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A criterion of compactness in the space of fuzzy numbers and applications

We propose a simple criterion of compactness in the space of fuzzy number on the space of finite dimension and apply to deal with a class of fuzzy intergral equations in the best condition.


Introduction
In recent decades, we have seen that the theory about fuzzy logic and fuzzy mathematics have strongly developed and more and more widely penetrated into many fields of application science such as: decision making, fuzzy control, neural networks, data analysis, risk assessment, optimization and transportation,...
A direction contributing to the development of the fuzzy mathematics is the research on the existence, uniqueness and properties of solution of fuzzy differential, integral equations, see for instance [2]. There, the core tools are fixed point theorems such as contraction principal, Schauder theorem see for example Ravi P Agarwal et al. [3], A Khastan et al. [4] etc.
In [3], q−1 f (s, y(s))ds, is continuous and compact. However, according to our group's opinion, this paper contains three problems to be necessarily discussed.
(1) In the step of proving continuity of operator T is continuous, we think that: It is necessary to add the hypothesis of uniform continuity of the function f .
(2) Besides, in the step of proving the level-equicontinuity of T (Ω)(t). Authors used the level-equicontinuity of y 0 (t). While this property is not deduced from the hypothesis which is only given as that y 0 ([0, a * ]) is compactsupported. (3) In the proof of the compact-supported property of T (Ω)(t) (the end of page 10). Authors argued that: by using Theorem 2.2 and the relative compact property of f ([0, a * ] × Ω) we deduce the compact-supported property of f ([0, a * ] × Ω) . This argument is not true with Theorem 2.2 only stating that: If A is a compact-supported subset of E n c , then the two following statements are equivalent.
Theorem 2.2 did not assert that: The compact-supported property is a consequence of the relative compact property.
Maybe authors A. Khanstan et al have cared for the shortcomings. So in [4], they investigated the existence of solution of fuzzy integral equation: which similar and simpler than (1.1). Besides, Theorem 2.2 of paper [3] is recalled in the label "Theorem 2.3" and right below they had a remark that: in fact, the following two statements F and level-equicontinuous on [0,1], are equivalent. This remark could deal with the third shortcomings in paper [3]. However, they did not say that it was cited from which resources?
Next, the authors in [4] also claimed that: the operator T defined by q−1 f (s, u(s))ds is continuous and compact.
As for the detail, they used a different technique from one in [3]. The operator T is decomposed into two operators A and N i.e, T = A • N , where To prove the continuity and compactness of T , they made the following assumptions (presented in Theorem 3.13) The strong point of paper [4] is the fact that they overcame all shortcomings of paper [3]. Further, the result of the existence of solution is still true in the case that the function f admits a singular at t = 0.
However to overcome the first shortcoming of [3], the authors in [4] must use the hypothesis of uniform continuity of the function f r .
Another observation is the fact that both paper [3] and [4] applied the result of Theorem 2.2 (stated in [3]). This theorem relates to a quite complex concept that is the compact -supported concept.
From the above motivation, in the present paper we wish to propose one simple approach without using compact -supported concept as well as uniform continuity of the function f but we still obtain the result of the existence of solution for equation (1.1).
The layout of this paper is organiged as follows: In Section 2, some necessary results and concepts are recalled. Specially, we use Section 3 to present a convenient criterion of compactness in the space of fuzzy number R F c . Section 4 is used to prove an existence result by using the above criterion. Finally, in Section 5, we extend the existence result for the space of fuzzy numbers E n c .

Preliminaries
In this section, we give some definitions and introduce the necessary notations which will be used throughout this paper.
Let us denote K c (R n ) as the family of all nonempty, compact and convex subsets of The distance between A and B is defined by the Hausdorff-Pompeiu metric K c (R n ) is a complete and separable metric space with respect to the Hausdorff -Pompeiu metric (see [6]). In the following, we give some basic notions and results on fuzzy set theory. We denote by E n the space of all fuzzy numbers in R n , that is, E n is the space of all functions u : R n → [0, 1], satisfying the following properties (see for example [1] The fuzzy null set is defined by If u ∈ E n , then the set We usually denote E 1 by R F and if u ∈ E 1 , we denote , for every α ∈ [0, 1]. According to Zadeh's extension principle, we have the addition and the scalar multiplication in fuzzy-number space E n as usual. It is well known that The metric in E n is defined by we have the following properties (see [9]): A strongly measurable and integrably bounded mapping We denote E n c the space of u ∈ E n with the property that the mapping The concept of semilinear space and related concepts were already considered, for instance in [3]. A semilinear metric space is a semilinear space S with a metric d : S × S → R which is translation invariant and positively homogeneous, that is, (a, b), for all a, b ∈ S, λ ≥ 0. If S is a semilinear metric space, then addition and scalar multiplication on S are continuous. Furthermore, if S is complete, then we say that S is a semilinear Banach space. We say that a semilinear space S has the cancellation property if ], E n c ) are semilinear Banach spaces having the cancellation property.
By using the fact that a semilinear metric space S having cancellation property can be isometrically embedded into a normed space we can prove Schauder's fixed point theorem for S. Theorem 2.5. (see [3]) Let B be a nonempty, closed, bounded and convex subset of a semilinear Banach space S having the cancellation property, and suppose that P : B → B is a compact operator. Then P has at least one fixed point in B.
Remark 2.6. Here, it is understood that: a compact operator is a continuous operator mapping bounded sets into relative compact sets.

