Some Additions to the Fuzzy Convergent and Fuzzy Bounded Sequence Spaces of Fuzzy Numbers

Abstract

Some properties of the fuzzy convergence and fuzzy boundedness of a sequence of fuzzy numbers were studied in Choi (1996). In this paper, we have consider, some important problems on these spaces and shown that these spaces are fuzzy complete module spaces. Also, the fuzzy 𝛼-, fuzzy 𝛽-, and fuzzy 𝛾-duals of the fuzzy module spaces of fuzzy numbers have been computeded, and some matrix transformations are given.

1. Introduction

As known, the ideas of fuzzy sets and fuzzy operations were first introduced by Zadeh [1], and after his innovation, many authors have studied different aspects of the fuzzy numbers theory and applications. One of them is the sequence spaces of the fuzzy numbers. A major direction in the study on sequence spaces of fuzzy numbers is the metric properties of these spaces (see [2–4]), but this direction has been altered by Talo and Başar [5].

Some important problems on sequence spaces of fuzzy numbers can be ordered as follows:(1)to construct a sequence space of fuzzy numbers and compute 𝛼-, 𝛽- and 𝛾-duals,(2)to find some isomorphic spaces of it,(3)to give some theorems about matrix transformation on sequence space of fuzzy numbers,(4)to study some inclusion problems and other properties.

By using the metric, 𝑑(𝑢,𝑣)=sup_𝑘𝑑(𝑢_𝑘,𝑣_𝑘), these problems have been nicely studied by Talo and Başar in [5]. But, as known, by defining different metrics on sequence spaces of fuzzy numbers, different metric spaces can be built up. By using 𝑑 metric, so many spaces of fuzzy sequences have been built and many metric properties have been investigated. In literature, one can easily reach many documents about sequence space of fuzzy numbers.

In this paper, using a fuzzy metric, we will deal with some problems about fuzzy convergent and fuzzy bounded sequence spaces of fuzzy numbers which did not come up in [6]. Thus, we believe that some gaps in this area will be filled.

The rest of this paper is structured as follows.

Some required definitions and consequences related with the fuzzy numbers given in Section 2. Besides, as a proposition, the complete fuzzy module space of fuzzy numbers is given, and the sequence spaces of fuzzy numbers with fuzzy metric are introduced in this section. In Section 3, we have stated and proved the theorems determining the fuzzy 𝛼-, fuzzy 𝛽-, and fuzzy 𝛾-duals of the fuzzy sequence space of fuzzy numbers. Finally, in Section 4, the fuzzy classes (ℓ_∞(𝐸¹)∶ℓ_∞(𝐸¹)) and (𝑐₀(𝐸¹)∶𝑐₀(𝐸¹)) of infinite matrix of fuzzy numbers are characterized.

2. Preliminaries

Lets suppose that ℕ is the set of all positive integer numbers, ℝ is the set of all real numbers, is the 𝐸_𝑖 be the set of all bounded and closed intervals on the real line ℝ, that is, 𝐸_𝑖={𝑎=[𝑎⁻,𝑎⁺]∶𝑎⁻≤𝑥≤𝑎⁺,𝑎⁻and𝑎⁺∈ℝ}. For 𝑎,𝑏∈𝐸_𝑖 and define𝑑(𝑎,𝑏)=max|𝑎⁻−𝑏⁻|,𝑎⁺−𝑏⁺. (2.1) Then, it can be seen easily that 𝑑 defines a metric on 𝐸_𝑖 and (𝐸_𝑖,𝑑) is a complete metric space [7]. Let 𝑋 be nonempty set. According to Zadeh, a fuzzy subset of 𝑋 is a nonempty subset {(𝑥,𝑢(𝑥))∶𝑥∈𝑋} of 𝑋×[0,1] for some function 𝑢∶𝑋→[0,1], [8]. Consider a function 𝑢∶ℝ→[0,1] as a subset of a nonempty base space ℝ and denote the totality of all such functions or fuzzy sets by 𝐸. A fuzzy number (FN) is a function 𝑢 from ℝ to [0,1], which satisfies the following properties:FN1 𝑢 is normal, that is, there exists an 𝑥₀∈ℝ such that 𝑢(𝑥₀)=1,FN2 𝑢 is fuzzy convex, that is, for any 𝑥,𝑦∈ℝ and 𝜇∈[0,1], 𝑢[𝜇𝑥+(1−𝜇)𝑦]≥min{𝑢(𝑥),𝑢(𝑦)},FN3 𝑢 is upper semicontinuous,FN4 the closure of {𝑥∈ℝ∶𝑢(𝑥)>0}, denoted by 𝑢⁰, is compact.

(FN1), (FN2), (FN3), and (FN4) imply that for each 𝛼∈[0,1], the 𝛼-level set defined by [𝑢]_𝛼={𝑥∈ℝ∶𝑢(𝑥)≥𝛼} is in 𝐸_𝑖, as well as the support 𝑢⁰, that is, 𝑢_𝛼=[𝑢⁻_𝛼,𝑢⁺_𝛼] for each 𝛼∈[0,1]. We denote the set of all fuzzy numbers by 𝐸¹.

Let us suppose that [𝑢]_𝜆=[𝑢⁻(𝜆),𝑢⁺(𝜆)] for 𝑢∈𝐸¹ and for each 𝜆∈[0,1]. Then the following statements are held:(1)𝑢⁻(𝜆) is a bounded and nondecreasing left continuous function on [0,1),(2)𝑢⁺(𝜆) is a bounded and nonincreasing left continuous function on (0,1],(3)The functions 𝑢⁺(𝜆) and 𝑢⁻(𝜆) are right continuous at the point 𝜆=0,(4)𝑢⁺(𝜆)≥𝑢⁻(𝜆).

