It was given a prototype constructing a new sequence space of fuzzy numbers by means of the matrix domain of a particular limitation method. That is we have constructed the Zweier sequence spaces of fuzzy numbers , , and consisting of all sequences such that in the spaces , , and , respectively. Also, we prove that , , and are linearly isomorphic to the spaces , , and , respectively. Additionally, the -, -, and -duals of the spaces , , and have been computed. Furthermore, two theorems concerning matrix map have been given. 1. Introduction and Preliminaries Let suppose that , , and are the set of all positive integers, all real numbers, and all bounded and closed intervals on the real line ; that is, and , respectively. For define It can easily be seen that defines a metric on and the pair is a complete metric space [1]. Let be nonempty set. According to Zadeh, a fuzzy subset of is a nonempty subset of for some function [2]. Consider a function as a subset of a nonempty base space and denote the family of all such functions or fuzzy sets by . Let us suppose that the function satisfies the following properties:(1) is normal; that is, there exists an such that ;(2) is fuzzy convex; that is, for any and , ;(3) is upper semicontinuous;(4)the closure of , denoted by , is compact. Then the function is called a fuzzy number [9]. Properties (1)–(4) imply that for each , the -cut set of the fuzzy number defined by is in ; that is, for each . We denote the set of all fuzzy numbers by . Also, the following statements hold:(5) is a bounded and nondecreasing left continuous function on ;(6) is a bounded and nonincreasing left continuous function on ;(7)the functions and are right continuous at the point ;(8) . Sometimes, the representation of fuzzy numbers with -cut sets is cause failures according to algebraic operations. For example, if is any fuzzy number, then is not equal to fuzzy zero. In this study, we have used another type representation of a fuzzy number to avoid this type algebraic failure which is used in [3, 4]. Furthermore, we know that shape similarity of the membership functions does not reflect the conception itself, but the context in which it is used. Whether a particular shape is suitable or not can be determined only in the context of a particular application. However, many applications are not overly sensitive to variations in the shape. In such cases, it is convenient to use a simple shape, such as the triangular shape of membership function. For example, let us consider any triangular fuzzy number . If the function is the membership function
References
[1]
R. E. Moore, “Automatic error analysis in digital computation,” Tech. Rep. LSMD-48421, Lockheed Missiles and Space Company, 1959.
[2]
P. Diamond and P. Kloeden, Metric Spaces of Fuzzy Sets, World Scientific, River Edge, NJ, USA, 1994.
[3]
D. Filev and R. Yager, “A generalized defuzzification method under BAD distributions,” International Journal of Intelligent Systems, vol. 6, pp. 689–697, 1991.
[4]
M. Sugeno, “An introductory survey of fuzzy control,” Information Sciences, vol. 36, no. 1-2, pp. 59–83, 1985.
[5]
Z. Mitrovic and S. Rusov, “Z similarity measure among fuzzy sets,” FME Transactions, vol. 34, pp. 115–119, 2006.
[6]
Z. Zarars？z and M. Seng？nül, Center of Gravity of Sequence Space of Fuzzy Numbers, AFMI, 2013.
[7]
S. Nanda, “On sequence spaces of fuzzy numbers,” Fuzzy Sets and Systems, vol. 33, pp. 123–126, 1989.
[8]
H. Alt？nok, R. ？olak, and M. Et, “ -difference sequence spaces of fuzzy numbers,” Fuzzy Sets and Systems, vol. 160, no. 21, pp. 3128–3139, 2009.
[9]
？. Talo and F. Ba？ar, “Determination of the duals of classical sets of sequences of fuzzy numbers and related matrix transformations,” Computers & Mathematics with Applications, vol. 58, no. 4, pp. 717–733, 2009.
[10]
B. C. Tripathy and A. J. Dutta, “Statistically convergent and Cesàro summable double sequences of fuzzy real numbers,” Soochow Journal of Mathematics, vol. 33, no. 4, pp. 835–848, 2007.
[11]
T. Bilgin, “ -statistical and strong -Cesaro convergence of sequences of fuzzy numbers,” Mathematical Communications, vol. 8, no. 1, pp. 95–100, 2003.
[12]
Y. Alt？n, M. Mursaleen, and H. Alt？nok, “Statistical summability for sequences of fuzzy real numbers and a Tauberian theorem,” Journal of Intelligent & Fuzzy Systems, vol. 21, no. 6, pp. 379–384, 2010.
[13]
R. ？olak, Y. Alt？n, and M. Mursaleen, “On some sets of difference sequences of fuzzy numbers,” Soft Computing, vol. 15, pp. 787–793, 2011.
[14]
B. Altay, F. Ba？ar, and M. Mursaleen, “On the Euler sequence spaces which include the spaces and . I,” Information Sciences, vol. 176, no. 10, pp. 1450–1462, 2006.
[15]
F. Ba？ar and B. Altay, “On the space of sequences of -bounded variation and related matrix mappings,” Ukrainian Mathematical Journal, vol. 55, no. 1, pp. 108–118, 2003.
[16]
E. Malkowsky, “Recent results in the theory of matrix transformations in sequence spaces,” Matematichki Vesnik, vol. 49, no. 3-4, pp. 187–196, 1997.
[17]
P. N. Ng and P. Y. Lee, “Cesàro sequence spaces of non-absolute type,” Commentationes Mathematicae. Prace Matematyczne, vol. 20, no. 2, pp. 429–433, 1978.
[18]
C. S. Wang, “On N？rlund sequence spaces,” Tamkang Journal of Mathematics, vol. 9, no. 2, pp. 269–274, 1978.
[19]
B. Altay and F. Ba？ar, “Some paranormed Riesz sequence spaces of non-absolute type,” Southeast Asian Bulletin of Mathematics, vol. 30, no. 4, pp. 591–608, 2006.
[20]
A. Wilansky, Summability Through Functional Analysis, vol. 85 of North-Holland Mathematics Studies, North-Holland, Amsterdam, The Netherlands, 1984.
[21]
B. C. Tripathy and A. Baruah, “N？rlund and Riesz mean of sequences of fuzzy real numbers,” Applied Mathematics Letters, vol. 23, no. 5, pp. 651–655, 2010.
[22]
M. ？eng？nül and F. Ba？ar, “Some new Cesàro sequence spaces of non-absolute type which include the spaces and ,” Soochow Journal of Mathematics, vol. 31, no. 1, pp. 107–119, 2005.
[23]
G. G. Lorentz, “über Limitierungsverfahren, die von einem Stieltjes-Integral abh？ngen,” Acta Mathematica, vol. 79, pp. 255–272, 1947.
[24]
F. Ba？ar, “Matrix transformations between certain sequence spaces of and ,” Soochow Journal of Mathematics, vol. 26, no. 2, pp. 191–204, 2000.
[25]
F. Ba？ar and R. ？olak, “Almost-conservative matrix transformations,” Do？a, vol. 13, no. 3, pp. 91–100, 1989.
[26]
B. Kuttner, “On dual summability methods,” Mathematical Proceedings of the Cambridge Philosophical Society, vol. 71, pp. 67–73, 1972.
[27]
G. G. Lorentz and K. Zeller, “Summation of sequences and summation of series,” Proceedings of the Cambridge Philosophical Society, vol. 71, pp. 67–73, 1972.
