Abstract
A new concept of inner product on the space of compact convex subsets of ℝ is introduced. Using this inner product we investigate the global existence and uniqueness of the solution of interval-valued functional differential equations. We apply these results to interval differential equations with distributed delays.
MSC:34K05, 34K30, 47G20.
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1 Introduction
Functional differential equations (or, as they are also called, delay differential equations) play an important role in an increasing number of system models in biology, engineering, physics, and other sciences. There exists an extensive literature dealing with functional differential equations and their applications. We refer to the monographs [1–7], and references therein.
The set-valued differential and integral equations are an important part of the theory of set-valued analysis, and they have high value for control theory and its application. They were first studied in 1969 by De Blasi and Iervolino [8]. Recently, set-valued differential equations have been studied by many scientists due to their applications in many areas. For many results in the theory of set-valued differential and integral equations, the readers are referred to the following books and papers [1, 9–17] and references therein. The interval-valued analysis and interval-valued differential equations (IDEs) are special cases of set-valued analysis and set-valued differential equations, respectively. In many cases, when modeling real-world phenomena, information about the behavior of a dynamical system is uncertain and one has to consider these uncertainties to gain better understanding of the full models. The interval-valued differential can be used to model dynamical systems subject to uncertainties. The papers [18–21] are focused on the interval-valued differential equations. These equations can be studied with a framework of the Hukuhara derivative [22]. However, it causes the solutions to have increasing lengths of their values. Stefanini and Bede [18] proposed to consider the so-called strongly generalized derivative of interval-valued functions. The interval-valued differential equations with this derivative can have solutions with decreasing lengths of their values. Some very important extensions of the interval-valued differential equations are the set differential equations [1, 6, 12–15, 19, 23–32], the fuzzy differential equations [10, 11, 33–45], and the fuzzy stochastic differential equations [46–51].
In the papers [18, 20, 21] one can find the studies on interval-valued differential equations under generalized Hukuhara differentiability, i.e., equations of the form
where is understood as the classical Hukuhara derivative and then as the second type Hukuhara derivative. The existence and uniqueness of a Cauchy problem is then obtained under the assumption that the coefficients satisfy a condition with the Lipschitz constant (see [18]). The proof is based on an application of the Banach fixed point theorem. In [21], under the generalized Lipschitz condition, Marek T Malinowski obtained the existence and uniqueness of solutions to both kinds of IDEs. In this paper, we study two kinds of solutions to IFDEs. The different types of solutions to IFDEs are generated by the usage of two different concepts of interval-valued derivative. Furthermore, in [4], Lupulescu established local and global existence and uniqueness results for fuzzy functional differential equations. Malinowski [6] studied the existence and uniqueness result of the solution to the delay set-valued differential equations under the condition that the right-hand side of the equation is Lipschitzian in the functional variable. Our direction of research is motivated by the results of Lupulescu [4, 5], Stefanini and Bede [18], and Malinowski [6, 20, 21, 52].
By using the concept of inner product on the fuzzy space which was introduced and studied in [5], we establish a new concept of inner product on the space (space of compact convex subsets of ℝ). With the help of this inner product we formulate some dissipative conditions for interval-valued functional differential equations and, under these conditions, we prove the global existence and uniqueness of the solution. The paper is organized as follows. Section 2 is devoted to the preliminaries of interval-valued analysis and interval-valued differential equations. In Section 3, we formulate the notion of inner product on and its properties. The global existence of a solution of IFDE is investigated in Section 4. In the last section, we give some examples being simple illustrations of the theory of the interval-valued functional differential equation.
2 Preliminaries
Let denote the family all nonempty, compact, and convex subsets of ℝ. The addition and scalar multiplication in we define as usual, i.e. for , , , where , , and , we have
Furthermore, let , , and , then we have and . Let as above. Then the Hausdorff metric H in is defined as follows:
It is well known that is a complete, separable, and locally compact metric space. We define the magnitude and the length of by
respectively, where is the zero element of , which is regarded as one point.
