© A. Khastan et al. 2009
Received: 30 April 2009
Accepted: 1 July 2009
Published: 20 July 2009
We firstly present a generalized concept of higher-order differentiability for fuzzy functions. Then we interpret th-order fuzzy differential equations using this concept. We introduce new definitions of solution to fuzzy differential equations. Some examples are provided for which both the new solutions and the former ones to the fuzzy initial value problems are presented and compared. We present an example of a linear second-order fuzzy differential equation with initial conditions having four different solutions.
The term "fuzzy differential equation" was coined in 1987 by Kandel and Byatt  and an extended version of this short note was published two years later . There are many suggestions to define a fuzzy derivative and in consequence, to study fuzzy differential equation . One of the earliest was to generalize the Hukuhara derivative of a set-valued function. This generalization was made by Puri and Ralescu  and studied by Kaleva . It soon appeared that the solution of fuzzy differential equation interpreted by Hukuhara derivative has a drawback: it became fuzzier as time goes by . Hence, the fuzzy solution behaves quite differently from the crisp solution. To alleviate the situation, Hüllermeier  interpreted fuzzy differential equation as a family of differential inclusions. The main shortcoming of using differential inclusions is that we do not have a derivative of a fuzzy-number-valued function.
The strongly generalized differentiability was introduced in  and studied in [9–11]. This concept allows us to solve the above-mentioned shortcoming. Indeed, the strongly generalized derivative is defined for a larger class of fuzzy-number-valued functions than the Hukuhara derivative. Hence, we use this differentiability concept in the present paper. Under this setting, we obtain some new results on existence of several solutions for th-order fuzzy differential equations. Higher-order fuzzy differential equation with Hukuhara differentiability is considered in  and the existence and uniqueness of solution for nonlinearities satisfying a Lipschitz condition is proved. Buckley and Feuring  presented two different approaches to the solvability of th-order linear fuzzy differential equations.
Here, using the concept of generalized derivative and its extension to higher-order derivatives, we show that we have several possibilities or types to define higher-order derivatives of fuzzy-number-valued functions. Then, we propose a new method to solve higher-order fuzzy differential equations based on the selection of derivative type covering all former solutions. With these ideas, the selection of derivative type in each step of derivation plays a crucial role.
In this section, we give some definitions and introduce the necessary notation which will be used throughout this paper. See, for example, .
Remark 2.2 (see ).
The following properties are wellknown:
3. Generalized Fuzzy Derivatives
The concept of the fuzzy derivative was first introduced by Chang and Zadeh ; it was followed up by Dubois and Prade  who used the extension principle in their approach. Other methods have been discussed by Puri and Ralescu , Goetschel and Voxman , Kandel and Byatt [1, 2]. Lakshmikantham and Nieto introduced the concept of fuzzy differential equation in a metric space . Puri and Ralescu in  introduced H-derivative (differentiability in the sense of Hukuhara) for fuzzy mappings and it is based on the -difference of sets, as follows. Henceforth, we suppose for
The above definition is a straightforward generalization of the Hukuhara differentiability of a set-valued function. From [6, Proposition 4.2.8], it follows that Hukuhara differentiable function has increasing length of support. Note that this definition of derivative is very restrictive; for instance, in , the authors showed that if where is a fuzzy number and is a function with , then is not differentiable. To avoid this difficulty, the authors  introduced a more general definition of derivative for fuzzy-number-valued function. In this paper, we consider the following definition .
In the previous definition, (1)-differentiability corresponds to the H-derivative introduced in , so this differentiability concept is a generalization of the H-derivative and obviously more general. For instance, in the previous example, for with we have .
In , the authors consider four cases for derivatives. Here we only consider the two first cases of [9, Definition 5]. In the other cases, the derivative is trivial because it is reduced to crisp element (more precisely, . For details, see [9, Theorem 7]).
