1. Introduction

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.

Often by consideration of the integral equations containing small parameter, the averaging method of Bogolyubov - Krylov ^{[1, 2]} is applied. Two approaches are thus used:

1) the integral equation is reduced to integro-differential by differentiation, and then one of the averaging schemes is applied ^{[3, 4, 5]};

2) one of the averaging schemes is applied directly to the integral equation ^{[3, 4, 5, 6, 7]}.

Recently the set-valued and fuzzy integral equations and inclusions began to be considered [6-14]^[6]. In this paper the substantiation of the averaging method for fuzzy integral equation using the second approach is considered.

2. Preliminaries

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

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

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

i) is normal, i.e. there exists such that ;

ii) is fuzzy convex, i.e.

for any and ;

iii) is upper semicontinuous, i.e. for any and there exists such that whenever ;

iv) is compact.

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 ^[15]:

1) is a complete metric space;

2) for all ;

3) for all and .

Definition 1. ^[15] A mapping is measurable [continuous] if for any the set-valued mapping defined by is Lebesgue measurable [continuous].

Definition 2. ^[15] A mapping is said to be integrably bounded if there exists an integrable function such that for all.

Definition 3. ^[15] The integral of a fuzzy mapping is defined levelwise by : is a measurable selection of for all .

Definition 4. ^[15] 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.

Lemma 1. Let be integrable and . Then

1) ;

2) ;

3) is integrable;

4) .

3. Main Results

Consider the fuzzy integral equation with a small parameter

(1)

where is time, is a phase variable, is a small parameter.

Consider the following full averaged fuzzy integral equation

(2)

where

(3)

The following theorem that proves the closeness of solutions of systems (1) and (2) on the finite interval holds:

Theorem 1. Let in the domain the following conditions fulfill:

1) the fuzzy mapping is continuous and satisfies the Lipschitz condition in with the constant i.e. for all

2) the fuzzy mapping is equicontinuous on for any solution of equation (1);

3) uniformly with respect to limit (3) exists;

4) the solution of equation (2) is defined for all and and belongs with some neighborhood to the domain

Then for any and there exists such that for and for all the inequality holds

(4)

where and are the solutions of equations (1) and (2).

Proof. First of all, let us notice that the fuzzy mapping satisfies the Lipschitz condition in with the constant

Really, using condition 3) of the Theorem we have that for any there exists such that for the following estimate holds:

Then

As is arbitrary small, we get

Let us estimate the distance between the solutions of the initial and the averaged fuzzy integral equations:

Using the Gronwall’s lemma we get

(5)

Denote by the module of continuity of the fuzzy mapping Then

for

Divide the interval with the step where

as Such choice of the step is possible in view of the properties of the module of continuity, and exactly as

and one can take for example equal to

Denote by

the division points, is the solution of equation (1) in the division points.

Estimate the distance

on the interval k =

Using condition 2) of the theorem we have

then

According to condition 3) of the theorem there exists such monotone decreasing function that tends to 0 as that for all

Then

Therefore,

So

(6)

Denote by Choose from the condition

Then for all using (5) and (6) we get

and the statement of the theorem is proved provided the solution belongs to the domain Q on the interval , and it follows from condition 4).

Theorem 2. Let in the domain the following conditions fulfill:

1) the fuzzy mapping is continuous, bounded with the constant M, periodic in and satisfies the Lipschitz condition in with the constant

2) the fuzzy mapping is equicontinuous on for any solution of equation (1);

3) the solution of equation

(7)

where

(8)

is defined for all and and belongs with some neighborhood to the domain

Then for any there exists such and that for and for all the inequality holds

(9)

where and are the solutions of equations (1) and (7).

The proof of this theorem is similar to the proof above taking the step equal to.

Proof. First of all, let us notice that the fuzzy mapping satisfies the Lipschitz condition:

Let us estimate the distance between the solutions of the initial and the averaged fuzzy integral equations. Similar to the proof of Theorem 1 we get

(10)

Denote by the module of continuity of the fuzzy mapping Then

for

Divide the interval with the step denote by

the division points, is the solution of equation (1) in the division points.

Estimate the distance

on the interval k = Similar to the proof of Theorem 1 we get

Using condition 2) of the theorem we have

then

According to (8)

Using the condition 1) of the theorem we have

Therefore,

(11)

Then for all using (10) we get

and the statement of the theorem is proved provided the solution belongs to the domain Q on the interval , and it follows from condition 3).

Also the scheme of partial averaging for the fuzzy integral equation with a small parameter (1) can be applied. Such variant of the averaging method happens to be useful when the average of some mappings does not exist or their presence in the equation does not complicate its research.

Consider the following partial averaged fuzzy integral equation

(12)

where

(13)

The following theorem that proves the closeness of solutions of systems (1) and (12) on the finite interval holds:

Theorem 3. Let in the domain the following conditions fulfill:

1) the fuzzy mappings are continuous and satisfy the Lipschitz condition in with the constant

2) the fuzzy mapping is equicontinuous on for any any solution of equation (1);

3) uniformly with respect to limit (13) exists;

4) the solution of equation (12) is defined for all and and belongs with some neighborhood to the domain

Then for any and there exists such that for and for all the inequality holds

where and are the solutions of equations (1) and (12).

The proof of this theorem is similar to the proof of Theorem 1.

Similar to Theorem 2 the periodic case can be considered.

4. Conclusion

Theorems 1-3 generalize the results of ^[16] on substantiation of the averaging method for fuzzy differential equations. When passing from the fuzzy differential equation to the equivalent fuzzy integral equation we have that conditions 1), 3), 4) of Theorems 1,3 hold owing to the corresponding conditions on the right hand sides of the initial fuzzy differential equation, and condition 2) is satisfied because the fuzzy mapping is Lipschitzian in t as the right-hand side of the fuzzy differential equation is uniformly bounded.

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