Conversely, if the pair of functions 𝛼 and 𝛽 satisfy the conditions (1)–(4), then there exists a unique 𝑢∈𝐸¹ such that [𝑢]_𝜆=[𝛼⁻(𝜆),𝛽⁺(𝜆)] for each 𝜆∈[0,1], [9]. The fuzzy number 𝑢 corresponding to the pair of functions 𝛼 and 𝛽 is defined by 𝑢∶ℝ→[0,1], 𝑢(𝑥)=sup{𝜆∶𝛼(𝑥)≤𝜆≤𝛽(𝑥)}, [5].

A sequence 𝑢=(𝑢_𝑘) of fuzzy numbers is a function 𝑢 from the set ℕ, the set of all positive integers, into 𝐸¹, and fuzzy number 𝑢_𝑘 denotes the value of the function at 𝑘 and is called the 𝑘th term of the sequence.

Let 𝑢,𝑣∈𝐸¹ and 𝜆∈ℝ, then the operations addition and scalar multiplication are defined on 𝐸¹ in terms of 𝛼-level sets by𝑢+𝑣=𝑤⟺[𝑤]_𝛼=[𝑢]_𝛼+[𝑣]_𝛼,[𝜆𝑢]_𝛼=𝜆[𝑢]_𝛼∀𝛼∈[0,1]. (2.2)

Define a map 𝑑∶𝐸¹×𝐸¹→ℝ by 𝑑(𝑢,𝑣)=sup_0≤𝛼≤1𝑑([𝑢]_𝛼,[𝑣]_𝛼). It is known that 𝐸¹ is a complete metric space with the metric 𝑑 [3]. Let us suppose that 𝑤(𝐸¹), 𝑐(𝐸¹), and ℓ_∞(𝐸¹) are set of all sequences space of all fuzzy numbers, convergent and bounded sequences of fuzzy numbers, respectively.

Let us suppose that 𝑢,𝑣∈𝐸¹ and 𝐺 is the set of all nonnegative fuzzy numbers. The function 𝑑_𝑓∶𝐸¹×𝐸¹→𝐺 is called fuzzy metric [6] which satisfies the following properties:(1)𝑑_𝑓(𝑢,𝑣)≥0,(2)𝑑_𝑓(𝑢,𝑣)=0 if and only if 𝑢=𝑣,(3)𝑑_𝑓(𝑢,𝑣)=𝑑_𝑓(𝑣,𝑢), (4)whenever 𝑤∈𝐸¹, we have 𝑑_𝑓(𝑢,𝑣)≤𝑑_𝑓(𝑢,𝑤)+𝑑_𝑓(𝑤,𝑣).

In [4], Nanda has studied the spaces of bounded and convergent sequences of fuzzy numbers and has shown that they are complete metric spaces with the metric 𝑑(𝑢,𝑣)=sup_𝑘𝑑(𝑢_𝑘,𝑣_𝑘).

By using this metric, 𝑑, so many spaces of fuzzy sequences have been built and published in famous maths journals. By reviewing the literature, one can reach them easily. However, another important metric which is called as fuzzy metric is used for measuring fuzzy distances among fuzzy numbers.

If 𝑑_𝑓 is a fuzzy metric on 𝐸¹, then the pair of (𝐸¹,𝑑_𝑓) is called as a fuzzy metric space. For any 𝑢,𝑣∈𝐸¹, the fuzzy metric of Zhang is [10–12] defined by𝑑_𝑓(𝑢,𝑣)=sup_[]𝜆∈0,1𝜆𝑑₁(𝑢,𝑣),𝑑_𝜆(𝑢,𝑣), (2.3) where 𝑑_𝜆(𝑢,𝑣)=sup_{𝛼∈[𝜆,1]}𝑑([𝑢]_𝛼,[𝑣]_𝛼) and 𝑑₁(𝑢,𝑣)=sup_𝛼=1𝑑([𝑢]_𝛼,[𝑣]_𝛼).

Also, fuzzy metric spaces have been studied in [13]. But the given metric space definition in [13] is different from our fuzzy metric space definition.

Theorem 2.1 (see [10–12]). The metric 𝑑_𝑓 defined by equality (2.3) is a fuzzy distance of fuzzy numbers; thus, (𝐸¹,𝑑_𝑓) is a fuzzy metric space.

Theorem 2.2. If 𝑢,𝑣∈𝐸¹, then 𝑑(𝑢,𝑣) is a point of the interval determined by the fuzzy metric 𝑑_𝑓(𝑢,𝑣).

Proof. Clearly, if 𝜆 run from 0, then the second side of 𝑑_𝑓, 𝑑_𝜆(𝑢,𝑣), is equal to 𝑑(𝑢,𝑣) since 𝑑_𝑓(𝑢,𝑣)=sup_[]𝜆∈0,1𝜆𝑑₁(𝑢,𝑣),𝑑_𝜆(𝑢,𝑣). (2.4)

Theorem 2.3 (see [12]). The fuzzy metric space (𝐸¹,𝑑_𝑓) is complete metric space.

Definition 2.4. Let 𝜆(𝐸¹) be the subset of all sequence spaces of fuzzy numbers and suppose that ‖⋅‖∶𝜆(𝐸¹)→𝐺 is a function. The function ‖⋅‖ is called fuzzy module or fuzzy norm if it has the following properties:
(N1) ‖𝑢‖=𝜃⇔𝑢=𝜃,
(N2) ‖𝛼𝑢‖=|𝛼|‖𝑢‖_𝐸¹,
(N3) ‖𝑢+𝑣‖≤‖𝑢‖+‖𝑣‖
If the function ‖⋅‖∶𝜆(𝐸¹)→𝐺 satisfies 𝑁1, 𝑁2, and 𝑁3, then 𝜆(𝐸¹) is called fuzzy module sequence space of the fuzzy numbers. And if 𝜆(𝐸¹) is complete with respect to the fuzzy module, then 𝜆(𝐸¹) is called complete fuzzy module sequence space of the fuzzy numbers.

