1. Introduction

As is generally known the absence of exact universal research methods for many important problems of analytical dynamics 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 began to be applied as the basic constructive tool for solving the complicated problems of analytical dynamics described by the differential equations. It became possible due to the works of N.M. Krylov, N.N. Bogolyubov, Yu.A. Mitropolskij, V.M. Volosov, N.N. Moiseev, etc. (see [1-8]).

In recent years the development of the calculus in metric spaces has attracted some attention [6-12]. Earlier, F.S. de Blasi, F. Iervolino [13] started the investigation of set differential equations (SDEs) in semilinear metric spaces. This has now evolved into the theory of SDEs as an independent discipline: properties of solutions [6-45], the impulse equations [6,7,41], control systems [42-45] and asymptotic methods [6-8,46-50]. On the other hand, SDEs are useful in other areas of mathematics. For example, SDEs are used, as an auxiliary tool, to prove existence results for differential inclusions [6,30,35,39]. Also, one can employ SDEs in the investigation of fuzzy differential equations [7,10,25-27, 29,30]. Moreover, SDEs are a natural generalization of the usual ordinary differential equations in finite (or infinite) dimensional Banach spaces.

In this article we prove the substantiation of the method of full averaging for the set integrodifferential equations with small parameter. 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 in. The set is the Hukuhara difference of and, if, i.e.

From Radstrom's Cancellation Lemma [51], it follows that if this difference exists, then it is unique.

Definition [52]. A mapping is differentiable in the sense of Hukuhara at if for some the Hukuhara differences

exists in for all and there exists an such that

and

.

Here is called the Hukuhara derivative of at.

3. The Scheme of Full Average

Consider the Cauchy problem with small parameter

(1)

where is a small parameter, , is a set mapping, is a set mapping, Here the integral is understood in the sense of[52] (the integral exists for example if is measurable and the real mapping is integrable on),.

Definition. A mapping is a solution to the problem (1) if and only if it is continuous and satisfies the integral equation

for all.

In the beginning this section we associate with the equation (1) the following full averaged integrodifferential equation

(2)

where

(3)

(4)

Remark. In the beginning we will consider a case when the limit (3) exists.

Definition. We say that the limit (3) exists uniformly in, if for any and any there exists such that.

for all.

Theorem. Let in domain.

the following hold:

1) is continuous in

2) is continuous in;

3) there exist continuous function, and constant such that

for any;

4) , where

5) there exist constants, such that

for any;

6) the limit (3) exist uniformly in;

7) for any and the solution of the equation (2) together with a -neighborhood belong to the domain.

Then for any and there exists such that for all and the following statement fulfill:

(5)

where, are the solutions of the initial and the full averaged equations.

Proof. Since.

we have

Hence

Then we obtain

(6)

In the beginning we will estimate last summand in (6). Divide the interval into partial intervals by the points

Then

where

As for all,; then

Hence

where.

Obviously, , and.

Now, we will estimate. From condition 6) of the theorem it follows that there exists the increasing function such that

1) ;

2) .

Then

Where

By (6), we have

Using Gronwall-Bellman's inequality, we obtain

Fix. Then for any, there exists such that the following estimate is true for:

Hence we obtain for all.

This concludes the proof.

Now we consider a case when the limit (3) not exists. Then we associate with the equation (1) the following integrodifferential equation

(7)

Theorem. Let in domain

the following hold:

1) is continuous in

2) is continuous in;

3) there exist continuous function, and constants such that

for any;

4) , where

5) for any and the solutions of the equations (1) and (7) together with a -neighborhood belong to the domain.

Then the statement (8) fulfill:

(8)

where,.

Proof. By (1) and (7) we have

Hence we obtain

(9)

As we have for all

then by (9) we obtain (8). This concludes the proof.

4. Conclusions

Here we used the approach of Hukuhara at definition of the derivative which has essential shortages. However the given approach is well investigated by many authors. Also in the literature exist other approaches to definition of the derivative[7,8,11,12,25,40], but also they have the shortages. It is easily possible to show that this outcome will be true for some other cases with little changes.