Global existence of solutions for interval-valued second-order differential equations under generalized Hukuhara derivative
© Ngo and Nguyen; licensee Springer. 2013
Received: 18 April 2013
Accepted: 27 August 2013
Published: 7 November 2013
In this paper, we present a generalized concept of second-order differentiability for interval-valued functions. The global existence of solutions for interval-valued second-order differential equations (ISDEs) under generalized H-differentiability is proved. We present some examples of linear interval-valued second-order differential equations with initial conditions having four different solutions. Finally, we propose a new algorithm based on the analysis of a crisp solution. Some examples are given by using our proposed algorithm.
MSC:34K05, 34K30, 47G20.
The set-valued differential and integral equations are an important part of the theory of set-valued analysis. They have the important value of 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 the basic theory on set-valued differential and integral equations, the readers can be referred to the good books and papers [2–10] and references therein. The interval-valued analysis and interval 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. Interval-valued differential and integro-differential equations are a natural way 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., [11–15]). There are several approaches to the study of interval differential equations. One popular approach is based on H-differentiability. The approach based on H-derivative has the disadvantage that it leads to solutions which have an increasing length of their support. Note that this definition of derivative is very restrictive; for instance, we can see that if , where A is an interval-valued constant and is a real function with , then is not differentiable. Recently, to avoid this difficulty, Stefanini and Bede  solved the above mentioned approach under strongly generalized differentiability of interval-valued functions. In this case, the derivative exists and the solution of an interval-valued differential equation may have decreasing length of the support, but the uniqueness is lost. The paper of Stefanini and Bede was a starting point for the topic of interval-valued differential equations (see [14–16]) and later also for fuzzy differential equations.
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. In  authors considered n th-order fuzzy differential equations with initial value conditions and proved the local existence and uniqueness of solution for nonlinearities satisfying a Lipschitz condition. In [12, 19] authors studied the existence of the solutions to interval-valued and fuzzy differential equations involving generalized differentiability. Based on the results in , authors studied the local existence and uniqueness of solutions for fuzzy second-order integro-differential equations with fuzzy kernel under strongly generalized H-differentiability . In n th-order fuzzy differential equations (NFDEs) under generalized differentiability were discussed and the existence and uniqueness theorem of solution of NFDE was proved by using the contraction principle in a Banach space. In  authors developed the fuzzy improved Runge-Kutta-Nystrom method for solving a second-order fuzzy differential based on the generalized concept of higher-order fuzzy differentiability. Also, a very important generalization and 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 the generalized Hukuhara derivative. Recently, several works, e.g., [13, 23–43], have been done on set-valued differential equations, fuzzy differential equations, and fuzzy integro-differential equations, fractional fuzzy differential equations, and some methods for solving fuzzy differential equations [44–49].
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.
The different types of solutions to ISDEs are generated by the usage of two different concepts of interval-valued second-order derivative. This direction of research is motivated by the results of Stefanini and Bede , Malinowski [14, 15] concerning deterministic IDEs with generalized interval-valued derivative.
This paper is organized as follows. In Section 2, we recall some basic concepts and notations about interval analysis and interval-valued differential equations. In Section 3, we present the global existence and uniqueness theorem of a solution to the interval-valued second-order differential equation under two kinds of the Hukuhara derivative. Some examples of linear second-order interval-valued differential equations with initial conditions having four different solutions are presented. In Section 4, we propose a new algorithm based on the analysis of a crisp solution.
We notice that is a complete, separable and locally compact metric space.
respectively, where is the zero element of , which is regarded as one point.
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 .
- (i)for all sufficiently small, , and
- (ii)for all sufficiently small, , and
- (iii)for all sufficiently small, , and
- (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.
on the interval , under the strong differentiability condition, (i) or (ii), respectively. We notice that the equivalence between two equations in this lemma means that any solution is a solution for the other one.
- (i)for all sufficiently small, , and the following limits hold (in the metric H)
- (ii)for all sufficiently small, , and the following limits hold (in the metric H)
- (iii)for all sufficiently small, , and the following limits hold (in the metric H)
- (iv)for all sufficiently small, , and the following limits hold (in the metric H)
where is an interval-valued function of t, is an interval-valued function and the interval-valued variables , are defined as the derivatives of and , respectively.
, where X and are (i)-differentiable;
, where X is (i)-differentiable and is (ii)-differentiable;
, where X is (ii)-differentiable and is (i)-differentiable;
, where X and are (i)-differentiable.
- (a)The space of continuous functions X by with the distance(2.3)
- (b)The space of continuous functions by with the distance(2.4)
where we assume that , exist and .
In this definition, we know that the space of continuous functions is a complete metric space with distance (2.3).
Lemma 2.2 is a complete metric space.
Based on Definition 2.3, there are two possible cases for each derivation, so there are four possible cases for the second-order derivation.
, where and are (i)-differentials, or
, where is (i)-differential and is (ii)-differential, or
, where is (ii)-differential and is (i)-differential, or
, where and are (ii)-differentials.
Definition 2.5 Let , be interval-valued functions which are (i)-differentiable. If X, and their derivatives satisfy problem (2.2), we say X is a (i-i)-solution of problem (2.2).