The Hausdorff metric (2.1) satisfies the following properties:
for all and . Let . If there exists an interval such that , then we call C the Hukuhara difference of A and B. We denote the interval C by . Note that . It is well known that exists in the case . Besides that, we can see the following properties for (see [20]):
-
If , exist, then .
-
If , exist, then .
-
If , exist, then there exist and .
-
If , , exist, then there exist and .
Definition 2.1 We say that the interval-valued mapping is continuous at the point if for every there exists such that, for all such that , one has .
The strongly generalized differentiability was introduced in [18] and studied in [20, 41–43].
Definition 2.2 Let and . We say that X is strongly generalized differentiable at t if there exists such that
-
(i)
for all sufficiently small, , and
or
-
(ii)
for all sufficiently small, , and
or
-
(iii)
for all sufficiently small, , and
or
-
(iv)
for all sufficiently small, , and the limits
(h at denominators means .) In this definition, case (i) ((i)-differentiability for short) corresponds to the classical H-derivative, so this differentiability concept is a generalization of the Hukuhara derivative. In this paper we consider only the first two of Definition 2.2. In the other cases, the derivative is trivial because it is reduced to a crisp element (more precisely, ). Further, we say that X is (i)-differentiable or that we have (ii)-differentiation on , if it is differentiable in the sense of (i) or (ii) of Definition 2.2, respectively.
Assume that is continuous. The interval-valued differential equation (1.1) is equivalent to one of the following integral equations:
if X is (i)-differentiable, and
if X is (ii)-differentiable, provided that the H-difference exists.
Corollary 2.1 (see [18, 20, 21])
Let be given. Denote for , where .
-
(i)
If the mapping X is (i)-differentiable (i.e. classically Hukuhara differentiable) at , then the real-valued functions , are differentiable at t and .
-
(ii)
If the mapping X is (ii)-differentiable at , then the real-valued functions , are differentiable at t and .
The following comparison principle is fundamental in the investigation of the global existence of solutions of interval-valued functional differential equations.
Theorem 2.1 ([53])
Let and satisfy the inequality
where , . Assume that is the maximal solution of the IVP
existing on . Then, if , we have , .
3 Inner product on space
In the following, for we consider the function , defined by
Lemma 3.1 ([5])
For every the following limit exists
In the following, for any we define the inner product product on by
From (3.3), we easily see that
Theorem 3.1 ([5])
The mappings are upper semicontinuous.
Lemma 3.2 For any , the following properties hold:
-
(i)
,
-
(ii)
,
-
(iii)
, for ,
-
(iv)
, for ,
-
(v)
, for .
Proof (i) For any and , we have
Passing to the limit when , we get (i). Analogously, for and , we obtain
So, by passing to the limit when , we get (ii). Equalities (iii), (iv), and (v) follow from (i). □
Based on Lemma 3.2, we easily obtain the following properties for the inner product on .
Corollary 3.1 For any , the following properties hold:
-
(i)
,
-
(ii)
, for ,
-
(iii)
, for ,
-
(iv)
.
Theorem 3.2 If is continuously differentiable on , then is continuously differentiable on and
for every .
Proof Since the way of the proof is similar for all four cases in Definition 2.2, we only consider the case of (i)-differentiability for X. Let be such that is fixed. Let , . Then we get
Since , we find that exists and
The proof is complete. □
Corollary 3.2 If is continuously differentiable on , then
for every .
Proof It is easy to prove this corollary with the result of Theorem 3.2 and (3.4). Indeed,
□
4 Global existence and uniqueness of solution of interval-valued functional differential equations
For let denote the space of continuous mappings from to . Define a metric in by
Let . Denote , . For any denote by the element of defined by for .
Let us consider the interval-valued functional differential equations (IFDEs) with generalized Hukuhara derivative of the form
where , and the symbol denotes the generalized Hukuhara derivative from Definition 2.2. By a solution to (4.1) we mean an interval-valued mapping that satisfies for , where X is differentiable on and for .
