4) (2012), 673–681.
• [21] M. R. Türkmen, E. Dündar, On Lacunary Statistical Convergence of Double Sequences and Some Properties in Fuzzy Normed Spaces, J. Intell. Fuzzy Syst., 36(1) (2019), 467-472.
• [22] R. Saadati, On the $\mathcal{I}$-fuzzy topological spaces, Chaos, Solitons Fractals, 37 (2008), 1419–1426.
• [23] T.Bag, S.K. Samanta, Fixed point theorems in Felbin’s type fuzzy normed linear spaces, J. Fuzzy Math., 16(1) (2008), 243–260.
• [24] B. Bede, S.G. Gal, Almost periodic fuzzy-number-valued functions, Fuzzy Sets Syst., 147(2004), 385–403.
• [25] P. Diamond, P. Kloeden, Metric Spaces of Fuzzy Sets-Theory and Applications, World Scientific Publishing, Singapore, (1994).
• [26] E. Dündar, B. Altay, $\mathcal{I}_2$-convergence and $\mathcal{I}_2$-Cauchy of double sequences, Acta Math. Sci., 34(2) (2014), 343–353.
• [27] E. Dündar, B. Altay, On some properties of $\mathcal{I}_2$-convergence and $\mathcal{I}_2$-Cauchy of double sequences, Gen. Math. Notes, 7(1) (2011), 1–12.
• [28] J.-X. Fang, H. Huang, On the level convergence of a sequence of fuzzy numbers, Fuzzy Sets and Systems, 147 (2004), 417–415.
• [29] C. Felbin, Finite-dimensional fuzzy normed linear space, Fuzzy Sets and Systems, 48(2) (1992), 239–248.
• [30] Ö. Kişi, $\mathcal{I}_2-\lambda^{-}$-statistically convergence of double sequences in fuzzy normed spaces, J. Intell. Fuzzy Syst., 36(2) (2019), 1–12.
• [31] M. Mizumoto, K. Tanaka, Some properties of fuzzy numbers, Advances in Fuzzy Set Theory and Applications, North- Holland (Amsterdam), 1979, 153–164.
• [32] A. Nabiev, S. Pehlivan, M. Gürdal, On $\mathcal{I}$-Cauchy sequences, Taiwanese J. Math. 11(2) (2007) 569–5764.
• [33] A. Pringsheim, Zur theorie der zweifach unendlichen Zahlenfolgen, Math. Ann. 53 (1900), 289–321.
• [34] E. Savaş, M. Mursaleen, On statistically convergent double sequences of fuzzy numbers, Inform. Sci. 162 (2004), 183–192
• [35] L.A. Zadeh, Fuzzy sets, Inform. Control 8(1965), 338–353.