International Journal of Control Science and Engineering

p-ISSN: 2168-4952    e-ISSN: 2168-4960

2012;  2(1): 1-6

doi:10.5923/j.control.20120201.01

Existence and Uniqueness Theorems for Generalized Set Differential Equations

Andrej Plotnikov1, 2, Natalia Skripnik1

1Department of Optimal Control & Economic Cybernetics, Odessa National University named after I.I. Mechnikov, Odessa, 65026, Ukraine

2Department of Applied Mathematics, Odessa State Academy of Civil Engineering and Architecture, Odessa, 65029, Ukraine

Correspondence to: Andrej Plotnikov, Department of Optimal Control & Economic Cybernetics, Odessa National University named after I.I. Mechnikov, Odessa, 65026, Ukraine.

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Copyright © 2012 Scientific & Academic Publishing. All Rights Reserved.

Abstract

In this paper the concept of generalized differentiability for set-valued mappings proposed by A.V. Plotnikov, N.V. Skripnik is used. The generalized set-valued differential equations with generalized derivative are considered and the existence and uniqueness theorems are proved.

Keywords: Set-valued mapping, Generalized derivative, Existence and uniqueness theorems

Cite this paper: Andrej Plotnikov, Natalia Skripnik, Existence and Uniqueness Theorems for Generalized Set Differential Equations, International Journal of Control Science and Engineering, Vol. 2 No. 1, 2012, pp. 1-6. doi: 10.5923/j.control.20120201.01.

Article Outline

1. Introduction
2. The Generalized Derivative
3. Generalized Differential Equations with the Generalized Derivative
4. Conclusions

1. Introduction

The concept of derivative for set-valued mapping was first entered by M. Hukuhara[1]. Then the problems of differentiability of fuzzy mappings were considered by T. F. Bridgland[2], J.N. Tyurin[3], H.T. Banks and M.Q. Jacobs[4], A.V. Plotnikov[5, 6], A.N. Vityuk[7], B. Bede and S.G. Gal[8], A.V. Plotnikov and N.V. Skripnik[9]. The properties of the these derivatives were considered in[10-18].
F.S. de Blasi and F. Iervolino begun studying of set-valued differential equations (SDEs) in semilinear metric spaces[12,19-21]. Now it developed in the theory of SDEs as an independent discipline. The properties of solutions, the impulsive SDEs, control systems and asymptotic methods for SDEs were considered[5,6,9-11,16-24]. On the other hand, SDEs are useful in other areas of mathematics. For example, SDEs are used as an auxiliary tool to prove the existence results for differential inclusions. Also, one can employ SDEs in the investigation of fuzzy differential equations. Moreover, SDEs are a natural generalization of usual ordinary differential equations in finite (or infinite) dimensional Banach spaces[19].
In[9] a new concept of a derivative of a set-valued mapping that generalizes the concept of Hukuhara derivative was entered and a new type of a set-valued differential equation such that the diameter of its solution can whether increase or decrease (for example, to be periodic) was considered. In the ideological sense this definition of the derivative is close to the definitions proposed in[5,6,8].
In this paper the generalized set-valued differential equations with generalized derivative are considered and the existence and uniqueness theorems are proved.

2. The Generalized Derivative

Let be a space of all nonempty convex closed sets of with Hausdorff metric,where .
Definition 1[1]. Let . A set such that is called a Hukuhara difference of the sets X and Y and is denoted by .
From Rådström's Embedding Lemma[25] it follows that if this difference exists, then it is unique.
Let ; be a set-valued mapping; be a -neighbourhood of a point ; .
For any consider the following Hukuhara differences if these differences exist.
(1)
(2)
(3)
(4)
The differences (1) and (2)[(3) and (4)] are called the right[left] differences. From the definition of the Hukuhara difference it follows that both one-sided differences exist only in the case when for or . If all differences (1)-(4) exist then in -neighbourhood of the point .
If for all there exists only one of the one-sided differences, then using the properties of the Hukuhara difference, we get that the mapping in the -neighbourhood of the point can be:
a) non-decreasing on ;
b) non-increasing on ;
c) non-decreasing on and non-increasing on ;
d) non-increasing on and non-decreasing on .
Hence, for each of the above mentioned cases only one of combinations of differences is possible:
a) (1) and (3); b) (2) and (4); c) (2) and (3); d) (1) and (4).
Consider four types of limits corresponding to one of the difference types:
(5)
(6)
(7)
(8)
So it is possible to say that in the point not more than two limits can exist (as we assumed that there exist only two of four Hukuhara differences).
Considering all above we have that there can exist only the following combinations of limits:
a) (5) and (7); b) (6) and (8);
c) (6) and (7); d) (5) and (8).
Definition 2[9]. If the corresponding two limits exist and are equal we will say that the mapping is differentiable in the generalized sense in the point and denote the generalized derivative by .
Let us say that the set-valued mapping is differentiable in the generalized sense on the interval if it is differentiable in the generalized sense at every point of this interval.
Remark 1. Properties of the generalized derivative have been considered in[9].
Definition 3[9]. The set-valued mapping is called absolutely continuous on the interval if there exist a measurable set-valued mapping and a system of intervals , , such that for all or .
Theorem 1[9]. Let a set-valued mapping is absolutely continuous on the interval .
Then the set-valued mapping is differentiable in the generalized sense almost everywhere on the interval and almost everywhere on .