Now we introduce definitions for higher-order derivatives based on the selection of derivative type in each step of differentiation. For the sake of convenience, we concentrate on the second-order case.
For a given fuzzy function , we have two possibilities (Definition 3.2) to obtain the derivative of ot : and . Then for each of these two derivatives, we have again two possibilities: and respectively.
This definition is consistent. For example, if is and -differentiable simultaneously at , then is (1)- and (2)-differentiable around . By remark in , is a crisp function in a neighborhood of .
We present the details only for the case (i), since the other cases are analogous.
This completes the proof of the theorem.
Let be a positive integer number, pursuing the above-cited idea, we write to denote the th-derivatives of at with for . Now we intend to compute the higher derivatives (in generalized differentiability sense) of the -difference of two fuzzy functions and the product of a crisp and a fuzzy function.
By Definition 3.2 the statement of the lemma follows easily.
By [9, Remark 16], let and define by , for all . If is differentiable on then is differentiable on , with . By Theorem 3.12, if then is (1)-differentiable on . Also if then is (2)-differentiable on . If , by [9, Theorem 10], we have . We can extend this result to second-order differentiability as follows.
Cases (i) and (iv) follow from Theorem 3.13. To prove (ii), since , by Remark 3.14, is (1)-differentiable and we have on . Also, since , then is (2)-differentiable and we conclude the result. Case (iii) is similar to previous one.
we have the following
(v)for : we have , then by [9, Theorem 10] we have , again by applying this theorem, we get
4. Second-Order Fuzzy Differential Equations
where , and is a continuous fuzzy function on some interval . The interval can be for some or . In this paper, we suppose Our strategy of solving (4.1) is based on the selection of derivative type in the fuzzy differential equation. We first give the following definition for the solutions of (4.1).
Let be an -solution for (4.1). To find it, utilizing Theorems 3.6 and 3.9 and considering the initial values, we can translate problem (4.1) to a system of second-order linear ordinary differential equations hereafter, called corresponding -system for problem (4.1).
Therefore, four ODEs systems are possible for problem (4.1), as follows:
Suppose is the -solution of problem (4.1). According to the Definition 4.1, then and exist and satisfy problem (4.1). By Theorems 3.6 and 3.9 and substituting and their derivatives in problem (4.1), we get the -system corresponding to -solution. This completes the proof.
The previous theorems illustrate the method to solve problem (4.1). We first choose the type of solution and translate problem (4.1) to a system of ordinary differential equations. Then, we solve the obtained ordinary differential equations system. Finally we find such a domain in which the solution and its derivatives have valid level sets and using Stacking Theorem  we can construct the solution of the fuzzy initial value problem (4.1).
We see that the solution of fuzzy differential equation (4.1) depends upon the selection of derivatives. It is clear that in this new procedure, the unicity of the solution is lost, an expected situation in the fuzzy context. Nonetheless, we can consider the existence of four solutions as shown in the following examples.
For (1,2)-solution, we get the following solutions for (1,2)-system:
Finally, (2-2)-system gives
Then we have an example of a second-order fuzzy initial value problem with four different solutions.
To find (1,1)-solution, we have
For (1,2)-solutions we deduce
For (2,1)-solutions we get
Finally, to find (2,2)-solution, we find
We then have a linear fuzzy differential equation with initial condition and two solutions.
5. Higher-Order Fuzzy Differential Equations
Selecting different types of derivatives, we get several solutions to fuzzy initial value problem for second-order fuzzy differential equations. Theorem 4.2 has a crucial role in our strategy. To extend the results to th-order fuzzy differential equation, we can follow the proof of Theorem 4.2 to get the same results for derivatives of higher order. Therefore, we can extend the presented argument for second-order fuzzy differential equation to th-order. Under generalized derivatives, we would expect at most solutions for an th-order fuzzy differential equation by choosing the different types of derivatives.
We thank Professor J. J. Nieto for his valuable remarks which improved the paper. This research is supported by a grant from University of Tabriz.
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