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Interval-valued functional integro-differential equations
Advances in Difference Equations volume 2014, Article number: 177 (2014)
This paper is devoted to studying the local and global existence and uniqueness results for interval-valued functional integro-differential equations (IFIDEs). In the paper, for the local existence and uniqueness, the method of successive approximations is used and for the global existence and uniqueness, the contraction principle is a good tool in investigating. Some examples are given to illustrate the results.
MSC:34G20, 34A12, 34K30.
Functional differential equations (or, as they are 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 amount of literature dealing with functional differential equations and their applications; the reader is referred to the monographs [1–6] and the references therein.
The set-valued differential and integral equations are an important part of the theory of set-valued analysis. They have an important value in theory and application in control theory; and they were studied in 1969 by De Blasi and Iervolino . Recently, set-valued differential equations have been studied by many authors due to their application in many areas. For many results in the theory of set-valued differential and integral equations, the readers can be referred to the following books and papers [8–23] and the references therein. The interval-valued analysis and interval-valued differential equations (IDEs) are the particular cases of the set-valued analysis and set differential equations, respectively. In many cases, when modeling real-world phenomena, information about the behavior of a dynamic system is uncertain, and we have to consider these uncertainties to gain more models. The interval-valued differential and integro-differential equations can be used to model dynamic systems subject to uncertainties. Recently, many works have been done by several authors in the theory of interval-valued differential equations (see, e.g., [24–26]). These equations can be studied with a framework of the Hukuhara derivative . However, it causes that the solutions have increasing length of their values. Stefanini and Bede  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 length of their values. This approach was the starting point for the topic of interval-valued differential equations (see [24, 25]). Besides that, some very important extensions of the interval-valued differential equations are the set differential equations (see [6, 8, 11–14, 16, 18, 20, 23, 28–31]).
The connection between the fuzzy analysis and the interval analysis is very well known (Moore and Lodwick ). Interval analysis and fuzzy analysis were introduced as an attempt to handle interval uncertainty that appears in many mathematical or computer models of some deterministic real-world phenomena. Based on the results in , there are some very important extensions, and the development related to the subject of the present paper is in the field of fuzzy sets, i.e., fuzzy calculus and fuzzy differential equations under generalized Hukuhara derivative. Recently, several works, e.g., [5, 9, 10, 30, 34–45], have been done on fuzzy differential equations and fuzzy integro-differential equations, the fuzzy stochastic differential equations [46–51], fractional fuzzy differential equations [30, 52–55], and some methods for solving fuzzy differential equations [56, 57].
where denotes two kinds of derivatives, namely the classical Hukuhara derivative and the second-type Hukuhara derivative (generalized Hukuhara differentiability). The existence and uniqueness of a Cauchy problem is then obtained under an assumption that the coefficients satisfy a condition with the Lipschitz constant (see ). The proof is based on the application of the Banach fixed point theorem. In , under the generalized Lipschitz condition, Malinowski obtained the existence and uniqueness of solutions to both kinds of IDEs. In this paper, we study two kinds of solutions to IFIDEs. The different types of solutions to IFIDEs are generated by the usage of two different concepts of interval-valued derivative. Furthermore, in , Lupulescu established the local and global existence and uniqueness results for fuzzy functional differential equations. Malinowski  studied the existence and uniqueness result of solution to the delay set-valued differential equations under the condition that the right-hand side of an equation is Lipschitzian in the functional variable. Inspired and motivated by the results of Stefanini and Bede , Malinowski [24, 25] and Lupulescu , we consider the interval-valued functional integro-differential equations under generalized Hukuhara derivative. The paper is organized as follows. As preliminaries, we recall some basic concepts and notations about interval analysis and interval-valued differential equations. In Section 3, we present the local and global existence and uniqueness theorem of solution of IFIDEs under generalized Hukuhara derivatives. In the last section, we give some examples as simple illustrations of the theory of interval-valued functional integro-differential equations.
Let us denote by the set of any nonempty compact intervals of the real line ℝ. The addition and scalar multiplication in are defined as usual, i.e., for , , , where , , and , then we have
Furthermore, let , and , then we have and . Let as above, the Hausdorff metric H in is defined as follows:
It is known that is a complete, separable and locally compact metric space. We define the magnitude and the length of by
respectively, where 0 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 known that exists in the case . Besides that, we can see the following properties for (see ):
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 with .
Definition 2.2 Let and . We say that X is strongly generalized differentiable at t if there exists such that
for all sufficiently small, , and
for all sufficiently small, , and
for all sufficiently small, , and
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 two first 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 (ii)-differentiable on , if it is differentiable in the sense (i) or (ii) of Definition 2.2, respectively.