A criterion of compactness in R c F
In this section we propose a convenient criterion of compactness for R c F to deal with a problem in Section 4.
First, we recall two conventional forms of Ascoli -Arzela Theorem.
Theorem 3.1. Let E be a complete metric space and K be a compact metric space. Then a subset A of C(K, E) is relatively compact iff i) for every r ∈ K, A(r) = {u(r) | u ∈ A} is relatively compact in E, ii) A is equicontinuous.
In particular for E = R m with m ∈ N, we have the following version of Ascoli -Arzela theorem ii) A is level -equicontinuous.
Next consider the following mapping j It is easy to check that (3.1) So the mapping j isometrically embeds R c F into X. From the above isometric embedding, it is easy to check that: A is relatively compact in R c F iff j(A) is relatively compact in X. Furthermore, it is not difficult to check that: • The equicontinuity of j(A) is just be level -equicontinuity of A, • The uniform boundedness of j(A) is just be the boundedness of A in R c F . Finally, we end the proof by applying Theorem 3.2 for j(A) in X.

Fuzzy integral equation
We consider the nonlinear fuzzy integral equation of the form where a > 0 and 0 < q < 1 are constants given.
We establish the following assumptions: F is a given continuous and compact mapping.
It is possible that f has a singular at t = 0.
We put M is well -defined because of compactness of f. Then, we have the following theorem. Proof. For η ∈ (0, a], we put Then it is not difficult to check that X is a semilinear complete metric space with respect to the metric We define the set Ω = u ∈ X | D X (u, 0) ≤ R , and B R = {x ∈ R c F : D(x, 0) ≤ R}. It is easy to see that Ω is a closed, bounded and convex subset of the semilinear Banach space X. On the set Ω, we define the operator We claim that the operator T is continuous and compact. It is sufficient to show that A is continuous and compact.
The proof consists of five steps.
By Appling Lebesgue's Dominated Convergence Theorem, we obtain that This implies that Au(t) ∈ R c F .
Step 3. We claim that T u ∈ Ω, for every u ∈ Ω. Indeed, for t ∈ [0, η] we have This shows that D X (T u, 0) ≤ R, for every u ∈ Ω. Thus, T maps the set Ω to itself.
Step 4. Let u, u n ∈ Ω : u n → u. We claim that Au n → Au.
This means that Au n → Au in X as n → ∞.
Step 5. Now we claim that A(Ω) is relatively compact in X. Indeed, first we observe that the equicontinuity of A(Ω) is a consequence of estimate (4.2). Therefore by using Theorem 3.1, it remains to prove that: For every t ∈ [0, η], A(Ω)(t) is relatively compact in R c F . From the same arguments as estimate (4.3), it is easy to check that the boundedness of A(Ω)(t) holds for every t ∈ [0, η]. By using Theorem 3.3 we only need to prove that A(Ω)(t) is level -equicontinuous, for every t ∈ [0, η].
So A(Ω) is relatively compact in X, i.e, A is a compact operator, we deduce that T is too.
The results obtained from the above steps allow us to conclude that T is continuous and compact. By Theorem 2.5 T has a fixed point in Ω i.e, the integral equation (4.1) has a solution.

Generalized problem
The solvability of the problem (4.1) still holds if the space R c F is replaced by E n c . Indeed the whole proof is completely similar by using a criterion of compactness for the space E n c which we propose as follows.
Proof. First we need the following lemmas Lemma 5.2. (see [6]) A subset Γ of K c (R n ) is relatively compact iff Γ is bounded.  By using Theorem 3.1 we obtain the following result: B ⊂ E n c is relatively compact in C([0, 1], K c (R n )) (i.e. in E n c because of the closedness of E n c ) iff a) for every α ∈ [0, 1], B(α) is relatively compact in K c (R n ), b) B is equicontinuous (just be level -equicontinuous). Using the Lemma 5.3 we see that the part a) is equivalent to the part i) of Theorem 5.1 and we end the proof of Theorem 5.1.
Remark 5.4. By the same technique solving fuzzy integral equation (4.1), we can deal with a number of fuzzy fractional differential equations.