Definition 2.5. The fuzzy module of the fuzzy number 𝑢 is defined which corresponds to the fuzzy distance from 𝑢 to 0, that is, ‖𝑢‖_𝐸¹∶=sup_𝜆∈[0,1]𝜆𝑑₁𝑢,0,𝑑_𝜆𝑢,0. (2.5)

Proposition 2.6. The set 𝐸¹ of the fuzzy numbers is fuzzy complete module space with the fuzzy module in (2.5).

Let 𝑢=(𝑢_𝑘) be a sequence of fuzzy numbers, and let ‖⋅‖ be a fuzzy module, then the sequence (𝑢_𝑘) is said to converge fuzzy to 𝑢₀∈𝐸¹ with the fuzzy module ‖⋅‖ if for any given 𝜖>0, there exists an integer 𝑛₀ such that ‖𝑢_𝑘−𝑢₀‖<[𝜖,𝜖]=𝜖 for 𝑘≥𝑛₀. The sequence (𝑢_𝑘) is said to be fuzzy bounded in fuzzy module ‖⋅‖ if sup_𝑘‖𝑢_𝑘‖<∞ for all 𝑘∈ℕ.

We will write 𝔏_∞(𝐸¹), ℭ(𝐸¹), and ℭ₀(𝐸¹) for the fuzzy sets of all fuzzy bounded, fuzzy convergent, fuzzy null sequences, respectively, that is,𝔏_∞𝐸¹∶=𝑢=𝑢_𝑘∈𝑤𝐸¹∶sup_𝑘sup_[]𝜆∈0,1𝜆𝑑₁𝑢_𝑘,0,𝑑_𝜆𝑢_𝑘,0,ℭ𝐸<∞₁∶=𝑢=𝑢_𝑘∈𝑤𝐸¹∶∃𝑢₀∈𝐸¹∋lim_𝑘sup_[]𝜆∈0,1𝜆𝑑₁𝑢_𝑘,𝑢₀,𝑑_𝜆𝑢_𝑘,𝑢₀,ℭ=𝜃₀𝐸¹∶=𝑢=𝑢_𝑘∈𝑤𝐸¹∶lim_𝑘sup_[]𝜆∈0,1𝜆𝑑₁𝑢_𝑘,0,𝑑_𝜆𝑢_𝑘,0=𝜃. (2.6) In [6], Kong and Cho has proved that the fuzzy convergent sequence spaces of fuzzy numbers ℭ(𝐸¹) and fuzzy bounded sequence spaces of fuzzy numbers 𝔏_∞(𝐸¹) are fuzzy complete metric spaces with fuzzy metric 𝑑_𝑓 defined by𝑑_𝑓(𝑢,𝑣)=sup_𝑘sup_𝜆∈[0,1]𝜆𝑑₁𝑢_𝑘,𝑣_𝑘,𝑑_𝜆𝑢_𝑘,𝑣_𝑘. (2.7)

Now, let us give some definitions that will facilitate our work.

A sequence 𝑢=(𝑢_𝑛) in 𝐸¹ is said to be a fuzzy fundamental sequence if forever 𝜀>0, there exists an integer 𝑛₀ such that 𝑑_𝑓(𝑢_𝑛,𝑢_𝑚)<𝜀 for 𝑛,𝑚>𝑛₀. A fuzzy metric space (𝐸¹,𝑑_𝑓) is called the fuzzy complete metric space if every fundamental sequence converges in 𝐸¹.

Theorem 2.7 (see [10–12]). The sequence 𝑢 in 𝐸¹ is fuzzy convergent in the metric 𝑑_𝑓 if and only if 𝑢 is a fuzzy fundamental sequence.

Theorem 2.8. The fuzzy metric spaces (ℭ(𝐸¹),𝑑_𝑓) and (𝔏_∞(𝐸¹),𝑑_𝑓) are fuzzy complete metric spaces.

Theorem 2.9 (see [14]). Let 𝒰⊂𝐸¹, and let (𝑢_𝑘) be a sequence of fuzzy functions from 𝒰 to 𝐸¹. If for each 𝑘, there exists a real number 𝑀_𝑘 such that max{|(𝑢_𝑘(𝑈))⁻_𝜆|,|(𝑢_𝑘(𝑈))⁺_𝜆|}≤𝑀_𝑘 for all 𝑈∈𝒰 and 𝜆∈(0,1], and if the series ∑_𝑘𝑀_𝑘 converges, then there is a fuzzy function 𝑢∶𝒰→𝐸¹ such that ∑_𝑘𝑢_𝑘 converges uniformly to 𝑢.

