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

Authors: M. Şengönül; Z. Zararsız
Journal: Abstract and Applied Analysis
 (2011)

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## 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 fuzzyα-, fuzzy β-, and fuzzy γ-duals of the fuzzy module spaces of fuzzy numbers have been computeded, and some matrix transformations are given.


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## Body

## 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 computeα-, 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,d̂(u,v)=sup⁡kd¯(uk,vk), 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 d̂ 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 afuzzy 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 Section2. 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 thatℕ is the set of all positive integer numbers, ℝ is the set of all real numbers, is the Ei be the set of all bounded and closed intervals on the real line ℝ, that is, Ei={a=[a-,a+]:a-≤x≤a+,a-anda+∈ℝ}. For a,b∈Ei and define(2.1)d(a,b)=max⁡⁡{|a--b-|,|a+-b+|}.
Then, it can be seen easily that d defines a metric on Ei and (Ei,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:FN1u is normal, that is, there exists an x0∈ℝ such that u(x0)=1,FN2u is fuzzy convex, that is, for any x,y∈ℝ and μ∈[0,1], u[μx+(1-μ)y]≥min⁡⁡{u(x),u(y)},FN3u 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 eachα∈[0,1], the α-level set defined by [u]α={x∈ℝ:u(x)≥α} is in Ei, as well as the support u0, that is, uα=[uα-,uα+] 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 functionsu+(λ) and u-(λ) are right continuous at the point λ=0,(4)
u+(λ)≥u-(λ).Conversely, if the pair of functionsα 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 sequenceu=(uk) of fuzzy numbers is a function u from the set ℕ, the set of all positive integers, into E1, and fuzzy number uk denotes the value of the function at k and is called the kth term of the sequence.Letu,v∈E1 and λ∈ℝ, then the operations addition and scalar multiplication are defined on E1 in terms of α-level sets by(2.2)u+v=w⟺[w]α=[u]α+[v]α,[λu]α=λ[u]α∀α∈[0,1].Define a mapd¯: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 thatu,v∈E1 and G is the set of all nonnegative fuzzy numbers. The function df:E1×E1→G is called fuzzy metric [6] which satisfies the following properties:(1)
df(u,v)≥0,(2)
df(u,v)=0 if and only if u=v,(3)
df(u,v)=df(v,u),(4)
wheneverw∈E1, we have df(u,v)≤df(u,w)+df(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 d̂(u,v)=sup⁡kd¯(uk,vk).By using this metric,d̂, 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.Ifdf is a fuzzy metric on E1, then the pair of (E1,df) is called as a fuzzy metric space. For any u,v∈E1, the fuzzy metric of Zhang is [10–12] defined by(2.3)df(u,v)=sup⁡λ∈[0,1]λ[d1(u,v),d̂λ(u,v)],
where d̂λ(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 metricdf defined by equality (2.3) is a fuzzy distance of fuzzy numbers; thus, (E1,df) is a fuzzy metric space.Theorem 2.2.
Ifu,v∈E1, then d¯(u,v) is a point of the interval determined by the fuzzy metric df(u,v).Proof.
Clearly, ifλ run from 0, then the second side of df, d̂λ(u,v), is equal to d¯(u,v) since
(2.4)df(u,v)=sup⁡λ∈[0,1]λ[d1(u,v),d̂λ(u,v)].Theorem 2.3 (see [12]).
The fuzzy metric space(E1,df) is complete metric space.Definition 2.4.
Letλ(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 numberu is defined which corresponds to the fuzzy distance from u to 0¯, that is,
(2.5)‖u‖E1∶=sup⁡λ∈[0,1]λ[d1(u,0¯),d̂λ(u,0¯)].Proposition 2.6.