Definition 2.6 Let , be interval-valued functions which are (i)-differentiable and (ii)-differentiable, respectively. If X, and their derivatives satisfy problem (2.2), we say X is a (i-ii)-solution of problem (2.2).
Definition 2.7 Let , be interval-valued functions which are (ii)-differentiable and (i)-differentiable, respectively. If X, and their derivatives satisfy problem (2.2), we say X is a (ii-i)-solution of problem (2.2).
Definition 2.8 Let , be interval-valued functions which are (ii)-differentiable. If X, and their derivatives satisfy problem (2.2), we say X is a (ii-ii)-solution of problem (2.2).
for all , . Then problem (2.2) has a unique local solution on some intervals () for each case.
3 Global existence of solutions for interval-valued second-order differential equations
for all , where .
Theorem 3.1 Assume that
(C1) is locally Lipschitzian in X, ;
exists throughout I;
(C3) , and .
Then the largest interval of the existence of any solution of (3.1) for each case is I. In addition, if is bounded on I, then exists.
Therefore can be extended beyond β, which contradicts our assumption. In addition, since is bounded and nondecreasing on I, it follows that exists and is finite. □
where we assume that are continuous functions, then the solutions of (3.3) for each case are on .
Therefore, the solutions of Example 3.2 are on .
Theorem 3.3 Assume that
(C4) , F is bounded on bounded sets, and there exists a local solution of problem (3.1) for every ;
(C6) the maximal solution of problem (3.2) exists on and is positive whenever .
Then, for every such that , problem (3.1) has a (i-i)-solution on , which satisfies the estimate , .
Proof Let denote the set of all functions X defined on with values in such that is a (i-i)-solution of problem (3.1) on and , . We define a partial order ≤ on as follows: the relation implies that and on . We shall first show that is nonempty. Indeed, by assumption (C4), there exists a (i-i)-solution of problem (3.1) defined on .
Therefore is a (i-i)-solution of problem (3.1) on , and, by repeating the arguments that were used to obtain (3.6), we get , . This contradicts the maximality of Z, and hence . The proof is complete. □
Proof One can obtain these results easily by using the same methods as in the proof of Theorem 3.3. □
where , .
It is easily seen that a little change of initial conditions or a little change of the right-hand side of the equation causes a little change of corresponding solutions.
where a, b are positive constants, , and is a continuous interval-valued function on , and . Our strategy of solving (3.14) is based on the choice of the derivative in the interval-valued differential equation. In order to solve (3.14), we have three steps: first we choose the type of derivative and change problem (3.14) to a system of ODE by using Theorem 2.1 and considering initial values. Second we solve the obtained ODE system. The final step is to find such a domain in which the solution and its derivatives have valid sets, i.e., we ensure that , and are valid sets.
By using Theorem 2.1, we obtain that four ODEs systems are possible for problem (3.14), as follows.
By solving (3.21), we get is not (i)-differentiable, there is no solution in this case.
By solving (3.23), we have is not (ii)-differentiable on . Therefore, there is no solution in this case.
By solving (3.25), we obtain . Since is not (i)-differentiable, there is no solution in this case.
By solving (3.26), we get . Since, is not (ii)-differentiable, there is no solution in this case.
4 A new method for solving interval-valued second-order differential equations
In  the author applied the modification of Laplace decomposition method (LDM) to solve nonlinear interval-valued Volterra integral equations. The modified LDM can be applied to derive the lower and upper solutions. In  Salahshour and Allahviranloo solved the second-order fuzzy differential equation under generalized H-differentiability by using fuzzy Laplace transforms. Now, in Section 3 we see that some second-order IDEs are solved under generalized H-differentiability. For the second-order IDEs, there exist at most four solutions, but all the obtained solutions may not be acceptable. It is an important result in the theory of IDEs of higher order. Notice that by applying the above approach (in Section 3), the original second-order IDEs are transformed to four ordinary differential systems. However, by applying Salahshour et al.’s  approach, each solution can be obtained directly. In this section, we propose a new algorithm based on the analysis of a crisp solution. We establish a synthesis of a crisp solution of an interval-valued initial value problem and the method proposed in Kaleva  and Akin .
From two min and max problems, we notice that it is not an easy task to determine the result of the min and max problems, so we propose the following method.
where is a real-valued solution of (4.3), , , , and .
Step 2: Next, we investigate the behavior of the crisp solution in (4.3) and the following four cases according to domains:
+ Let the domain where , be .
+ Let the domain where , be .
+ Let the domain where , be .
+ Let the domain where , be .
Step 3: After we obtain the four domains above, we set:
+ , and in domain .
+ , and in domain .
+ , and in domain .
+ , and in domain .
Step 4: Substitute the obtained domains in (4.1). Then we obtain that four ODEs systems are possible for problem (4.1), as follows:
In this paper, we have obtained a global existence result for a solution to interval-valued second-order differential equations. Also, we have proposed a new algorithm based on the analysis of a crisp solution. The proposed method in this paper may be useful if the coefficients, initial values and forcing terms are interval. If the crisp problem is not solvable explicitly, we cannot determine the mentioned domains precisely.
The authors would like to express their gratitude to the anonymous referees for their helpful comments and suggestions, which have greatly improved the paper.
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