Now, let us define the sequence sets ℭ𝑠(𝐸¹), 𝔅𝑠(𝐸¹), and 𝔏_𝑝(𝐸¹) as the set of all fuzzy convergent series of FNs, the set of all fuzzy bounded series of FNs, and the set of 𝑝-absolutely fuzzy convergent series of the FNs, respectively, that is,ℭ𝑠𝐸¹=𝑢=𝑢_𝑘∈𝑤𝐸¹∶lim_𝑛sup_[]𝜆∈0,1𝜆𝑑₁^𝑛_𝑘=1𝑢_𝑘,𝑢₀,𝑑_𝜆^𝑛_𝑘=1𝑢_𝑘,𝑢₀,𝐸=0𝔅𝑠¹=𝑢=𝑢_𝑘∈𝑤𝐸¹∶sup_𝑛sup_[]𝜆∈0,1𝜆𝑑₁^𝑛_𝑘=1𝑢_𝑘,0,𝑑_𝜆^𝑛_𝑘=1𝑢_𝑘,0,𝔏<∞_𝑝𝐸¹=𝑢=𝑢_𝑘∈𝑤𝐸¹∶_𝑘(sup_[]𝜆∈0,1𝜆[𝑑₁(𝑢_𝑘,0),𝑑_𝜆(𝑢_𝑘,0)])^𝑝1/𝑝<∞,1≤𝑝<∞. (2.8) Let us see the following theorems which about the sets 𝔏_∞(𝐸¹), ℭ(𝐸¹), and ℭ₀(𝐸¹).

Theorem 2.10. The sets 𝔏_∞(𝐸¹), ℭ(𝐸¹), and ℭ₀(𝐸¹) are fuzzy complete module spaces defined by fuzzy module ‖𝑢‖=sup_𝑘sup_𝜆∈[0,1]𝜆𝑑₁𝑢_𝑘,0,𝑑_𝜆𝑢_𝑘,0. (2.9)

Proof. Since the proof for ℭ(𝐸¹) and ℭ₀(𝐸¹) can be proved in a similar way, we will consider only 𝔏_∞(𝐸¹). Clearly, it is straightforward to see that ‖𝑢‖ is a fuzzy module on 𝔏_∞(𝐸¹). To show that 𝔏_∞(𝐸¹) is fuzzy complete in this fuzzy module, let us suppose that (𝑢_𝑘) is a fuzzy fundamental sequence in 𝔏_∞(𝐸¹) where (𝑢_𝑘)=(𝑢₀^(𝑖),𝑢₁^(𝑖),𝑢₂^(𝑖),…), then, for any 𝜖>0, there exists an integer 𝑛₀ such that 𝑢^𝑖_𝑘−𝑢^𝑗_𝑘=sup_𝑘sup_[]𝜆∈0,1𝜆𝑑₁𝑢^𝑖_𝑘(1)−𝑢^𝑗_𝑘(1),0,𝑑_𝜆𝑢^𝑖_𝑘(𝛼)−𝑢^𝑗_𝑘(𝛼),0≤[𝜖,𝜖]=𝜖, (2.10) for 𝑖,𝑗≥𝑛₀. Hence, we obtain |𝑢^𝑖_𝑘(1)⁻−𝑢^𝑗_𝑘(1)⁻|<𝜖/𝜆 and sup_{𝛼∈[𝜆,1]}max{|𝑢_𝑘^𝑖−(𝛼)−𝑢_𝑘^𝑗−(𝛼)|,|𝑢_𝑘^𝑖+(𝛼)−𝑢_𝑘^𝑗+(𝛼)|}<𝜖/𝜆. This shows that (𝑢^𝑖_𝑘(1)⁻) is fundamental sequence of real numbers in ℝ, and (𝑢^𝑖_𝑘(𝛼)) is fundamental sequence of fuzzy numbers in 𝐸¹. Since ℝ and 𝐸¹ are complete, so (𝑢^𝑖_𝑘(1)) converges in ℝ and (𝑢^𝑖_𝑘(𝛼)) converges in 𝐸¹ for all 𝑘∈ℕ.
Let us suppose that lim_𝑖𝑢^𝑖_𝑘(1)⁻=𝑢_𝑘(1)⁻ and lim_𝑖𝑢^𝑖_𝑘(𝛼)=𝑢_𝑘(𝛼) for each 𝑘∈ℕ. Put 𝑢(1)⁻=(𝑢_𝑘(1)⁻) and 𝑢(𝛼)=(𝑢_𝑘(𝛼)). Now, we shall show that lim_𝑖𝑢^𝑖(1)⁻=𝑢(1)⁻ and 𝑢(1)⁻∈ℝ, lim_𝑖𝑢^𝑖(𝛼)=𝑢(𝛼) and 𝑢(𝛼)∈𝐸¹. Since (𝑢^𝑖(1)) is a fundamental sequence in ℝ, given 𝜖>0, there exists 𝑛₀∈ℕ such that for 𝑖,𝑗>𝑛₀, |𝑢^𝑖(1)−𝑢^𝑗(1)|<𝜖/𝜆, and if we take limit over 𝑗, we get |𝑢^𝑖(1)−𝑢(1)|<𝜖/𝜆. Therefore, lim_𝑖𝑢^𝑖(1)=𝑢(1). Similarly, since (𝑢^𝑖_𝑘(𝛼)) is a fundamental sequence in 𝐸¹, given 𝜖>0, there exists 𝑛₀∈ℕ such that for 𝑖,𝑗>𝑛₀, sup_{𝛼∈[𝜆,1]}max{|𝑢_𝑘^𝑖−(𝛼)−𝑢_𝑘^𝑗−(𝛼)|,|𝑢_𝑘^𝑖+(𝛼)−𝑢_𝑘^𝑗+(𝛼)|}<𝜖/𝜆, and if we take limit over 𝑗 we get sup_{𝛼∈[𝜆,1]}max{|𝑢_𝑘^𝑖−(𝛼)−𝑢⁻_𝑘(𝛼)|,|𝑢_𝑘^𝑖+(𝛼)−𝑢⁺_𝑘(𝛼)|}<𝜖/𝜆<𝜖. Therefore, lim_𝑖𝑢^𝑖(𝛼)=𝑢_𝑘(𝛼). Let us show that 𝑢_𝑘(1)⁻∈ℝ and 𝑢_𝑘(𝛼)∈𝐸¹. Also, since |𝑢(1)⁻|≤|𝑢(1)⁻−𝑢^𝑖_𝑘(1)⁻|+|𝑢^𝑖_𝑘(1)⁻|<∞ and sup_{𝛼∈[𝜆,1]}max{|𝑢⁻_𝑘(𝛼)|,|𝑢⁺_𝑘(𝛼)|}≤sup_{𝛼∈[𝜆,1]}max{|𝑢⁻_𝑘(𝛼)−𝑢_𝑘^𝑖−(𝛼)|,|𝑢⁺_𝑘(𝛼)−𝑢_𝑘^𝑖+(𝛼)|}+sup_{𝛼∈[𝜆,1]}max{|𝑢_𝑘^𝑖−(𝛼)|,|𝑢_𝑘^𝑖+(𝛼)|}<∞, this shows that 𝑢∈𝔏_∞(𝐸¹).