The setE1 of the fuzzy numbers is fuzzy complete module space with the fuzzy module in (2.5).Letu=(uk) be a sequence of fuzzy numbers, and let ∥·∥ be a fuzzy module, then the sequence (uk) 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 ∥uk-u0∥<[ϵ,ϵ]=ϵ for k≥n0. The sequence (uk) is said to be fuzzy bounded in fuzzy module ∥·∥ if sup⁡k∥uk∥<∞ for all k∈ℕ.We will write𝔏∞(E1), ℭ(E1), and ℭ0(E1) for the fuzzy sets of all fuzzy bounded, fuzzy convergent, fuzzy null sequences, respectively, that is,(2.6)L∞(E1)∶={u=(uk)∈w(E1):sup⁡ksup⁡λ∈[0,1]λ[d1(uk,0¯),d̂λ(uk,0¯)]<∞},C(E1)∶={u=(uk)∈w(E1):∃u0∈E1∋lim⁡ksup⁡λ∈[0,1]λ[d1(uk,u0),d̂λ(uk,u0)]=θ},C0(E1)∶={u=(uk)∈w(E1):lim⁡ksup⁡λ∈[0,1]λ[d1(uk,0¯),d̂λ(uk,0¯)]=θ}.
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 d̂f defined by(2.7)d̂f(u,v)=sup⁡ksup⁡λ∈[0,1]λ[d1(uk,vk),d̂λ(uk,vk)].Now, let us give some definitions that will facilitate our work.A sequenceu=(un) in E1 is said to be a fuzzy fundamental sequence if forever ɛ>0, there exists an integer n0 such that df(un,um)<ɛ for n,m>n0. A fuzzy metric space (E1,df) is called the fuzzy complete metric space if every fundamental sequence converges in E1.Theorem 2.7 (see [10–12]).
The sequenceu in E1 is fuzzy convergent in the metric df if and only if u is a fuzzy fundamental sequence.Theorem 2.8.
The fuzzy metric spaces(ℭ(E1),d̂f) and (𝔏∞(E1),d̂f) are fuzzy complete metric spaces.Theorem 2.9 (see [14]).
Let𝒰⊂E1, and let (uk) be a sequence of fuzzy functions from 𝒰 to E1. If for each k, there exists a real number Mk such that max⁡⁡{|(uk(U))λ-|,|(uk(U))λ+|}≤Mk for all U∈𝒰 and λ∈(0,1], and if the series ∑kMk converges, then there is a fuzzy function u:𝒰→E1 such that ∑kuk converges uniformly to u.Now, let us define the sequence setsℭs(E1), 𝔅s(E1), and 𝔏p(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,(2.8)Cs(E1)={u=(uk)∈w(E1):lim⁡nsup⁡λ∈[0,1]λ[d1(∑k=1nuk,u0),d̂λ(∑k=1nuk,u0)]=0},Bs(E1)={u=(uk)∈w(E1):sup⁡nsup⁡λ∈[0,1]λ[d1(∑k=1nuk,0¯),d̂λ(∑k=1nuk,0¯)]<∞},Lp(E1)={u=(uk)∈w(E1):(∑k(sup⁡λ∈[0,1]λ[d1(uk,0¯),d̂λ(uk,0¯)])p)1/p<∞,1≤p<∞}.
Let us see the  following theorems which about the sets 𝔏∞(E1), ℭ(E1), and ℭ0(E1).Theorem 2.10.
The sets𝔏∞(E1), ℭ(E1), and ℭ0(E1) are fuzzy complete module spaces defined by fuzzy module
(2.9)‖u‖=sup⁡ksup⁡λ∈[0,1]λ[d1(uk,0¯),d̂λ(uk,0¯)].Proof.
Since the proof forℭ(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 (uk) is a fuzzy fundamental sequence in 𝔏∞(E1) where (uk)=(u0(i),u1(i),u2(i),…), then, for any ϵ>0, there exists an integer n0 such that
(2.10)‖uki-ukj‖=sup⁡ksup⁡λ∈[0,1]λ[d1(uki(1)-ukj(1),0¯),d̂λ(uki(α)-ukj(α),0¯)]≤[ϵ,ϵ]=ϵ,
for i,j≥n0. Hence, we obtain |uki(1)--ukj(1)-|<ϵ/λ and sup⁡α∈[λ,1]max⁡{|uki-(α)-ukj-(α)|,|uki+(α)-ukj+(α)|}<ϵ/λ. This shows that (uki(1)-) is fundamental sequence of real numbers in ℝ, and (uki(α)) is fundamental sequence of fuzzy numbers in E1. Since ℝ and E1 are complete, so (uki(1)) converges in ℝ and (uki(α)) converges in E1 for all k∈ℕ.
Let us suppose thatlim⁡iuki(1)-=uk(1)- and lim⁡iuki(α)=uk(α) for each k∈ℕ. Put u(1)-=(uk(1)-) and u(α)=(uk(α)). Now, we shall show that lim⁡iui(1)-=u(1)- and u(1)-∈ℝ, lim⁡iui(α)=u(α) and u(α)∈E1. Since (ui(1)) is a fundamental sequence in ℝ, given ϵ>0, there exists n0∈ℕ such that for i,j>n0, |ui(1)-uj(1)|<ϵ/λ, and if we take limit over j, we get |ui(1)-u(1)|<ϵ/λ. Therefore, lim⁡iui(1)=u(1). Similarly, since (uki(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⁡{|uki-(α)-ukj-(α)|,|uki+(α)-ukj+(α)|}<ϵ/λ, and if we take limit over j we get sup⁡α∈[λ,1]max⁡{|uki-(α)-uk-(α)|,|uki+(α)-uk+(α)|}<ϵ/λ<ϵ. Therefore, lim⁡iui(α)=uk(α). Let us show that uk(1)-∈ℝ and uk(α)∈E1. Also, since |u(1)-|≤|u(1)--uki(1)-|+|uki(1)-|<∞ and sup⁡α∈[λ,1]max⁡{|uk-(α)|,|uk+(α)|}≤sup⁡α∈[λ,1]max⁡{|uk-(α)-uki-(α)|,|uk+(α)-uki+(α)|}+sup⁡α∈[λ,1]max⁡{|uki-(α)|,|uki+(α)|}<∞, this shows that u∈𝔏∞(E1).Theorem 2.11.
The space𝔏p(E1) is fuzzy complete module space defined by module
(2.11)∥u∥Lp(E1)=(∑k(sup⁡λ∈[0,1]λ[d1(uk,0¯),d̂λ(uk,0¯)])p)1/p<∞,1≤p<∞.Proof.
Since the proof is similar to the proof of the Theorem2.10, we omit it.


