International Journal of Systems Science and Applied Mathematics
Volume 1, Issue 4, November 2016, Pages: 42-49

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

(A.V. Plotnikov)

*Corresponding author

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

Contents

1. Introduction

In 1990 J.P. Aubin [6] 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 [34]. 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 [23], D. G. Korenevskii [24] for the ordinary hyperbolic differential equations.

2. Preliminaries

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 [31]) 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 [41]:

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)

(1)

where .

We interpret fuzzy Darboux problem (1) as a family of set-valued Darboux problems

(2)

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

(3)

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 [40] and [50],  for every  and

By [40] and [51], if  then

for every .

Consider any solutions  and any . Let  be such that

for every .

Then

i.e.  for every  and .

By [40] and [51], 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

(4)

where  - small parameters,

In this work, we associate with the problem (4) the following full averaged fuzzy Darboux problem

(5)

where  such that

(6)

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

(7)

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 [52], 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 [52], for any  there exists  such that

(8)

(9)

(10)

for any   and .

Combining (8), (9) and (10), choosing  and  we obtain

The theorem is proved.

5. Conclusion

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

(11)

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 .

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