Theorem 2.1 Let be (i)-differentiable or (ii)-differentiable on , and assume that the derivative is integrable over . We have
if X is (i)-differentiable on , then ;
if X is (ii)-differentiable on , then .
Provided that, the above Hukuhara differences exist.
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.
The following well-known result is useful in the next section.
Lemma 2.2 Let , and be real-valued nonnegative continuous functions defined on , is a constant for which the inequality
holds for all . Then
Let be given. Denote for , where .
If the mapping X is (i)-differentiable (i.e., classical Hukuhara differentiable) at , then the real-valued functions , are differentiable at t and .
If the mapping X is (ii)-differentiable at , then the real-valued functions , are differentiable at t and .
3 Main results
For a positive number σ, we denote by 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 integro-differential equations (IFIDEs) with the generalized Hukuhara derivative under the form
where , , and the symbol denotes the generalized Hukuhara derivative from Definition 2.2. By a solution to equation (3.1) we mean an interval-valued mapping that satisfies for , X is differentiable on and for . We note that the solution in this sense is considered just one-side differentiable at (specifically, right-differentiable at ).
Lemma 3.1 Assume that , and . Then the interval-valued mapping belongs to .
Remark 3.1 Under assumptions of the lemma above, the mapping is integrable over the interval I.
Remark 3.2 If , are continuous and , then the mapping is bounded on I. Also, the function is bounded on I.
Lemma 3.2 Assume that , are continuous. An interval-valued mapping is called a local solution to problem (3.1) 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. We remark that in (3.3), the following statement is hidden: there exists the Hukuhara difference .
Proof We prove the case of (ii)-differentiability, the proof of the other case being similar. Assume that is a solution to problem (3.1). Hence X is (ii)-differentiable on and is integrable as a continuous function. Applying Theorem 2.1, we obtain that
for every . Since and for , we easily obtain
To show that the opposite implication is true, let us assume that is a continuous interval-valued mapping and it satisfies equation (3.3). Equation (3.3) allows us to claim that and that there exists the Hukuhara difference
Now, let and h be such that . We observe that
Indeed, we have by direct computation
Similarly to (3.4), we can obtain
for . Multiplying (3.4)-(3.5) by and passing to limit with , we have by Definition 2.2 that X is (ii)-differentiable, and consequently
Indeed, we have, for every ,
Since F, G are continuous, for , we obtain
Proceeding as above, we can obtain
The proof is complete. □
Definition 3.1 Let be an interval-valued function which is (i)-differentiable. If X and its derivative satisfy problem (3.1), we say that X is (i)-solution of problem (3.1). (i)-solution is unique if it holds for any which is (i)-solution of (3.1).
Definition 3.2 Let be an interval-valued function which is (ii)-differentiable. If X and its derivative satisfy problem (3.1), we say that X is (ii)-solution of problem (3.1). (ii)-solution is unique if it holds for any which is (ii)-solution of (3.1).
Theorem 3.1 Let and suppose that , satisfy the conditions: there exists a constant such that
for every , and . Moreover, there exists such that . Then the following successive approximations given by
for the case of (i)-differentiability, and
for the case of (ii)-differentiability (where such that equation (3.7) is well defined, i.e., the foregoing Hukuhara differences do exist), converge uniformly to two unique solutions and of (3.1), respectively, on where .
Proof We prove that for the case of (ii)-differentiability, the proof of the other case is similar. From assumptions of the theorem, we have
for . Further, for every and , we get
In particular, from (3.7) it follows that
Therefore, by mathematical induction, for every and ,
In inequality (3.8), are balancing constants. We observe that for every , the function is continuous. Indeed, since , is continuous on . We see that
Thus, by mathematical induction, for every , we deduce that
as . A similar inequality is obtained for as . In the sequel, we shall show that for the Cauchy convergence condition is satisfied uniformly in t, and as a consequence is uniformly convergent. For , from (3.8) we obtain
The convergence of this series implies that for any we find large enough such that for ,
Since is a complete metric space and (3.9) holds, the sequence is uniformly convergent to a mapping . We shall show that is (ii)-solution to (3.1). Since for every and every , we easily have . For and ,
as for any . Consequently, we have
We infer that
for every . Therefore, is the (ii)-solution of (3.1), due to Lemma 3.2 it follows that is the (ii)-solution of (3.1). For the uniqueness of the (ii)-solution , let us assume that are two solutions of (3.3). By definition of the solution, if . Note that for ,
If we put , , then we obtain
and by Lemma 2.2 we obtain that on . This proves the uniqueness of the (ii)-solution for (3.1) □
Remark 3.3 The existence and uniqueness results for solutions of problem (3.1) can be obtained by using the contraction principle.
