On the Zweier Sequence Spaces of Fuzzy Numbers
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 . Let be nonempty set. According to Zadeh, a fuzzy subset of is a nonempty subset of for some function . 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 .
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.
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 of the triangular fuzzy number , then the fuzzy number can represent the notation . Clearly this representation is unique for and on the contrary for every , there is an which is unique. The triangular fuzzy numbers will be denoted shortly by and the set of all triangular fuzzy numbers will be denoted by ; that is, The , , and are called first, middle, and end points of triangular fuzzy number, respectively. Also the notation says to us that, the height of the fuzzy number is 1 at the .
Let and . The operations addition and scalar multiplication are defined on as follows: In fact, it is necessary to make some assumptions on the set of fuzzy numbers for constructing an algebraic structure. For example, we must redefine fuzzy identity elements according to addition and multiplication.
Let us suppose that , , and . If we accept that any fuzzy number is a fuzzy zero, then the difference is acceptable fuzzy zero for us. Therefore, we say that the inverse of the fuzzy number is , according to addition and the determines a fuzzy number. With this idea, we can solve equations in the form which may be given by inexact data.
If we sum fuzzy number with fuzzy zero, then the spread of fuzziness of may increase or decrease. This is normal because we are summing a “fuzzy number” and “fuzzy zero.” This situation is valid for fuzzy one; that is, if the fuzzy is one, then spread of fuzziness of the products or will be increase or decrease. This is normal, too.
The “fuzzy number a” can be denoted by the notation and should be less than 1. Theoretically, the assumption “ should be less than 1” that is not necessary but in the practical, it has to be. For example, the “approximately 5” can be taken as but in the applications, generally, “approximately 5” is taken as , . This choice is more accurate than . We know that, generally, in the practical applications, the spread of fuzziness is not very large. It is fact that the set is not linear space according to algebraic operations sense of (4).
The second important matter is the topology on the set . If we want construct a topology on, then we can use the metric defined by , We can easily show that the set is a complete metric space with the metric .
The function , is called a sequence of fuzzy numbers and is represented by , .
Let us denote the set of all sequences of fuzzy numbers by ; that is, where and for all , . The , , and are called first, middle, and end points of general term of sequences of triangular fuzzy numbers, respectively. If degree of membership at is equal to 1, then is a sequence of fuzzy number and if it is not, then the sequence is a sequence of the fuzzy sets.
Each subspace of is called a sequence space of fuzzy numbers. We will write , , and for the spaces of all bounded, convergent, and null sequences of fuzzy numbers, respectively; that is,
In , Nanda has studied the spaces of bounded and convergent sequences of fuzzy numbers and shown that they are complete metric spaces with the metric , .
Definition 1. Let and algebraic operations on sense of (4). The function . The function is called module on the set if it has the following properties:(9),
If the function satisfies (9)–(11) on , then is called module sequence space of the fuzzy numbers. If is complete with respect to the module , then is called complete module sequence space of the fuzzy numbers.
By Definition 1, we have “Every normed space is a module space but converse is not true.”
Let and be two spaces of fuzzy valued sequences and let be an infinite matrix of positive real numbers , where . Then, we say that defines a real-matrix mapping from to , and we denote it by writing , if for every sequence , the sequence is in where and the series are convergent for all . By , we denote the class of matrices such that . Thus, if and only if the series on the right side of (9) are convergent for each and every ; we have for all .
Let where is a sequence space of fuzzy numbers. Then, the set of sequences of fuzzy numbers is called the domain of an infinite matrix in .
Let the infinite matrix of positive real numbers be the Cesàro matrix of one order; that is,
Then the -transform of the sequence of fuzzy numbers is equal to which means that we can obtain a limit with this way. From here, we realize that any limitation method can transform sequences of fuzzy numbers into set of all sequences of fuzzy numbers. It is also very important to construct new sequence space of fuzzy numbers.
The idea to construct a new sequence space of real or complex numbers using by matrix domain of a particular limitation method has been employed many authors; for example, you can see Altay et al. , Başar and Altay , Malkowsky , Ng and Lee , and Wang . They introduced the sequence spaces with , , and in , in , in , and in , where , , , , and denote the Euler, means order , Riesz means with respect to the sequence , backward difference matrix, Cesàro means of order one, and Nörlund means with respect to the sequence , respectively, and . In the present paper, following [14–18], we will define matrix domain of sequence spaces of fuzzy numbers and introduce the sequence spaces of fuzzy numbers and and derive some results related to those sequence spaces. Furthermore, we have computed the -, - and -duals of the spaces and . Latter, we have characterized the matrix mappings from the spaces and to and from to and by employing the suitable relations between the related matrix classes, where is a given sequence space. Now we will give a lemma.
Lemma 2. The sets , , and are complete module spaces with the module defined by
Proof. We consider only . It is easy to see that is a module on . To show that is complete in this module, suppose that is a fundamental sequence in for each . Then we have
Hence we can write
From here, we realize that , , and are fundamental sequences of real numbers in . Since is complete, the sequences , , and are convergent in for all .
Let us suppose that for each and . Since for all , we have This means that, for , as ; that is, in . On the other hand, since we realize that .
2. The Zweier Sequence Spaces , , and
In this section, we wish to introduce the , , and spaces, as the set of all sequences such that -transforms of them are in the spaces , , and , respectively; that is Define the sequence of fuzzy numbers which will be frequently used, as the -transform of a sequence of fuzzy numbers ; that is, where , and the matrix defined by
As we said above in (5), if we assume that each fuzzy number in the form is a fuzzy zero, then can be left alone in equality (19). This assumption is important for our idea, which is always used in this paper.