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

For the fuzzy sequence spacesλ(E1) and μ(E1), define the set S(λ(E1),μ(E1)) by(3.1)S(λ(E1),μ(E1))={u∈w(E1):uv=(ukvk)∈μ(E1)∀u=(uk)∈λ(E1)}.
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 λα(E1), λβ(E1), and λγ(E1), are defined by(3.2)λα(E1)=S(λ(E1),L1(E1)),λβ(E1)=S(λ(E1),Cs(E1)),λγ(E1)=S(λ(E1),Bs(E1)).Definition 3.1.
Let us suppose thatλ(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 (uk)∈λ(E1) there is (vk)∈μ(E1) such that ∥uk∥≤∥vk∥ for all k∈ℕ.
Ifλ(E1) is fuzzy cofinal in μ(E1), then λα(E1)=μα(E1); the converse of this assertion is not true.Now, we may give results concerning the fuzzyα-dual, fuzzy β-dual, and fuzzy γ-dual of the sets ℭ0(E1), ℭ(E1), 𝔏∞(E1), and 𝔏p(E1).Theorem 3.2.
The fuzzyα-dual of the set 𝔏∞(E1) of sequence spaces of FNs is the set 𝔏1(E1).Proof.
Let(uk)∈𝔏∞α(E1). If we consider (vk)=([1¯,1¯])∈𝔏∞(E1), then the series
(3.3)∑k(sup⁡λ∈[0,1]λ[d1(ukvk,0¯),d̂λ(ukvk,0¯)])=∑k(sup⁡λ∈[0,1]λ[|uk-(1)vk-(1)|,sup⁡α∈[λ,1]max⁡⁡{|uk-(α)vk-(α)|,|uk+(α)vk+(α)|}])≤∑k(sup⁡λ∈[0,1]λ[|uk-(1)|,sup⁡α∈[λ,1]max⁡⁡{|uk-(α)|,|uk+(α)|}]sup⁡λ∈[0,1]λ[|vk-(1)|,sup⁡α∈[λ,1]max⁡⁡{|vk-(α)|,|vk+(α)|}])=∑k(sup⁡λ∈[0,1]λ[d1(uk,0¯),d̂λ(uk,0¯)])
converges, that is to say, (uk)∈𝔏1(E1). Therefore, we have
(3.4)L∞α(E1)⊆L1(E1).
Conversely, let us suppose that(uk)∈𝔏∞(E1) and (vk)∈𝔏1(E1), then there exists a 0¯<K∈E1 such that K=sup⁡ksup⁡λ∈[0,1]λ[|d1(uk(1),0¯),d̂λ(uk(α),0¯)]<∞. From here, we have
(3.5)∑k(sup⁡λ∈[0,1]λ[d1(uk(1)vk(1),0¯),d̂λ(uk(α)vk(α),0¯)])=∑k(sup⁡λ∈[0,1]λ[d1(vk(1),0¯),d̂λ(vk(α),0¯)])‖u‖≤K∑k(sup⁡λ∈[0,1]λ[d1(vk(1),0¯),d̂λ(vk(α),0¯)])=K‖v‖L1(E1)<∞,
which gives that
(3.6)L1(E1)⊆L∞α(E1).