Theorem 2.11. The space 𝔏_𝑝(𝐸¹) is fuzzy complete module space defined by module ‖𝑢‖_{𝔏𝑝(𝐸}¹₎=_𝑘sup_[]𝜆∈0,1𝜆𝑑₁𝑢_𝑘,0,𝑑_𝜆𝑢_𝑘,0^𝑝1/𝑝<∞,1≤𝑝<∞. (2.11)

Proof. Since the proof is similar to the proof of the Theorem 2.10, we omit it.

3. Construction of the Fuzzy Duals of the Fuzzy Module Sequence Spaces

For the fuzzy sequence spaces 𝜆(𝐸¹) and 𝜇(𝐸¹), define the set 𝑆(𝜆(𝐸¹),𝜇(𝐸¹)) by𝑆𝜆𝐸¹,𝜇𝐸¹=𝑢∈𝑤𝐸¹∶𝑢𝑣=𝑢_𝑘𝑣_𝑘∈𝜇𝐸¹∀𝑢=𝑢_𝑘∈𝜆𝐸¹. (3.1) With the notation of (3.1), the fuzzy 𝛼-, fuzzy 𝛽-, and fuzzy 𝛾-duals of a fuzzy sequence space 𝜆(𝐸¹), which are, respectively, denoted by 𝜆^𝛼(𝐸¹), 𝜆^𝛽(𝐸¹), and 𝜆^𝛾(𝐸¹), are defined by𝜆^𝛼𝐸¹=𝑆𝜆𝐸¹,𝔏₁𝐸¹,𝜆^𝛽𝐸¹=𝑆𝜆𝐸¹,ℭ𝑠𝐸¹,𝜆^𝛾𝐸¹=𝑆𝜆𝐸¹,𝔅𝑠𝐸¹. (3.2)

Definition 3.1. Let us suppose that 𝜆(𝐸¹), 𝜇(𝐸¹) are sets of the fuzzy sequences of FNs and 𝜆(𝐸¹)⊂𝜇(𝐸¹), then 𝜆(𝐸¹) is called fuzzy cofinal in 𝜇(𝐸¹) if for (𝑢_𝑘)∈𝜆(𝐸¹) there is (𝑣_𝑘)∈𝜇(𝐸¹) such that ‖𝑢_𝑘‖≤‖𝑣_𝑘‖ for all 𝑘∈ℕ.
If 𝜆(𝐸¹) is fuzzy cofinal in 𝜇(𝐸¹), then 𝜆^𝛼(𝐸¹)=𝜇^𝛼(𝐸¹); the converse of this assertion is not true.

Now, we may give results concerning the fuzzy 𝛼-dual, fuzzy 𝛽-dual, and fuzzy 𝛾-dual of the sets ℭ₀(𝐸¹), ℭ(𝐸¹), 𝔏_∞(𝐸¹), and 𝔏_𝑝(𝐸¹).

Theorem 3.2. The fuzzy 𝛼-dual of the set 𝔏_∞(𝐸¹) of sequence spaces of FNs is the set 𝔏₁(𝐸¹).

Proof. Let (𝑢_𝑘)∈𝔏^𝛼_∞(𝐸¹). If we consider (𝑣_𝑘)=([1,1])∈𝔏_∞(𝐸¹), then the series _𝑘sup_[]𝜆∈0,1𝜆𝑑₁𝑢_𝑘𝑣_𝑘,0,𝑑_𝜆𝑢_𝑘𝑣_𝑘,0=_𝑘sup_[]𝜆∈0,1𝜆𝑢⁻_𝑘(1)𝑣⁻_𝑘(1),sup_{[]𝛼∈𝜆,1}max𝑢⁻_𝑘(𝛼)𝑣⁻_𝑘(𝛼),𝑢⁺_𝑘(𝛼)𝑣⁺_𝑘(𝛼)≤_𝑘sup_[]𝜆∈0,1𝜆𝑢⁻_𝑘(1),sup_{[]𝛼∈𝜆,1}max𝑢⁻_𝑘(𝛼),𝑢⁺_𝑘(𝛼)sup_𝜆∈[0,1]𝜆𝑣⁻_𝑘(1),sup_{[]𝛼∈𝜆,1}max𝑣⁻_𝑘(𝛼),𝑣⁺_𝑘(𝛼)=_𝑘sup_[]𝜆∈0,1𝜆𝑑₁𝑢_𝑘,0,𝑑_𝜆𝑢_𝑘,0 (3.3) converges, that is to say, (𝑢_𝑘)∈𝔏₁(𝐸¹). Therefore, we have 𝔏^𝛼_∞𝐸¹⊆𝔏₁𝐸¹. (3.4)
Conversely, let us suppose that (𝑢_𝑘)∈𝔏_∞(𝐸¹) and (𝑣_𝑘)∈𝔏₁(𝐸¹), then there exists a 0<𝐾∈𝐸¹ such that 𝐾=sup_𝑘sup_𝜆∈[0,1]𝜆[|𝑑₁(𝑢_𝑘(1),0),𝑑_𝜆(𝑢_𝑘(𝛼),0)]<∞. From here, we have _𝑘sup_[]𝜆∈0,1𝜆𝑑₁𝑢_𝑘(1)𝑣_𝑘(1),0,𝑑_𝜆𝑢_𝑘(𝛼)𝑣_𝑘(𝛼),0=_𝑘sup_[]𝜆∈0,1𝜆𝑑₁𝑣_𝑘(1),0,𝑑_𝜆𝑣_𝑘(𝛼),0‖𝑢‖≤𝐾_𝑘sup_[]𝜆∈0,1𝜆𝑑₁𝑣_𝑘(1),0,𝑑_𝜆𝑣_𝑘(𝛼),0=𝐾‖𝑣‖_𝔏1(𝐸¹₎<∞, (3.5) which gives that 𝔏₁𝐸¹⊆𝔏^𝛼_∞𝐸¹. (3.6) From (3.4) and (3.6), we see that 𝔏₁(𝐸¹)=𝔏^𝛼_∞(𝐸¹).