Now, we may begin with the following theorem which is essential in the text.
Theorem 3. The sequence spaces , , and are linearly isomorphic to the spaces , , and , respectively; that is, , , and .
Proof. We will only consider with . To prove this, we should show the existence of a linear isometry between the spaces and . Consider the transformation defined , with the notation of (19), from to by . Equality is clear. Let us suppose that ; then,
that is, have the property homogeneity. Thus, is linear.
Let , , and define the sequence by Then, we have That is, is norm preserving. Consequently, the spaces and are linearly isometric. It is clear here that if the spaces and are, respectively, replaced by the spaces and , , and , then we obtain the fact that and . This completes the proof.
Theorem 4. The sets , , and are complete module sequence space of the fuzzy numbers with the module defined by
Proof. It was seen that in Theorem 3 that the sequence spaces of fuzzy numbers , , and are linearly isometric to the spaces , , and , respectively. Also, since the matrix is normal (see, ) and , , and are complete module spaces (see Lemma 2), we see that the sequence spaces , , and are complete module spaces with the module defined in (25).
Let be a sequence of nonnegative real numbers which are not all zero and , for all . The matrix is defined by that is called the Nörlund matrix.
Other one of the best known matrix is , the Riesz matrix which is a lower triangular matrix defined by for all , where is real sequence with , .
Let be a sequence of fuzzy numbers. If as , then as and as , (see [21, Theorem 3.1 and Theorem 3.2]). That is, the matrices and preserve the limits on the set of all convergent sequences of fuzzy numbers.
Theorem 5. Let be a sequence of fuzzy numbers. If as , then as . That is, the matrix is regular.
Proof. Let be a sequence of fuzzy numbers and as . Then for a given , there exists a positive integer such that Thus we can write If we take , then we see that . That is
Theorem 6. The inclusions and strictly hold.
Proof. To prove the validity of the inclusion , let us take any . Then, bearing in mind the regularity of the method (see, Theorem 5), we immediately observe that which means that . Hence, the inclusion holds. Furthermore, let us consider the sequence defined by for all . Then, we have which implies that is in but . This shows that the inclusion is strict. One can see by analogy that the strict inclusion also holds. This completes the proof.
Let , and be an infinite matrix of fuzzy numbers and consider following expressions: In , some matrix classes are characterized by Talo and Başar which are given in the following lemma.
Lemma 7 (see ). Let us suppose that(12), , if and only if (31) holds,(13) if and only if (34) holds,(14) if and only if (31) and (33) hold,(15) if and only if (31) and (33) also hold with for all ,(16) if and only if (32) and (33) hold,(17) if and only if (32) and (33) also hold with for all ,(18) if and only if (31), (35), and (33) also hold with for all .
Analogously to Talo and Başar, we can prove following proposition.
Proposition 9. if and only if .
We know that each can be considered as a fuzzy number in the form of
3. Real Duals of the Spaces , , and
In this section, we state and prove the theorems determining the -, -, and -duals of the spaces , , and . For the sequence spaces and , define the set by where denotes all real valued sequences space. With the notation of (38), the -, -, and -duals of a sequence space , which are, respectively, denoted by , , and , are defined by We will use a technique, in the proof of the Theorems 10 and 13, which is used in [14, 15, 22].
Theorem 10. The -dual of the spaces , , and is the set
Proof. Since the proof is similar for the rest of the spaces, we determine only -dual of the set . Let and define the matrix via the sequence by Bearing in mind relation (19), we immediately derive that From (42), we realize that whenever if and only if whenever . Then, we derive by Part (12) of Lemma 7, and we have which yields the consequence that .
Theorem 11. is the set , where
Proof. This is clear from Proposition 9.
Theorem 12. Define the sets and by Then,.
Proof. Consider where is defined by Thus, we deduce from Proposition 8 with (47) that whenever if and only if whenever . It is obvious that the columns of that matrix , defined by (47) are in the space . Therefore, we derive the consequences from Proposition 8 that .
Theorem 13. The -dual of the space is the set .
Proof. The proof of this theorem is similar to the proof of Theorem 10, so we omit it.
4. Matrix Transformations
For the first time, Lorentz introduced the concept of dual summability methods for the limitation which depends on a Stieltjes integral and passed to the discontinuous matrix methods by means of a suitable step function in . Later, many authors such as Başar , Başar and Çolak , Kuttner , Lorentz and Zeller , and Şengönül and Başar  worked on the dual summability methods.
Let us suppose that the set is any of the sets , , and . In this section, we characterize the matrix mappings from into any given sequence space of fuzzy numbers via the concept of the dual summability methods and vice versa, so we call it the Zweier dual summability methods.
Let us suppose that the sequences and are connected with (19) and let the -transform of the sequence be and let the -transform of the sequence be ; that is,
It is clear here that the method is applied to the -transform of the sequence while the method is directly applied to the terms of the sequence . Therefore the methods and are essentially different (see, ).
Let us assume the existence of the matrix product which is a much weaker assumption than the conditions on the matrix belonging to any matrix class, in general. If becomes , under the application of the formal summation by parts, then the methods and in (48) are called Zweier dual type matrices. This leads us to fact that exists and is equal to and formally holds. This statement is equivalent to the relation for all .
Now we may give the following theorem concerning to the Zweier dual matrices.
Theorem 14. Let and be the Zweier dual type matrices, any given sequence space, and . Then, if and only if .
Proof. Suppose that and are Zweier dual type matrices, dual type matrices that is, to say that (49) holds and is any given sequence space and take account that the spaces and are linearly isomorphic.
Let and take any . Then, exists and which yields for each . Hence exists for each and thus letting in the equality