Lemma 4.1 ([6])
Assume that and . Then the interval-valued mapping belongs to .
Remark 4.1 ([6])
Under the assumptions of the lemma above we find that the mapping is integrable over the interval I.
Lemma 4.2 (cf. [6])
Assume that is continuous. An interval-valued mapping is called a local solution to the problem (4.1) on J if and only if X is a continuous interval-valued mapping and it satisfies one of the following interval-valued integral equations:
if X is (i)-differentiable.
if X is (ii)-differentiable.
Let us remark that in (4.3) is hidden the following statement: there exists a Hukuhara difference .
Definition 4.1 Let be an interval-valued function which is (i)-differentiable. If X and its derivative satisfy problem (4.1), we say X is a (i)-solution of problem (4.1). An (i)-solution is unique if we have , for any which is a (i)-solution of (4.1).
Definition 4.2 Let be an interval-valued function which is (ii)-differentiable. If X and its derivative satisfy problem (4.1), we say that X is a (ii)-solution of problem (4.1). An (ii)-solution is unique if we have , for any which is a (ii)-solution of (4.1).
Next, we shall establish the global existence and uniqueness result for IFDE (4.1). For the global existence and uniqueness we use the dissipative conditions. We now prove a comparison theorem, which is a useful tool in proving the global existence and uniqueness theorem.
Theorem 4.1 Assume that and for , , where . Let be the maximal solution of
for . Then, if , are any (ii)-solutions (or (i)-solutions) of IFDE (4.1) such that exist for , we have , provided that .
Proof We prove that for the case of (ii)-differentiability, the proof of the other case is similar. Since X, Y are solutions of IFDE (4.1) and are (ii)-differentiable, we find that, for small enough, there exist Hukuhara differences , . Now for , set , and we have
from which we get
Taking lim inf as yields
which together with the fact that , and by using Theorem 2.1, results in
The proof is complete. □
Remark 4.2 Under the assumptions of Theorem 4.1 with satisfying , where , we obtain .
Theorem 4.2 Assume that , F maps bounded sets on to bounded sets and F satisfies the following dissipative condition: there exists a continuous function and such that
for every and with . Then:
-
(i)
the interval-valued functional differential equation (4.1) has a (i)-solution on ;
-
(ii)
the interval-valued functional differential equation (4.1) has a (ii)-solution on if the following conditions hold:
Proof We prove that for the case of (i)-differentiability (i.e., case (i)), the proof of the other case is similar. For fixed, we consider the sets and , where and . Since F maps bounded sets onto bounded sets, we deduce that there exists such that , for every . Therefore, there exists such that the problem (4.1) has at least locally a (i)-solution on some intervals . Let
Then . Next, we define a partial order ≲ on as follows: if and only if and on . Then standard application of Zorn’s lemma assures the existence of the maximal element Z in . The proof is complete if we show that . Suppose that it is not true, so that .
Step 1: We show that there exists γ such that for every . Define . Using Corollary 3.2 we have
which implies
and so
Now, we consider the sets and . Then we have
for every . On the other hand, since F maps bounded sets onto bounded sets, there exists such that
and so, by Corollary 3.1, it follows that
From (4.7), (4.8) we obtain
and so (4.6) becomes
If we let , then we have
By Gronwall’s lemma, from the inequality (4.9) we obtain
for every . Here . Therefore, there exists such that on . We infer that , for every .
Step 2: We show that is Lipschitzian on . Indeed, for all with , we have
Therefore has a continuous extension on . By continuity, we have
This implies that is a (ii)-solution of (4.1) on . Further, we consider IFDE with the new initial function at
Then the last IFDE has a (ii)-solution on , . If we define
then it is clear that is a (ii)-solution of (4.1) on . This contradicts the maximality of and hence . □
Theorem 4.3 Assume that the assumptions of Theorem 4.2 hold. In addition, we assume that and for , , where , , and is only a solution of
for . Then, for every , the interval-valued functional differential equation (4.1) has a unique solution on for each case.