Theorem 3.3. The fuzzy sequence spaces ℭ₀(𝐸¹), ℭ(𝐸¹) are cofinal in 𝔏₁(𝐸¹).

Proof. Denote any of the spaces ℭ₀(𝐸¹) and ℭ(𝐸¹) by 𝜆(𝐸¹), and suppose that ‖𝑢_𝑘‖≤‖𝑣_𝑘‖ holds for some (𝑣_𝑘)∈𝔏₁(𝐸¹), then we can easily see that sup_𝑘‖𝑢_𝑘‖≤sup_𝑘‖𝑣_𝑘‖, lim_𝑘‖𝑢_𝑘‖≤sup_𝑘‖𝑣_𝑘‖ which lead us to the desired results.

Theorem 3.4. The fuzzy 𝛼-dual of the sets ℭ₀(𝐸¹) and ℭ(𝐸¹) of sequence spaces of FNs is the set 𝔏₁(𝐸¹).

Proof. Since the sets 𝒞₀(𝐸¹) and ℭ(𝐸¹) are cofinal in 𝔏_∞(𝐸¹) (see Theorem 3.3), the proof is clear.

Theorem 3.5. The fuzzy 𝛽-dual of the sets ℭ(𝐸¹) and 𝔏_∞(𝐸¹) of sequence spaces of FNs is the set 𝔏₁(𝐸¹).

Proof. We give the proof only for the set 𝔏_∞(𝐸¹). Let us suppose that (𝑢_𝑘)∈𝔏_∞(𝐸¹) and (𝑣_𝑘)∈𝑤(𝐸¹), then there is a 𝐾>0 such that 𝐾=[𝐾₁,𝐾₂]=sup_𝑘sup_𝜆∈[0,1]𝜆[𝑑₁(𝑢_𝑘(1),0),𝑑_𝜆(𝑢_𝑘(𝛼),0)]<∞, also since 𝑢⁻_𝑘(1)𝑣⁻_𝑘(1)=𝑑₁𝑢_𝑘𝑣_𝑘,0≤𝑑₁𝑢_𝑘,0𝑑₁𝑣_𝑘,0≤𝐾₁𝑑₁𝑣_𝑘,0,𝑢⁻_𝑘(𝛼)𝑣⁻_𝑘(𝛼)≤𝑑_𝜆𝑢_𝑘𝑣_𝑘,0≤𝑑_𝜆𝑢_𝑘,0𝑑_𝜆𝑣_𝑘,0≤𝐾₁𝑑_𝜆𝑣_𝑘,0,𝑢⁺_𝑘(𝛼)𝑣⁺_𝑘(𝛼)≤𝑑_𝜆𝑢_𝑘𝑣_𝑘,0≤𝑑_𝜆𝑢_𝑘,0𝑑_𝜆𝑣_𝑘,0≤𝐾₁𝑑_𝜆𝑣_𝑘,0. (3.7)
The series ∑_𝑘(𝑢⁻_𝑘(𝛼)𝑣⁻_𝑘(𝛼)), ∑_𝑘(𝑢⁺_𝑘(𝛼)𝑣⁺_𝑘(𝛼)), and ∑_𝑘(𝑢⁻_𝑘(1)𝑣⁻_𝑘(1)) are convergent uniformly from Theorem 2.9; therefore, ∑_𝑘𝑢_𝑘𝑣_𝑘 converges whenever ∑_𝑘𝑑₁(𝑣_𝑘,0) and ∑_𝑘𝑑_𝜆(𝑣_𝑘,0) converge. From here, we can write 𝔏₁(𝐸¹)⊆𝔏^𝛽_∞(𝐸¹).
Finally, we will show that the inclusion 𝔏^𝛽_∞(𝐸¹)⊆𝔏₁(𝐸¹) is held. Let us suppose that (𝑣_𝑘)∈𝔏^𝛽_∞(𝐸¹), then we have ∑_𝑘(sup_𝜆∈[0,1]𝜆[𝑑₁(𝑢_𝑘(1)𝑣_𝑘(1),0),𝑑_𝜆(𝑢_𝑘(𝛼)𝑣_𝑘(𝛼),0)])<∞ for all (𝑢_𝑘)∈𝔏_∞(𝐸¹). This holds for the sequence (𝑢_𝑘)=([1,1])∈𝔏_∞(𝐸¹), then we can write _𝑘sup_[]𝜆∈0,1𝜆𝑑₁𝑢_𝑘(1)𝑣_𝑘(1),0,𝑑_𝜆𝑢_𝑘(𝛼)𝑣_𝑘(𝛼),0=𝐾_𝑘sup_[]𝜆∈0,1𝜆𝑑₁𝑣_𝑘(1),0,𝑑_𝜆𝑣_𝑘(𝛼),0=‖𝑣‖_𝔏1(𝐸¹₎. (3.8) This shows that 𝑣∈𝔏₁(𝐸¹).