From (3.4) and (3.6), we see that 𝔏1(E1)=𝔏∞α(E1).Theorem 3.3.
The fuzzy sequence spacesℭ0(E1), ℭ(E1) are cofinal in 𝔏1(E1).Proof.
Denote any of the spacesℭ0(E1) and ℭ(E1) by λ(E1), and suppose that ∥uk∥≤∥vk∥ holds for some (vk)∈𝔏1(E1), then we can easily see that sup⁡k∥uk∥≤sup⁡k∥vk∥, lim⁡k∥uk∥≤sup⁡k∥vk∥ which lead us to the desired results.Theorem 3.4.
The fuzzyα-dual of the sets ℭ0(E1) and ℭ(E1) of sequence spaces of FNs is the set 𝔏1(E1).Proof.
Since the sets𝒞0(E1) and ℭ(E1) are cofinal in 𝔏∞(E1) (see Theorem 3.3), the proof is clear.Theorem 3.5.
The fuzzyβ-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 set𝔏∞(E1). Let us suppose that (uk)∈𝔏∞(E1) and (vk)∈w(E1), then there is a K>0 such that K=[K1,K2]=sup⁡ksup⁡λ∈[0,1]λ[d1(uk(1),0¯),d̂λ(uk(α),0¯)]<∞, also since
(3.7)|uk-(1)vk-(1)|=d1(ukvk,0¯)≤d1(uk,0¯)d1(vk,0¯)≤K1d1(vk,0¯),|uk-(α)vk-(α)|≤dλ(ukvk,0¯)≤dλ(uk,0¯)dλ(vk,0¯)≤K1dλ(vk,0¯),|uk+(α)vk+(α)|≤dλ(ukvk,0¯)≤dλ(uk,0¯)dλ(vk,0¯)≤K1dλ(vk,0¯).
The series∑k(uk-(α)vk-(α)), ∑k(uk+(α)vk+(α)), and ∑k(uk-(1)vk-(1)) are convergent uniformly from Theorem 2.9; therefore, ∑kukvk converges whenever ∑kd1(vk,0¯) and ∑kdλ(vk,0¯) converge. From here, we can write 𝔏1(E1)⊆𝔏∞β(E1).
Finally, we will show that the inclusion𝔏∞β(E1)⊆𝔏1(E1) is held. Let us suppose that (vk)∈𝔏∞β(E1), then we have ∑k(sup⁡λ∈[0,1]λ[d1(uk(1)vk(1),0),d̂λ(uk(α)vk(α),0¯)])<∞ for all (uk)∈𝔏∞(E1). This holds for the sequence (uk)=([1¯,1¯])∈𝔏∞(E1), then we can write
(3.8)∑k(sup⁡λ∈[0,1]λ[d1(uk(1)vk(1),0¯),d̂λ(uk(α)vk(α),0¯)])=K∑k(sup⁡λ∈[0,1]λ[d1(vk(1),0¯),d̂λ(vk(α),0¯)])=‖v‖L1(E1).
This shows that v∈𝔏1(E1).Proposition 3.6.
Theγ-dual of the set 𝔏∞(E1) of sequence spaces fuzzy numbers is the set 𝔏1(E1).