Proof We prove it for the case of (ii)-differentiability, the proof of the other case is similar. Let X and Y be two (ii)-solutions of IFDE (4.1) and be (ii)-differentiable, then we find that, for small enough, there exist Hukuhara differences , . Now for , set and noting that , we get
and so, by the comparison theorem for ordinary delay differential equations, it gives
where is the solution of (4.10) on . Since by assumption , we obtain on . The proof is complete. □
Corollary 4.1 Under the assumptions of Theorem 4.2, if we suppose in addition that there exists such that
on , then the IFDE (4.1) has a unique global solution for each case.
5 Illustrations
Let us consider again the interval-valued functional differential equation with the initial value condition
where is a continuous interval-valued function and . We denote
and
In (5.1), we shall solve it by two types of Hukuhara derivative, which are defined in Definition 2.2. Consequently, based on the type of differentiability, we have the following two cases.
If we consider the derivative of by using (i)-differentiability, then from Corollary 2.1, we have , for . Therefore, (5.1) is translated into the following delay differential system:
If we consider the derivative of by using (ii)-differentiability, then from Corollary 2.1, we have , for . Therefore, (5.1) is translated into the following delay differential system:
Remark 5.1 If we ensure that the solutions of the systems (5.2) and (5.3), respectively, are valid sets of interval-valued functions and if the derivatives are valid sets of interval-valued functions with two kinds of differentiability, respectively, then we can construct the solution of the interval-valued functional differential equation (5.1).
Next, we shall consider some examples being simple illustrations of the theory of interval-valued functional differential equations.
Example 5.1 Let us consider the linear interval-valued functional differential equation under two kinds Hukuhara derivative,
where , . In this example we shall solve (5.4) on .
Case 1: Considering (i)-differentiability, problem (5.4) is translated into the following delay system:
Solving delay system (5.5) by using the method of steps, we obtain a unique (i)-solution to (5.4) defined on and it is of the form
The (i)-solution is illustrated in Figure 1.
Case 2: Considering (ii)-differentiability, problem (5.4) is translated into the following delay system:
We obtain a unique (ii)-solution to (5.4) defined on and it is of the form
The (ii)-solution is illustrated in Figure 2.
Example 5.2 Let us consider the linear interval-valued functional differential equation under two kinds of Hukuhara derivative,
where , , , . In this example we shall solve (5.7) on .
One can obtain the (i)-solution and (ii)-solution by using the methods as in the above examples.
The (i)-solution to (5.7) defined on and it is of the form
The (ii)-solution to (5.7) is defined on and it is of the form
In Figures 3 and 4, the (i)-solution and (ii)-solution curves of (5.7) are given.
Example 5.3 Consider the interval-valued functional differential equation
where is given by and , , .
Case 1: Consider . If we consider in the sense of (i)-differentiability, we have to solve the following delay differential system:
Following the method of steps, we can obtain the expression of the (i)-solution on , which is represented in Figure 5.
On the other hand, if is (ii)-differentiable, then we have to solve the following delay differential system:
By the method of steps, we can obtain the expression of the (ii)-solution on corresponding to the initial condition . This solution is illustrated in Figure 6.
Case 2: Consider . If we consider in the form where it is (i)-differentiable, we have to solve the following delay differential system:
This solution is shown in Figure 7.
If we consider in the sense of (ii)-differentiability, we have to solve the following delay differential system:
The (ii)-solution is illustrated in Figure 8.
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The authors would like to express their gratitude to the anonymous referees for their helpful comments and suggestions, which have greatly improved the paper. The second named author would like to thank the University of Technical Education, Ho Chi Minh City, Vietnam.
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Phu, N.D., An, T.V., Hoa, N.V. et al. Interval-valued functional differential equations under dissipative conditions. Adv Differ Equ 2014, 198 (2014). https://doi.org/10.1186/1687-1847-2014-198
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DOI: https://doi.org/10.1186/1687-1847-2014-198