Proposition 3.6. The 𝛾-dual of the set 𝔏_∞(𝐸¹) of sequence spaces fuzzy numbers is the set 𝔏₁(𝐸¹).

4. Fuzzy Matrix Transformations

Let 𝜆(𝐸¹) and 𝜇(𝐸¹) be two sequence spaces of fuzzy numbers, and let 𝐴=(𝑎_𝑛𝑘) be an infinite matrix of fuzzy numbers 𝑎_𝑛𝑘 and 𝑢=(𝑢_𝑘)∈𝜆(𝐸¹), where 𝑛,𝑘∈ℕ={0,1,2,…}, then we can say that 𝐴 defines a matrix mapping from 𝜆(𝐸¹) to 𝜇(𝐸¹), and we denote it by writing 𝐴∶𝜆(𝐸¹)→𝜇(𝐸¹) if for every sequence 𝑢=(𝑢_𝑘)∈𝜆(𝐸¹), the sequence 𝐴𝑢={(𝐴𝑢)_𝑛}, the 𝐴-transform of 𝑢, is in 𝜇(𝐸¹), where𝐴𝑢=_𝑘𝑎_𝑛𝑘𝑢_𝑘. (4.1) For simplicity in notation, here and in what follows, the summation without limits runs from 0 to ∞. By (𝜆(𝐸¹)∶𝜇(𝐸¹)), we denote the class of matrices 𝐴 such that 𝐴∶𝜆(𝐸¹)→𝜇(𝐸¹). Thus, 𝐴∈(𝜆(𝐸¹)∶𝜇(𝐸¹)) if and only if the series on the right side of (4.1) converges for each 𝑛∈ℕ.

When does 𝐴∈(𝔏_∞(𝐸¹)∶𝔏_∞(𝐸¹))? It is obvious that sufficient and efficient conditions for this are in following theorem.

Theorem 4.1. 𝐴∈(𝔏_∞(𝐸¹)∶𝔏_∞(𝐸¹)) if and only if ‖𝐴‖=sup_𝑛𝑘sup_𝜆∈[0,1]𝜆𝑑₁𝑎_𝑛𝑘(1),0,𝑑_𝜆𝑎_𝑛𝑘(𝛼),0<∞. (4.2)

Proof. Let us suppose that (4.2) holds and 𝑢∈𝔏_∞(𝐸¹), then ‖𝐴𝑢‖_𝔏∞(𝐸¹₎=sup_𝑛sup_𝜆∈[0,1]𝜆𝑑_1𝑘𝑎_𝑛𝑘𝑢_𝑘(1),0,𝑑_𝜆𝑘𝑎_𝑛𝑘𝑢_𝑘(𝛼),0≤sup_𝑛𝑘sup_𝜆∈[0,1]𝜆𝑑₁𝑎_𝑛𝑘𝑢_𝑘(1),0,𝑑_𝜆𝑎_𝑛𝑘𝑢_𝑘(𝛼),0≤𝑀‖𝑢‖_𝔏∞(𝐸¹₎<∞, (4.3) that is, 𝐴𝑢∈ℓ_∞(𝐸¹).
Conversely, let us suppose that 𝐴∈(𝔏_∞(𝐸¹)∶𝔏_∞(𝐸¹)) and 𝑢∈𝔏_∞(𝐸¹), then, since 𝐴𝑢∈𝔏_∞(𝐸¹) exists, the series ∑_𝑘𝑎_𝑛𝑘𝑢_𝑘 converges for each fixed 𝑛∈ℕ, and hence 𝐴∈𝔏^𝛽_∞(𝐸¹). This holds for the sequence (𝑢_𝑘)=([1,1])∈𝔏_∞(𝐸¹), then, we can write ‖𝐴𝑢‖_𝔏∞(𝐸¹₎=sup_𝑛sup_𝜆∈[0,1]𝜆𝑑_1𝑘𝑎_𝑛𝑘𝑢_𝑘(1),0,𝑑_𝜆𝑘𝑎_𝑛𝑘𝑢_𝑘(𝛼),0≤sup_𝑛sup_𝜆∈[0,1]𝜆𝑑_1𝑘𝑎_𝑛𝑘(1),0,𝑑_𝜆𝑘𝑎_𝑛𝑘(𝛼),0sup_𝑛sup_𝜆∈[0,1]𝜆𝑑₁𝑢_𝑛(1),0,𝑑_𝜆𝑢_𝑛(𝛼),0≤𝑀sup_𝑛sup_𝜆∈[0,1]𝜆𝑑_1𝑘𝑎_𝑛𝑘(1),0,𝑑_𝜆𝑘𝑎_𝑛𝑘(𝛼),0<∞ (4.4) which means that (4.2) holds.

Theorem 4.2. Let lim_𝑛sup_𝜆∈[0,1]𝜆[𝑑₁(𝑎_𝑛𝑘(1),0),𝑑_𝜆(𝑎_𝑛𝑘(𝛼),0)]=0 (𝑘 fixed), and suppose that (4.2) is held, then 𝐴=(𝑎_𝑛𝑘) defines a bounded operator on ℭ₀(𝐸¹) into itself, where 𝑎_𝑛𝑘∈𝐸¹ for all 𝑛,𝑘∈ℕ.

