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International Journal of Differential Equations
Volume 2014 (2014), Article ID 307941, 5 pages
The Partial Averaging of Fuzzy Differential Inclusions on Finite Interval
1Department of Applied Mathematics, Odessa State Academy of Civil Engineering and Architecture, Didrihsona Street 4, Odessa 65029, Ukraine
2Department of Mathematics, Odessa State Academy of Civil Engineering and Architecture, Didrihsona Street 4, Odessa 65029, Ukraine
Received 12 February 2014; Accepted 23 April 2014; Published 4 May 2014
Academic Editor: Nikolai N. Leonenko
Copyright © 2014 Andrej V. Plotnikov and Tatyana A. Komleva. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
The substantiation of a possibility of application of partial averaging method on finite interval for differential inclusions with the fuzzy right-hand side with a small parameter is considered.
In 1990, Aubin  and Baidosov [2, 3] introduced differential inclusions with the fuzzy right-hand side. Their approach is based on usual differential inclusions. In 1995, Hüllermeier [4–6] introduced the concept of -solution similar to how it has been done in . Further in [8–20], various properties of solutions of fuzzy differential inclusions and their applications at modeling of various natural-science processes were considered.
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. After the systematic researches done by N. M. Krylov, N. N. Bogoliubov, Yu. A. Mitropolsky, and so forth, in 1930s, the averaging method gradually became one of the classical methods in analyzing nonlinear oscillations.
In works [21, 22], the possibility of application of schemes of full and partial averaging for differential inclusions with the fuzzy right-hand side, containing a small parameter, was proved. By proving these theorems, the scheme offered by Plotnikov et al. for a substantiation of schemes of an average of usual differential inclusions [23–28] was used. In this work, the possibility of application of partial averaging method for fuzzy differential inclusions without passage to reviewing of separate solutions is proved; that is, all estimations are spent for -solution corresponding fuzzy systems.
Let be a family of all nonempty (convex) compact subsets from the space with the Hausdorff metric: where and is -neighborhood of set .
Let be a family of all such that satisfies the following conditions:(1)is normal; that is, there exists an such that ;(2) is fuzzy convex; that is, for any and ;(3) is upper semicontinuous; that is, for any and exists such that whenever , ;(4)the closure of the set is compact.
If , then is called a fuzzy number and is said to be a fuzzy number space.
Definition 1. The set is called the -level of a fuzzy number , for . The closure of the set is called the -level of a fuzzy number .
It is clear that the set , for all .
Theorem 2 (see  (stacking theorem)). If , then (1), for all ;(2), for all ;(3)if is a nondecreasing sequence converging to , then .Conversely, if is the family of subsets of satisfying conditions (1)–(3), then there exists such that for and .
Let be the fuzzy number defined by , if and .
Define by the relation Then, is a metric in . Further, we know that (i) is a complete metric space;(ii), for all ;(iii), for all and .
3. Fuzzy Differential Inclusion: -Solution
Consider the fuzzy differential inclusion where , , , .
An -solution of (4) is understood to be an absolutely continuous function which satisfies (5) almost everywhere. Let denote the -solution set of (5) and let . Clearly, a family of subsets cannot satisfy the conditions of Theorem 2 (see [5, 6, 9]).
Therefore, we will consider an -solution of fuzzy differential inclusion (4).
Definition 3. The upper semicontinuous fuzzy mapping which satisfies the system is called the -solution of differential inclusion (4), where .
Theorem 4. Suppose that the following conditions hold:(1)fuzzy mapping is measurable, for all ;(2)there exists such that, for all , for almost every ;(3)there exists such that , for almost every and every ;(4)for all , and almost every , Then, there exists a unique -solution of fuzzy system (4) defined on the interval .
Proof. Let and .
By [5, 6], it follows that a family of subsets satisfy the conditions of Theorem 2; that is, , for every .
Divide the interval into partial intervals by the points , , , . We use Euler algorithm; let the mapping be given by where , , , .
By [7, 28], it follows that the sequence is equicontinuous and fundamental and its limit is a unique -solution of differential inclusion (5) and , for every and . This concludes the proof.
Also, we consider the differential inclusion where , , , .
Lemma 5. Let and satisfy conditions (1)–(4) of Theorem 4 and there exist and such that
for every and , .
Then , for every .
Proof. Divide the interval into partial intervals by the points , , , . By Definition 3, we have for every . This concludes the proof.
Remark 6. If , then , for every .
4. The Method of Partial Averaging
Now, consider the fuzzy differential inclusion with a small parameter where , , , , and is a small parameter.
In this work, we associate the following partial averaged fuzzy differential inclusion with the inclusion (10): where such that
Theorem 7. Let in domain the following conditions hold:(1)mappings , are measurable on ;(2)mappings , satisfy a Lipschitz condition
with a Lipschitz constant ;(3)there exists such that
for almost every and every ;(4)for all , and almost every ,
(5)limit (15) exists uniformly with respect to x in the domain ;(6)for any , , and , the -solution of the inclusion (10) together with a -neighborhood belongs to the domain G; that is, , for every .
Then, for any and L > 0, there exists such that, for all and , the following inequality holds: where , are the -solutions of initial and partial averaged inclusions.
Proof. Divide the interval on the partial intervals by the points , . We denote fuzzy mappings and such that
for every , , .
Also, we take
As for ,
Using estimates (22)–(25), for any , there exists such that, for , we have
Taking into account Lemma 5, for any , there exists such that, for all , the following inequality holds:
By combining (26) and (27) and choosing and , we obtain (19). The theorem is proved.
If is continuous on , then, instead of (5), it is possible to consider the following more simple equation: and, similarly, we can prove all the results received earlier.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
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