1. Introduction

Many important problems of analytical dynamics are described by the nonlinear mathematical models that as a rule are presented by the nonlinear differential or the integrodifferential equations. The absence of exact universal research methods for nonlinear systems has caused the development of numerous approximate analytic and numerically-analytic methods that can be realized in effective computer algorithms.

The averaging methods combined with the asymptotic representations (in Poincare sense) began to be applied as the basic constructive tool for solving the complicated problems of analytical dynamics described by the differential equations. Averaging theory for ordinary differential equations has a rich history, dating to back to the work of N.M. Krylov and N.N. Bogoliubov[1], and has been used extensively in engineering applications[2-6]. Books that cover averaging theory for differential equations and inclusions include[7-10].

In recent years, the fuzzy set theory introduced by Zadeh[11] 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[12-16], fuzzy integrodifferential equations[17-20], differential inclusions with fuzzy right-hand side[21-24] and fuzzy differential inclusions[25-27] as well as in the theory of control fuzzy differential equations[28-30], control fuzzyintegrodifferential equations[31-35], controlfuzzy differentialinclusions[36-39], and control fuzzy integrodifferentialinclusions[40]. In works[41-48] various schemes of an averagefor the fuzzy differential equations and inclusions have been considered. In this article we prove the substantiation of the method of full averaging for the integrodifferential inclusions with small parameter on the metric space. Thereby we expand a circle of systems to which it is possible to apply Krylov-Bogolyubov method of averaging.

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:

• is normal, that is, there exists an such that;

• is fuzzy convex, that is,

• for any and;

• is upper semicontinuous,

• is compact.

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

.

Then from 1)-4), it follows that the -level set for all.

Let be the fuzzy mapping defined by if and.

Define by the relation

where is the Hausdorff metric defined in. Then is a metric in.

Further we know that[49]:

1. is a complete metric space,

2. for all,

3. for all and.

Definition 1.[15] A mapping

is measurable if for all the set-valued map

defined by is

Lebesgue measurable.

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

Definition 3.[15] The integral of a fuzzy mapping

is defined levelwise by

. The set of all such that

is a measurable selection for

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.

Now we consider following fuzzy integrodifferential inclusion

(1)

where means; is the state;is a fuzzy mapping, is a fuzzy mapping, ,.

We interpret[21-24] the fuzzy differential inclusion (1) as a family of integrodifferential inclusions

(2)

where the subscript indicates that the -level set of a fuzzy set is involved (the system (2) can only have any significance as a replacement for (1) if the solutions generate fuzzy sets (fuzzy R-solution)[24]).

Let denotes the fuzzy R-solution of the fuzzy integrodifferential inclusion (1).

3. The Scheme of an Average

In this section we consider the Cauchy problem with small parameter

(3)

where is a small parameter.

In this section we associate with the equation (3) the following averaged integrodifferential equation

(4)

where

(5)

(6)

Remark. In this paper we will consider a case when the limits (5), (6) exist.

Theorem. Let in domain

the following hold:

1) is continuous in;

2) is continuous in;

3) there exist constants such that

for all, and any ;

4) there exist continuous functions, , and constant such that

for any, and any

5) there exist constants, and such that

for any;

6) there exist continuous functions , and constants such that

for any and

7) the limits (5),(6) exist uniformly in;

8) for any and the fuzzy R-solution of the system (4) together with a -neighbourhood belong to the domain.

Then for any and there exists

such that for all and

the following statement fulfil:

(7)

Where , are the fuzzy R-solutions of the initial and the full averaged equations.

Proof. Let is any constant. In the beginning, us

prove the validity of the inclusion for all

(8)

where for all. Let is arbitrary and we prove the validity of the inclusion

for all.

We consider the integrodifferential inclusion

. (9)

Let is any solution of the system (9). Then

(10)

Where

.

Divide the interval into partial intervals by the points We consider the function

where

.

By conditions 3), and 4) of the theorem, we have

for all and.

Also,

.

Let. Then, we get

(12)

By (12), and conditions 3), 5) of the theorem, we obtain

for all.

It follows, we have, for all.

As

for all, then we get

where

Therefore

for all.

Then we obtain

Now we have function such that

where

.

By condition 7) of the theorem, for any, there exists such that the following estimate is true for:

Hence,

Then

Also we have

for all.

From here and by conditions 3), 6) of the theorem, we obtain

where

Therefore,

(13)

From (13) it follows the existence of such a solution of inclusion

that for all

where

Than we obtain

Also , then

there exist such that,

for all. Hence, for all is arbitrary, then

, for all.

Now, we proof, that, for all. Also, let is arbitrary and we prove the validity of the inclusion for all.

We consider the integrodifferential inclusion

(14)

Let is any solution of the system (14). Then

(15)

where

Divide the interval into partial intervals by the points . We consider the function

where

By conditions 6), and 8) of the theorem, we have

for all .

Let. Then (similarly (12)), we get

and

As

for all, then we get

where

Therefore for all

where

Then, we obtain

Now we have function such that

where

By condition 7) of the theorem, for any, there exists such that the following estimate is truefor:

Hence,

Then

Also we have

for all.

From here and by conditions 3), and 6) of the theorem, we obtain

where

Therefore,

(16)

From (16) it follows the existence of such a solution of inclusion

that for all

Where

Than we obtain

Also thenthere exist such that ,

for all . Hence , for all . As is arbitrary, then

, for all . This concludes he proof.

4. Conclusions

In this article we prove the substantiation of the method of full averaging for the fuzzy integrodifferential inclusions with small parameter. Thereby we expand a circle of systems to which it is possible to apply Krylov-Bogolyubov method of averaging. In a case if limits (5) and (6) do not exist it is possible to receive only result similar[46].