1. Introduction

In recent years, the fuzzy set theory introduced by Zadeh ^[1] has emerged as an interesting and fascinating branch of pure and applied sciences. The applications of fuzzy set theory can be found in many branches of science as physical, mathematical, differential equations and engineering sciences. Recently there have been new advances in the theory of fuzzy differential equations [2-7]^[2], fuzzy integral equations [8-16]^[8], fuzzy integrodifferential equations ^{[17, 18, 19, 20]}, differential inclusions with fuzzy right-hand side [21-24]^[21] and fuzzy differential inclusions ^{[25, 26, 27]} as well as in the theory of control fuzzy differential equations ^{[28, 29, 30]}, control fuzzy integrodifferential equations [31-36]^[31], control fuzzy differential inclusions ^{[37, 38, 39, 40]}, and control fuzzy integrodifferential inclusions ^[41].

Almost in all papers mentioned above the authors also consideres equivalent fuzzy integral equations. However, integral equations are encountered in various fields of science and in numerous applications, including elasticity, plasticity, heat and mass transfer, oscillation theory, fluid dynamics, filtration theory, electrostatics, electrodynamics, biomechanics, game theory, control, queuing theory, electrical engineering, economics, and medicine. Therefore, in this article we consider fuzzy integral equations and prove the existence and uniqueness theorem.

2. Preliminaries

Let be a set of all nonempty (convex) compact subsets from the space ,

be Hausdorff distance between sets and , is -neighborhood of set .

Let be the set of all such that u satisfies the following conditions:

i) is normal, that is, there exists an such that ;

ii) is fuzzy convex, that is,

for any and ;

iii) is upper semicontinuous,

iv) is compact.

If , then is called a fuzzy number, and is said to be a fuzzy number space. For , denote

Then from i)-iv), it follows that the -level set for all .

Let be the fuzzy mapping defined by if and .

Define by the relation

Then is a metric in . Further we know that ^[42]:

1) is a complete metric space,

2) for all ,

3) for all and .

Definition 1. ^[5] A mapping is measurable (continuous) if for all the set-valued map defined by is Lebesgue measurable (continuous).

Definition 2. ^[5] A mapping is said to be integrably bounded if there is an integrable function such that for every .

Definition 3. ^[5] The integral of a fuzzy mapping is defined levelwise by : is a measurable selection of for all .

Definition 4. ^[5] A measurable and integrably bounded mapping is said to be integrable over if .

Note that if is measurable and integrably bounded, then is integrable. Further if is continuous, then it is integrable.

Proposition 1. ^[2] Let be integrable and . Then

1) ;

2) ;

3) is integrable;

4) .

3. Main Result

Consider the fuzzy integral equation

(1)

where is time, is a phase variable, is -dimensional matrix-valued function, is a fuzzy mapping, .

Definition 5. A fuzzy mapping is called a solution of integral equation (1) if it is continuous and satisfies integral equation (1) on interval .

Theorem. Let in the domain the following conditions hold:

i) for any fixed the fuzzy mapping is continuous;

ii) there exists a positive constant such that

for all ;

iii) there exists a positive constant such that

for all ;

iv) the matrix-valued functions are continuous;

v) there exist positive constants such that

for all .

Then equation (1) has a unique solution on the interval .

Proof. Let us build the successive approximations of the solution:

for ,

for .

By conditions i), ii) and iv) of the theorem is continuous on for all . Besides

;

;

and so on.

Therefore,

Then

.

Hence, it follows that the sequence of the fuzzy mappings in uniformly bounded:

for all .

Let us show that the sequence of the fuzzy mappings is a Cauchy sequence. For any we have

.

Hence,

Therefore, the sequence is a Cauchy sequence. Its limit is a continuous fuzzy mapping that we will denote by . Owing to the theorem conditions in (1) it is possible to pass to the limit under the sign of the integral. We receive that the fuzzy mapping satisfies equation (1), i.e. is the solution of (1) on the interval .

To prove the uniqueness, suppose that there exist at least two different solutions and of (1) on . Then .

As

then

.

So

,

.

Then for any that contradicts .

This concludes the proof.

Remark 1. If then fuzzy integral equation (1) is equivalent to the Cauchy problem

where is the fuzzy Hukuhara derivative of a fuzzy mapping ^[2].

Remark 2. Solutions of integral equation (1) can be not fuzzy differentiable in the sense of Hukuhara. For example, if

, ,

, where such, that for all , then we get the fuzzy integral equation

It is obvious that its solution is and is not fuzzy differentiable in the sense of Hukuhara for all . However fuzzy integral system (1) will be equivalent to the following fuzzy hybrid system

4. Conclusion

In 1982, D. Dubois and H. Prade ^{[43, 44]} first introduced the concept of integration of fuzzy functions. O. Kaleva ^[2] studied the measurability and integrability for the fuzzy set-valued mappings of a real variable whose values are normal, convex, upper semicontinuous, and compactly supported by fuzzy sets in. Existence of solutions of fuzzy integral equations has been studied by several authors. They have used the embedding theorem of Kaleva, which is a generalization of the classical Rådström embedding theorem, and the Darbo fixed point theorem in the convex cone. In this article we prove the existence and uniqueness theorem without using the embedding theorem of Kaleva.

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