3. Generalized Differential Equations with the Generalized Derivative

First consider a differential equation with the generalized derivative that is similar to a differential equation with the Hukuhara derivative, i.e.
(9)
where is the generalized derivative of a set-valued mapping , is a set-valued mapping, .
Definition 4. A set-valued mapping is said to be solution of differential equation (9) if it is absolutely continuous and satisfies (9) almost everywhere on .
Remark 2. Unlike the case of differential equations with Hukuhara derivative, if a differential equation with the generalized derivative (9) has a solution then there exists an infinite number of solutions irrespective of the conditions on the right-hand side of the equation.
Example 1. Consider the following differential equation with the generalized derivative
(10)
It is easy to check that the following set-valued mappings are the solutions of equation (10):
Also it is possible to construct other solutions, thus only will be the solution of the corresponding differential equation with the Hukuhara derivative, and and are solutions of the differential equation with the generalized derivative (in the sense of[8]).
Therefore we will consider the other differential equation with the generalized derivative:
(11)
where ;;; are set-valued mappings; is a continuous function; function
Definition 5. A set-valued mapping is called the solution of differential equation (11) if it is continuous and on any subinterval , where function of constant signs, satisfies the integral equation
If on the interval the function , then satisfies the integral equation
for and increases.
If on the interval the function , then we have
i.e. and decreases.
If on the interval the function , then we have .
So we can enter the other equivalent definition of a solution of equation (11).
Definition 6. A set-valued mapping is called the solution of differential equation (11) if it is absolutely continuous, satisfies (11) almost everywhere on and
Example 2. Consider the following differential equation with generalized derivative
(12)
As for we have for .
So for we get .
Further as for we have
.
So for we get .
Figure 1. The graph of a solution of system (12)
It means that the solution exists only for (see fig. 1).
Example 3. Consider the same differential equation with generalized derivative but with :
(13)
As for then we have
for .
Further as for then we get
.
Further as for then we get
So for we have .
Figure 2. The graph of a solution of system (13)
It means that the solution exists only for (see fig. 2).
Remark 3. It is obvious that the mappings , define only on “how much” the mapping changes in case of its "decrease"() or "increase"() and function defines what will occur to ["decrease" or "increase"]. If irrespective of and the mapping will be constant.
Example 4. Consider the differential equation from Example 2 with for . Then for .
Remark 4. If we take then we will have
.
Then for we get . So the solution exists for (see fig. 3).
Figure 3. The graph of a solution of system (12) for
So for all we can guarantee the existence of solution of the differential equation on the interval .
Let be a space of all nonempty strictly convex closed sets of and all element of [27].
The following theorem of existence of the solution of equation (11) for case holds:
Theorem 2. Let the set-valued mappings , in the domain
satisfy the following conditions:
i) for any fixed the set-valued mappings , are measurable;
ii) for almost every fixed t the set-valued mappings , are continuous;
iii) , , where , are summable on ;
iv) is continuous and has the finite number of intervals where ,
v) .
Then there exists a solution of equation (11) defined on the interval , where satisfies the conditions
a) ;
b) , where, ;
c)
where
Proof. Let us consider some cases.
1) for . Then equation (11) is the ordinary differential equation with Hukuhara derivative
(14)
Therefore, using[17] we get that the equation (11) has a solution defined on , where satisfies the condition , .
2) for . Then equation (11) is the ordinary differential equation with Hukuhara derivative and therefore, is the solution of (11) on .
3) for . Then equation (11) is the equation with the generalized derivative
(15)
According to Definition 5 consider the following integral equation
(16)
for and prove the existence of solution on the some interval .
3a) As for , then
,
where
So .
Define by .
It is obviously, that if , then.
As and , then there exists such that the set can be embedded in the set for all (i.e. there exists such that ) and is not embedded for . And, it is obviously, that can be found out from the equation .
Therefore, for all
the set is embedded in the set .
3b) As for all , then for all. Therefore, as and that the set can be embedded in the set for all , then the Hukuhara difference exists for all [27].
3c) Let us find such that and consider .
3d) Choose any natural . Sequentially on the intervals
, , let us build the successive approximations of the solution
(17)
By 3b) is exist and for all and . Also by conditions i) and ii) of the theorem is continuous on for all .
Besides
Hence, it follows that the sequence of the set-valued mappings in uniformly bounded:.
Let us show that the set-valued mappings are equicontinuous. For any and any natural the inequality holds
The function is absolutely continuous on as the integral of the summable function with a variable top limit. Hence, for any there exists such that for all such that the inequality is fair, the sequence is equicontinuous.
According to Askoli theorem[28] we can choose a uniformly converging subsequence of the sequence . Its limit is a continuous set-valued mapping that we will denote by . As and the first summand is less than for in view of the equicontinuity of the set-valued mappings , then along the chosen subsequence converges to . Owing to the theorem conditions in (15) it is possible to pass to the limit under the sign of the integral. We receive that the set-valued mapping satisfies equation (16) and , i.e. is the solution of (15) on the interval .
,
4) In case when the function changes sign on the interval , the existence of the solution is proved combining cases 1)-3). The theorem is proved.
Theorem 3. Let the set-valued mappings , in the domainsatisfy the conditions of Theorem 2 and satisfy the conditions ,for all .
Then there exists the unique solution of equation (11) defined on the interval .
The proof is similar to[17,24].
Finally we consider example for case .
Example 6. Consider the following differential equation with generalized derivative
(18)
where .
It is obvious that is the solution of differential equation (18) (see fig. 4).
Figure 4. The graph of a solution of system (18)
Remark 5. Also it is possible to prove the similar results if be a space of all nonempty M- strongly convex closed sets of and all element of [29].
Remark 6. Let's notice that we considered some continuous function but it is also possible to take , for example, where - is the diameter of some etalon set-valued mapping.

4. Conclusions

In this paper the concept of generalized differentiability (proposed in[9]) for set-valued mappings is used. The new type of the set-valued differential equation – generalized set differential equations – is considered. The existence and uniqueness theorems for set-valued differential equations with generalized derivative are proved.

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