Proof. Let 𝑢=(𝑢_𝑘)∈ℭ₀(𝐸¹). If 𝑢=(𝑢_𝑘)=0, then 𝐴_𝑛(𝑢)=∑_𝑘𝑎_𝑛𝑘𝑢_𝑘=0, for all 𝑛∈ℕ. Hence, 𝐴(𝑢)∈ℭ₀(𝐸¹). Now, we suppose that 𝑢≠0. Then, under conditions of hypothesis, since 𝑢∈ℭ₀(𝐸¹) and ∑_𝑘sup_𝜆∈[0,1]𝜆[𝑑₁(𝑎_𝑛𝑘(1),0),𝑑_𝜆(𝑎_𝑛𝑘(𝛼),0)]<∞, for all 𝑛∈ℕ, the series ∑_𝑘𝑎_𝑛𝑘𝑢_𝑘 is fuzzy absolute convergent for all 𝑛∈ℕ, 𝐴_𝑛(𝑢)_ℭ0(𝐸1)=sup_𝑛sup_[]𝜆∈0,1𝜆𝑑_1𝑘𝑎_𝑛𝑘(1)𝑢_𝑘(1),0,𝑑_𝜆𝑘𝑎_𝑛𝑘(𝛼)𝑢_𝑘(𝛼),0≤sup_𝑛𝑘sup_[]𝜆∈0,1𝜆𝑑₁𝑎_𝑛𝑘(1)𝑢_𝑘(1),0,𝑑_𝜆𝑎_𝑛𝑘(𝛼)𝑢_𝑘(𝛼),0=sup_𝑛^𝑚_𝑘=0sup_[]𝜆∈0,1𝜆𝑑₁𝑎_𝑛𝑘(1)𝑢_𝑘(1),0,𝑑_𝜆𝑎_𝑛𝑘(𝛼)𝑢_𝑘(𝛼),0+sup_{𝑛𝑘≥𝑚+1}sup_[]𝜆∈0,1𝜆𝑑₁𝑎_𝑛𝑘(1)𝑢_𝑘(1),0,𝑑_𝜆𝑎_𝑛𝑘(𝛼)𝑢_𝑘(𝛼),0≤sup_𝑛^𝑚_𝑘=0sup_[]𝜆∈0,1𝜆𝑑₁𝑎_𝑛𝑘(1),0,𝑑_𝜆𝑎_𝑛𝑘(𝛼),0sup_[]𝜆∈0,1𝜆𝑑₁𝑢_𝑘(1),0,𝑑_𝜆𝑢_𝑘(𝛼),0+sup_{𝑛𝑘≥𝑚+1}sup_[]𝜆∈0,1𝜆𝑑₁𝑎_𝑛𝑘(1),0,𝑑_𝜆𝑎_𝑛𝑘(𝛼),0sup_[]𝜆∈0,1𝜆𝑑₁𝑢_𝑘(1),0,𝑑_𝜆𝑢_𝑘(𝛼),0≤‖𝑢‖sup_𝑛^𝑚_𝑘=0sup_[]𝜆∈0,1𝜆𝑑₁𝑎_𝑛𝑘(1),0,𝑑_𝜆𝑎_𝑛𝑘(𝛼),0+𝑀sup_[]𝜆∈0,1𝜆𝑑₁𝑢_𝑘(1),0,𝑑_𝜆𝑢_𝑘(𝛼),0. (4.5)
Since 𝑢∈ℭ₀(𝐸¹), for enough big 𝑘>𝑛₀, we can write sup_𝜆∈[0,1]𝜆𝑑₁𝑢_𝑘(1),0,𝑑_𝜆𝑢_𝑘(𝛼),0<𝜖2𝑀, (4.6) and from lim_𝑛sup_𝜆∈[0,1]𝜆[𝑑₁(𝑎_𝑛𝑘(1),0),𝑑_𝜆(𝑎_𝑛𝑘(𝛼),0)]=0 (𝑘 fixed), we can choose 𝑛 so large that ^𝑚_𝑘=0sup_[]𝜆∈0,1𝜆𝑑₁𝑎_𝑛𝑘(1),0,𝑑_𝜆𝑎_𝑛𝑘(𝛼),0≤𝜖2sup_𝑘sup_𝜆∈[0,1]𝜆𝑑₁𝑢_𝑘(1),0,𝑑_𝜆𝑢_𝑘(𝛼),0. (4.7) From (4.6) and (4.7), we see that 𝐴𝑢∈ℭ₀(𝐸¹). Finally, we will show that 𝐴 is bounded as follows: ‖𝐴𝑢‖_ℭ0(𝐸¹₎=sup_𝑛sup_𝜆∈[0,1]𝜆𝑑_1𝑘𝑎_𝑛𝑘(1)𝑢_𝑘(1),0,𝑑_𝜆𝑘𝑎_𝑛𝑘(𝛼)𝑢_𝑘(𝛼),0≤sup_𝑛𝑘sup_𝜆∈[0,1]𝜆𝑑₁𝑎_𝑛𝑘,0,𝑑_𝜆𝑎_𝑛𝑘(𝛼),0sup_𝑛sup_[]𝜆∈0,1𝜆𝑑₁𝑢_𝑘(1),0,𝑑_𝜆𝑢_𝑘(𝛼),0≤𝑀‖𝑢‖_ℭ0(𝐸¹₎. (4.8)

The above theorem shows that a certain type of matrix defines a linear operator on ℭ₀(𝐸¹) into itself.

Acknowledgment

The authors would like to thank the referee(s) for much constructive criticism and attention for detail.