%0 Journal Article
%T On the Zweier Sequence Spaces of Fuzzy Numbers
%A Mehmet £¿eng£¿n¨¹l
%J International Journal of Mathematics and Mathematical Sciences
%D 2014
%I Hindawi Publishing Corporation
%R 10.1155/2014/439169
%X 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)¨C(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
%U http://www.hindawi.com/journals/ijmms/2014/439169/