Canak, I., Totur, U., Onder, Z. (2017). A Tauberian theorem for $(C,1,1)$ summable double sequences of fuzzy numbers. Iranian Journal of Fuzzy Systems, 14(1), 61-75. doi: 10.22111/ijfs.2017.3037

Ibrahim Canak; Umit Totur; Zerrin Onder. "A Tauberian theorem for $(C,1,1)$ summable double sequences of fuzzy numbers". Iranian Journal of Fuzzy Systems, 14, 1, 2017, 61-75. doi: 10.22111/ijfs.2017.3037

Canak, I., Totur, U., Onder, Z. (2017). 'A Tauberian theorem for $(C,1,1)$ summable double sequences of fuzzy numbers', Iranian Journal of Fuzzy Systems, 14(1), pp. 61-75. doi: 10.22111/ijfs.2017.3037

Canak, I., Totur, U., Onder, Z. A Tauberian theorem for $(C,1,1)$ summable double sequences of fuzzy numbers. Iranian Journal of Fuzzy Systems, 2017; 14(1): 61-75. doi: 10.22111/ijfs.2017.3037

A Tauberian theorem for $(C,1,1)$ summable double sequences of fuzzy numbers

^{1}Department of Mathematics, Ege University, 35100, Izmir, Turkey

^{2}Department of Mathematics, Adnan Menderes University, 09100, Aydin, Turkey

Abstract

In this paper, we determine necessary and sufficient Tauberian conditions under which convergence in Pringsheim's sense of a double sequence of fuzzy numbers follows from its $(C,1,1)$ summability. These conditions are satisfied if the double sequence of fuzzy numbers is slowly oscillating in different senses. We also construct some interesting double sequences of fuzzy numbers.

[1] B. Bede, Mathematics of fuzzy sets and fuzzy logic, Springer, Berlin, 2013. [2] _I. C. anak, Tauberian theorems for Cesaro summability of sequences of fuzzy number, J. Intell. Fuzzy Syst., 27(2) (2014), 937{942. [3] _I. C. anak, On Tauberian theorems for Cesaro summability of sequences of fuzzy numbers, J. Intell. Fuzzy Syst., 30(5) (2016), 2657{2662. [4] _I. C. anak, Holder summability method of fuzzy numbers and a Tauberian theorem, Iranian Journal of Fuzzy Systems, 11(4) (2014), 87{93. [5] _I.C. anak, Some conditions under which slow oscillation of a sequence of fuzzy numbers follows from Cesaro summability of its generator sequence, Iranian Journal of Fuzzy Systems, 11(4) (2014) 15{22. [6] D. Dubois and H. Prade, Fuzzy sets and systems: Theory and applications, Academic Press, New York-London, 1980. [7] R. Goetschel and W. Voxman, Elementary fuzzy calculus, Fuzzy Sets Syst., 18(1) (1986) 31{43. [8] M. Matlako, Sequences of fuzzy numbers, BUSEFAL, 28 (1986), 28{37. [9] F. Moricz, Tauberian theorems for Cesaro summable double sequences, Studia Math., 110 (1994), 83{96. [10] F. Moricz, Necessary and sufficient Tauberian conditions, under which convergence follows from summability (C; 1), Bull. London Math. Soc., 26 (1994), 288{294. [11] S. Nanda, On sequences of fuzzy numbers, Fuzzy Sets Syst., 33 (1989), 123{126. [12] E. Savas. , A note on double sequences of fuzzy numbers, Turkish J. Math., 20 (1996), 175{178. [13] P. V. Subrahmanyam, Cesaro summability of fuzzy real numbers, J. Anal., 7 (1999), 159{168. [14] B. C. Tripathy and A. J. Dutta, On fuzzy real-valued double sequence spaces, Soochow J. Math., 32(4) (2006), 509{520. [15]O. Talo and F. Bas.ar, On the slowly decreasing sequences of fuzzy numbers, Abstr. Appl. Anal., Article ID 891986 (2013), doi:10.1155/2013/891986, 1-7. [16] O. Talo and C. C. akan, On the Cesaro convergence of sequences of fuzzy numbers, Appl. Math. Lett., 25 (2012), 676{681. [17] O. Talo and C. C. akan, Tauberian theorems for statistically (C; 1)-convergent sequences of fuzzy numbers, Filomat, 28(4) (2014), 849{858. [18] B. C. Tripathy and A. Baruah, Norlund and Riesz mean of sequences of fuzzy real numbers, Appl. Math. Lett., 23 (2010), 651{655.

[19] B. C. Tripathy and A. J. Dutta, Bounded variation double sequence space of fuzzy real numbers, Comput. Math. Appl., 59(2) (2010), 1031{1037. [20] B. C. Tripathy and B. Sarma, Double sequence spaces of fuzzy numbers dened by Orlicz function, Acta Math. Sci. Ser. B Engl. Ed., 31(1) (2011), 134{140. [21] B. C. Tripathy and M. Sen, On lacunary strongly almost convergent double sequences of fuzzy numbers, An. Univ. Craiova Ser. Mat. Inform., 42(2) (2015), 254{259. [22] L. A. Zadeh, Fuzzy sets, Inf. Control, 8 (1965), 29{44.