## 4. Fuzzy Matrix Transformations

Letλ(E1) and μ(E1) be two sequence spaces of fuzzy numbers, and let A=(ank) be an infinite matrix of fuzzy numbers ank and u=(uk)∈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=(uk)∈λ(E1), the sequence Au={(Au)n}, the A-transform of u, is in μ(E1), where(4.1)Au=∑kankuk.
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 doesA∈(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
(4.2)‖A‖=sup⁡n∑ksup⁡λ∈[0,1]λ[d1(ank(1),0¯),d̂λ(ank(α),0¯)]<∞.Proof.
Let us suppose that (4.2) holds and u∈𝔏∞(E1), then
(4.3)‖Au‖L∞(E1)=sup⁡nsup⁡λ∈[0,1]λ[d1(∑kankuk(1),0¯),d̂λ(∑kankuk(α),0¯)]≤sup⁡n∑ksup⁡λ∈[0,1]λ[d1(ankuk(1),0¯),d̂λ(ankuk(α),0¯)]≤M‖u‖L∞(E1)<∞,
that is, Au∈ℓ∞(E1).
Conversely, let us suppose thatA∈(𝔏∞(E1):𝔏∞(E1)) and u∈𝔏∞(E1), then, since Au∈𝔏∞(E1) exists, the series ∑kankuk converges for each fixed n∈ℕ, and hence A∈𝔏∞β(E1). This holds for the sequence (uk)=([1¯,1¯])∈𝔏∞(E1), then, we can write
(4.4)‖Au‖L∞(E1)=sup⁡nsup⁡λ∈[0,1]λ[d1(∑kankuk(1),0¯),d̂λ(∑kankuk(α),0¯)]≤sup⁡nsup⁡λ∈[0,1]λ[d1(∑kank(1),0¯),d̂λ(∑kank(α),0¯)]sup⁡nsup⁡λ∈[0,1]λ[d1(un(1),0¯),d̂λ(un(α),0¯)]≤Msup⁡nsup⁡λ∈[0,1]λ[d1(∑kank(1),0¯),d̂λ(∑kank(α),0¯)]<∞
which means that (4.2) holds.Theorem 4.2.
Letlim⁡nsup⁡λ∈[0,1]λ[d1(ank(1),0¯),d̂λ(ank(α),0¯)]=0 (k fixed), and suppose that (4.2) is held, then A=(ank) defines a bounded operator on ℭ0(E1) into itself, where ank∈E1 for all n,k∈ℕ.Proof.
Letu=(uk)∈ℭ0(E1). If u=(uk)=0¯, then An(u)=∑kankuk=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(ank(1),0¯),d̂λ(ank(α),0¯)]<∞, for all n∈ℕ, the series ∑kankuk is fuzzy absolute convergent for all n∈ℕ,
(4.5)‖An(u)‖C0(E1)=sup⁡nsup⁡λ∈[0,1]λ[d1(∑kank(1)uk(1),0¯),d̂λ(∑kank(α)uk(α),0¯)]≤sup⁡n∑ksup⁡λ∈[0,1]λ[d1(ank(1)uk(1),0¯),d̂λ(ank(α)uk(α),0¯)]=sup⁡n∑k=0msup⁡λ∈[0,1]λ[d1(ank(1)uk(1),0¯),d̂λ(ank(α)uk(α),0¯)]+sup⁡n∑k≥m+1sup⁡λ∈[0,1]λ[d1(ank(1)uk(1),0¯),d̂λ(ank(α)uk(α),0¯)]≤sup⁡n∑k=0msup⁡λ∈[0,1]λ[d1(ank(1),0¯),d̂λ(ank(α),0¯)]sup⁡λ∈[0,1]λ[d1(uk(1),0¯),d̂λ(uk(α),0¯)]+sup⁡n∑k≥m+1sup⁡λ∈[0,1]λ[d1(ank(1),0¯),d̂λ(ank(α),0¯)]sup⁡λ∈[0,1]λ[d1(uk(1),0¯),d̂λ(uk(α),0¯)]≤‖u‖sup⁡n∑k=0msup⁡λ∈[0,1]λ[d1(ank(1),0¯),d̂λ(ank(α),0¯)]+Msup⁡λ∈[0,1]λ[d1(uk(1),0¯),d̂λ(uk(α),0¯)].
Sinceu∈ℭ0(E1), for enough big k>n0, we can write
(4.6)sup⁡λ∈[0,1]λ[d1(uk(1),0¯),d̂λ(uk(α),0¯)]<ϵ2M,
and from lim⁡nsup⁡λ∈[0,1]λ[d1(ank(1),0¯),d̂λ(ank(α),0¯)]=0 (k fixed), we can choose n so large that
(4.7)∑k=0msup⁡λ∈[0,1]λ[d1(ank(1),0¯),d̂λ(ank(α),0¯)]≤ϵ2sup⁡ksup⁡λ∈[0,1]λ[d1(uk(1),0¯),d̂λ(uk(α),0¯)].
From (4.6) and (4.7), we see that Au∈ℭ0(E1). Finally, we will show that A is bounded as follows:
(4.8)‖Au‖C0(E1)=sup⁡nsup⁡λ∈[0,1]λ[d1(∑kank(1)uk(1),0¯),d̂λ(∑kank(α)uk(α),0¯)]≤sup⁡n∑ksup⁡λ∈[0,1]λ[d1(ank,0¯),d̂λ(ank(α),0¯)]sup⁡nsup⁡λ∈[0,1]λ[d1(uk(1),0¯),d̂λ(uk(α),0¯)]≤M‖u‖C0(E1).The above theorem shows that a certain type of matrix defines a linear operator onℭ0(E1) into itself.



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*Source: 837584-2011-12-06.xml*