Now, we present the studies and results concerning the global existence and uniqueness of two solutions for (3.1), each one corresponding to a different type of differentiability, by using the contraction principle, which was studied in  for fuzzy functional differential equations. In the following, for a given , we consider the set of all continuous interval-valued functions such that on and . On we can define the following metric:
where is chosen suitably later. It is easy to prove that the space of continuous interval-valued functions is a complete metric space with distance (3.10).
Theorem 3.2 Assume that
, and there exists a constant such that
for all and ;
there exist and such that
for all , where .
the interval-valued functional integro-differential equation (3.1) has (i)-solution on ;
the interval-valued functional integro-differential equation (3.1) has (ii)-solution on if the following condition holds:(3.11)
Proof Since the way of the proof is similar for both cases, we only consider the case of (ii)-differentiability for X. Note that the space under inequality (3.11) depends on the positive constant k, the functions F, G and the initial condition φ. In , the continuity of F, G guarantees that under inequality (3.11) is a closed set in , so that under inequality (3.11) is a complete metric space considering the distance . We consider the complete metric space and define an operator
We can choose a big enough value for k such that is a contraction, so the Banach fixed point theorem provides the existence of a unique fixed point for , that is, a unique solution for (3.1).
First, we shall prove that , i.e., the operator T is well defined, with assumption . Indeed, let . For each , we get
Since , there exists such that for all . Therefore, for all , we obtain
We infer that .
Next, we shall prove that is a contraction by metric . Let . Then, for , . For each , we have
Choosing and , it follows that the operator on is a contraction. Using the Banach fixed point theorem provides the existence of a unique fixed point for , and the unique fixed point of is in the space , that is, a unique solution for (3.1) in the case of (ii)-differentiability. □
In this part, some simple examples are given to illustrate the theory of IFIDEs. We shall consider IFIDEs (3.1) with (i) and (ii) derivatives, respectively. Let us start the illustrations with considering the following interval-valued functional integro-differential equation:
where , . Let . By using Corollary 2.1, 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, (4.1) is translated into the following delay integro-differential system:
If we consider the derivative of by using (ii)-differentiability, then from Corollary 2.1, we have for . Therefore, (4.1) is translated into the following delay integro-differential system:
Remark 4.1 If we ensure that the solutions of systems (4.2) and (4.3) 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 interval-valued functional differential equation (4.1).
Next, we shall consider two examples being a simple illustration for the theory of interval-valued functional integro-differential equations.
Example 4.1 Let us consider the linear interval-valued functional integro-differential equation (with ) under two kinds of Hukuhara derivatives
where , . In this example we shall solve (4.4) on .
Case 1. Considering (i)-differentiability, problem (4.4) is translated into the following delay system:
Solving delay system (4.5) by using the method of steps, we obtain a unique (i)-solution to (4.4) defined on and it is of the form
The (i)-solution is illustrated in Figure 1.
Case 2. Considering (ii)-differentiability, problem (4.4) is translated into the following delay system:
We obtain a unique (ii)-solution to (4.4) defined on and it is of the form
The (ii)-solution is illustrated in Figure 2.
Example 4.2 Let us consider the linear interval-valued functional integro-differential equation under two kinds of Hukuhara derivatives
where , . In this example we shall solve (4.7) on .
Case 1. () From (4.2), we get
Following the method of steps, we obtain the (i)-solution to (4.7) defined on and it is of the form
From (4.3) we obtain
The (ii)-solution to (4.7) defined on is of the form
Case 2. () From (4.2) we get
By solving delay integro-differential system (4.10), we obtain (i)-solution
The (i)-solution of (4.7) on is illustrated in Figure 5.
From (4.3) we obtain
By solving delay integro-differential systems (4.11), we obtain (ii)-solution
The (ii)-solution of (4.7) on is illustrated in Figure 6.
In this study, we have established the local and global existence and uniqueness results of two solutions for interval-valued functional integro-differential equations. For the local existence and uniqueness, we use the method of successive approximations under the Lipschitz condition, and for global existence and uniqueness, we use the contraction principle under suitable conditions. In our further work, we would like to use these results to study the local and global existence and uniqueness results of solutions for interval-valued functional integro-differential equations under Caputo-type interval-valued fractional derivatives.
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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 authors declare that they have no competing interests.
All authors contributed equally to the writing of this paper. All authors read and approved the final version of the manuscript.
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Hoa, N.V., Phu, N.D., Tung, T.T. et al. Interval-valued functional integro-differential equations. Adv Differ Equ 2014, 177 (2014). https://doi.org/10.1186/1687-1847-2014-177
- interval-valued differential equations
- generalized Hukuhara derivative
- functional integro-differential equations