Partial Averaging of Fuzzy Hyperbolic Differential Inclusions
Tatyana Alexandrovna Komleva1, Irina Vladimirovna Molchanyuk2,
Andrej Viktorovich Plotnikov2, *, Liliya Ivanovna Plotnikova3
1Department of Mathematics, Odessa State Academy of Civil Engineering and Architecture, Odessa, Ukraine
2Department of Applied Mathematics, Odessa State Academy of Civil Engineering and Architecture, Odessa, Ukraine
3Department of Mathematics, Odessa National Polytechnic University, Odessa, Ukraine
To cite this article:
Tatyana Alexandrovna Komleva, Irina Vladimirovna Molchanyuk, Andrej Viktorovich Plotnikov, Liliya Ivanovna Plotnikova. Partial Averaging of Fuzzy Hyperbolic Differential Inclusions. International Journal of Systems Science and Applied Mathematics. Vol. 1, No. 4, 2016, pp. 42-49. doi: 10.11648/j.ijssam.20160104.12
Received: September 19, 2016; Accepted: September 28, 2016; Published: October 19, 2016
Abstract: In this article, we considered the fuzzy hyperbolic differential inclusions (fuzzy Darboux problem), introduced the concept of R-solution and proved the existence of such a solution. Also the substantiation of a possibility of application of partial averaging method for hyperbolic differential inclusions with the fuzzy right-hand side with the small parameters is considered.
Keywords: Hyperbolic Differential Inclusion, Fuzzy, Averaging, R-solution
In 1990 J.P. Aubin  and V.A. Baidosov [7, 8] introduced differential inclusions with the fuzzy right-hand side. Their approach is based on usual differential inclusions. E. Hüllermeier [20, 21] introduced the concept of R-solution similarly how it has been done in . Later, the various properties of fuzzy solutions of differential inclusions, and their use in modeling various natural science processes were considered (see [1, 4, 5, 17, 18, 26, 27] and the references therein).
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 etc, in 1930s, the averaging method gradually became one of the classical methods in analyzing nonlinear oscillations (see [10, 25, 40, 42] and references therein). In works [36-39], the possibility of application of schemes of full and partial averaging for fuzzy differential inclusions with a small parameter was proved.
In papers [2, 3, 9, 11, 13, 18, 22, 30, 32, 33, 35], authors investigate classical models of partial differential equations with uncertain parameters, considering the parameters as fuzzy numbers. It was an obvious step in the mathematical modeling of physical processes. Study of fuzzy partial differential equations means the generalization of partial differential equations in fuzzy sense. While doing modelling of real situation in terms of partial differential equation, we see that the variables and parameters involve in the equations are uncertain (in the sense that they are not completely known or inexact or imprecise). Many times common initial or boundary condition of ambient temperature is a fuzzy condition since ambient temperature is prone to variation in a range. We express this impreciseness and uncertainties in terms of fuzzy numbers. So we come across with fuzzy partial differential equations. Also obviously, these equations can be written in as fuzzy partial differential inclusions.
In this work we consider fuzzy hyperbolic differential inclusions (fuzzy Darboux problem) and introduce the concept of R-solution similarly how it has been done in [36, 40, 50, 52, 53]. Also we ground the possibility of application of partial averaging method for fuzzy Darboux problem. This result generalize the results of A. N. Vityuk [40, 52] for the ordinary hyperbolic differential inclusions and M. Kiselevich , D. G. Korenevskii  for the ordinary hyperbolic differential equations.
Let be a family of all nonempty (convex) compact subsets from the space with the Hausdorff metric
where , is -neighborhood of set .
Let be a family of all such that satisfies the following conditions:
1) is normal, i.e. there exists an such that ;
2) is fuzzy convex, i.e. for any and ;
3) is upper semicontinuous, i.e. for any and where 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 clearly that the set for all .
Theorem 1. (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 Hyperbolic Differential Inclusion. R-solution
Consider the fuzzy hyperbolic differential inclusion (or in other words, fuzzy Darboux problem)
We interpret fuzzy Darboux problem (1) as a family of set-valued Darboux problems
Qualitative properties and structure of the set of solutions of the set-valued Darboux problem have been studied by many authors, for instance [12, 14-16, 28, 29, 40, 44-53] and references therein.
Definition 2 [28, 43]. A function is said to be absolutely continuous on () if there exist absolutely continuous functions and , and Lebesgue integrable function such that
Definition 3. An solution of (1) is understood to be an absolutely continuous function that satisfies (2) for almost every and the boundary conditions for any and
Let denote the solution set of (2) and . Clearly a family of subsets may not satisfy to conditions of Theorem 1, i.e. For example, and for any Therefore, we introduce the definition of R-solutions for fuzzy Darboux problem (1).
Definition 3. The upper semicontinuous fuzzy mapping that satisfies to the following system
is called the R-solution of fuzzy Darboux problem (1), where , ,
Now we are interested in the following question: Under what conditions, there exists a unique R-solution to (1). In the next theorem we find the existence result for a unique R-solution of fuzzy Darboux problem (1).
Theorem 2. Suppose the following conditions hold:
1) fuzzy mapping is measurable, for all ;
2) there exists such that for all
for every ;
3) there exists such that for every ;
4) for all and every
5) functions and are absolutely continuous functions on and .
Then there exists a unique R-solution of fuzzy Darboux problem (1) defined on the set .
Proof. By [29,49], every set-valued Darboux problem of family (2) has solution on the set , i.e. for every and
Also by  and , for every and
By  and , if then
for every .
Consider any solutions and any . Let be such that
for every .
i.e. for every and .
By  and , function is solution of set-valued Darboux problem (2), i.e. for every . Consequently for every and
Since, for all and , then for all and .
By [50, 52], every Darboux problem of family (2) has one R-solution on the set and we have for every and .
By [20, 53], we get that a family of subsets satisfies to conditions of Theorem 1, i.e. for every . This concludes the proof.
4. The Method of Partial Averaging
Now consider fuzzy Darboux problem with the small parameters
where - small parameters,
In this work, we associate with the problem (4) the following full averaged fuzzy Darboux problem
where such that
The main theorem of this section is on averaging for fuzzy Darboux problem with the small parameters. It establishes nearness of R-solutions of (4) and (5), and reads as follows.
Theorem 3. Let in the domain the following conditions hold:
1) fuzzy mappings and is continuous on ;
2) fuzzy mappings and satisfy a Lipschitz condition
with a Lipschitz constant ;
3) there exists such that
for every and every ;
4) for all and every
5) limit (6) exists uniformly with respect to in the domain
6) functions and are absolutely continuous functions on and for all where
7) the R-solution of the Darboux problem
together with a neighborhood belong to the domain B for .
Then for any and L 0 there exists such that for all and the following inequality holds
where are the R-solutions of initial and partial averaged Darboux problems.
Proof. By theorem 2, we have unit R-solution of Darboux problem (4) on and unit R-solution of Darboux problem (5) on .
Let , , ,
and . We denote fuzzy mappings and such that
where , , , , , , , , , , , , , .
By , it follows that the sequences , and are equicontinuous and fundamental and their limits are levels of R-solutions and of the problems (4) and (5).
Consequently, the sequences and meet by and .
By , for any there exists such that
for any and .
Combining (8), (9) and (10), choosing and we obtain
The theorem is proved.
We conclude with a few remarks.
Remark 1. In this work, we considered the fuzzy differential inclusion, when fuzzy mapping is measurable on . If is continuous on then instead of the equation (1) it is possible to consider the following more simple equation
and similarly we can prove all results received earlier.
Remark 2. If the condition 4) of Theorem 3 is not true, then the R-solutions can not exist. But there are valid the following conditions:
1) for any -solution of inclusion (4) there exists a -solution of inclusion (5) such that for all and ;
2) for any -solution of inclusion (5) there exists a -solution of inclusion (4) such that for all and .