We firstly present a generalized concept of higherorder differentiability for fuzzy functions. Then we interpret thorder 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 secondorder fuzzy differential equation with initial conditions having four different solutions.
1. Introduction
The term "fuzzy differential equation" was coined in 1987 by Kandel and Byatt [1] and an extended version of this short note was published two years later [2]. There are many suggestions to define a fuzzy derivative and in consequence, to study fuzzy differential equation [3]. One of the earliest was to generalize the Hukuhara derivative of a setvalued function. This generalization was made by Puri and Ralescu [4] and studied by Kaleva [5]. It soon appeared that the solution of fuzzy differential equation interpreted by Hukuhara derivative has a drawback: it became fuzzier as time goes by [6]. Hence, the fuzzy solution behaves quite differently from the crisp solution. To alleviate the situation, Hüllermeier [7] 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 fuzzynumbervalued function.
The strongly generalized differentiability was introduced in [8] and studied in [9–11]. This concept allows us to solve the abovementioned shortcoming. Indeed, the strongly generalized derivative is defined for a larger class of fuzzynumbervalued 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 thorder fuzzy differential equations. Higherorder fuzzy differential equation with Hukuhara differentiability is considered in [12] and the existence and uniqueness of solution for nonlinearities satisfying a Lipschitz condition is proved. Buckley and Feuring [13] presented two different approaches to the solvability of thorder linear fuzzy differential equations.
Here, using the concept of generalized derivative and its extension to higherorder derivatives, we show that we have several possibilities or types to define higherorder derivatives of fuzzynumbervalued functions. Then, we propose a new method to solve higherorder 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.
2. Preliminaries
In this section, we give some definitions and introduce the necessary notation which will be used throughout this paper. See, for example, [6].
Definition 2.1.
Let be a nonempty set. A fuzzy set in is characterized by its membership function Thus, is interpreted as the degree of membership of an element in the fuzzy set for each
Let us denote by the class of fuzzy subsets of the real axis (i.e., ) satisfying the following properties:
(i) is normal, that is, there exists such that
(ii) is convex fuzzy set (i.e., ),
(iii) is upper semicontinuous on ,
(iv) is compact where denotes the closure of a subset.
Then is called the space of fuzzy numbers. Obviously, . For denote and . If belongs to then level set is a nonempty compact interval for all . The notation
denotes explicitly the level set of . One refers to and as the lower and upper branches of , respectively. The following remark shows when is a valid level set.
Remark 2.2 (see [6]).
The sufficient conditions for to define the parametric form of a fuzzy number are as follows:
(i) is a bounded monotonic increasing (nondecreasing) leftcontinuous function on and rightcontinuous for ,
(ii) is a bounded monotonic decreasing (nonincreasing) leftcontinuous function on and rightcontinuous for ,
(iii)
For and , the sum and the product are defined by where means the usual addition of two intervals (subsets) of and means the usual product between a scalar and a subset of
The metric structure is given by the Hausdorff distance:
by
The following properties are wellknown:
(i)
(ii)
(iii)
and is a complete metric space.
Definition 2.3.
Let . If there exists such that then is called the difference of and it is denoted .
In this paper the sign "" stands always for difference and let us remark that in general. Usually we denote by , while stands for the difference.
3. Generalized Fuzzy Derivatives
The concept of the fuzzy derivative was first introduced by Chang and Zadeh [14]; it was followed up by Dubois and Prade [15] who used the extension principle in their approach. Other methods have been discussed by Puri and Ralescu [4], Goetschel and Voxman [16], Kandel and Byatt [1, 2]. Lakshmikantham and Nieto introduced the concept of fuzzy differential equation in a metric space [17]. Puri and Ralescu in [4] introduced Hderivative (differentiability in the sense of Hukuhara) for fuzzy mappings and it is based on the difference of sets, as follows. Henceforth, we suppose for
Definition 3.1.
Let be a fuzzy function. One says, is differentiable at if there exists an element such that the limits
exist and are equal to Here the limits are taken in the metric space
The above definition is a straightforward generalization of the Hukuhara differentiability of a setvalued 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 [9], 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 [9] introduced a more general definition of derivative for fuzzynumbervalued function. In this paper, we consider the following definition [11].
Definition 3.2.
Let and fix One says is (1)differentiable at , if there exists an element such that for all sufficiently near to , there exist and the limits (in the metric )
is (2)differentiable if for all sufficiently near to , there exist and the limits (in the metric )
If is differentiable at , we denote its first derivatives by , for
Example 3.3.
Let and define by for all . If is differentiable at , then is generalized differentiable on and we have . For instance, if , is (1)differentiable. If then is (2)differentiable.
Remark 3.4.
In the previous definition, (1)differentiability corresponds to the Hderivative introduced in [4], so this differentiability concept is a generalization of the Hderivative and obviously more general. For instance, in the previous example, for with we have .
Remark 3.5.
In [9], 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]).
Theorem 3.6.
Let be fuzzy function, where for each .
(i)If is (1)differentiable, then and are differentiable functions and .
(ii) If is (2)differentiable, then and are differentiable functions and .
Proof.
See [11].
Now we introduce definitions for higherorder derivatives based on the selection of derivative type in each step of differentiation. For the sake of convenience, we concentrate on the secondorder 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.
Definition 3.7.
Let and . One says say is differentiable at , if exists on a neighborhood of as a fuzzy function and it is differentiable at . The second derivatives of are denoted by for .
Remark 3.8.
This definition is consistent. For example, if is and differentiable simultaneously at , then is (1) and (2)differentiable around . By remark in [9], is a crisp function in a neighborhood of .
Theorem 3.9.
Let or be fuzzy functions, where .
(i)If is (1)differentiable, then and are differentiable functions and .
(ii) If is (2)differentiable, then and are differentiable functions and .
(iii) If is (1)differentiable, then and are differentiable functions and .
(iv) If is (2)differentiable, then and are differentiable functions and .
Proof.
We present the details only for the case (i), since the other cases are analogous.
If and , we have
and multiplying by we have
Similarly, we obtain
Passing to the limit, we have
This completes the proof of the theorem.
Let be a positive integer number, pursuing the abovecited idea, we write to denote the thderivatives 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.
Lemma 3.10.
If are thorder generalized differentiable at in the same case of differentiability, then is generalized differentiable of order at and . (The sum of two functions is defined pointwise.)
Proof.
By Definition 3.2 the statement of the lemma follows easily.
Theorem 3.11.
Let be secondorder generalized differentiable such that is (1,1)differentiable and is (2,1)differentiable or is (1,2)differentiable and is (2,2)differentiable or is (2,1)differentiable and is (1,1)differentiable or is (2,2)differentiable and is (1,2)differentiable on . If the difference exists for then is secondorder generalized differentiable and
for all .
Proof.
We prove the first case and other cases are similar. Since is (1)differentiable and is (2)differentiable on , by [10, Theorem 4], is (1)differentiable and we have . By differentiation as (1)differentiability in Definition 3.2 and using Lemma 3.10, we get is (1,1)differentiable and we deduce
The difference of two functions is understood pointwise.
Theorem 3.12.
Let and be two differentiable functions ( is generalized differentiable as in Definition 3.2).
(i)If and is (1)differentiable, then is (1)differentiable and
(ii) If and is (2)differentiable, then is (2)differentiable and
Proof.
See [10].
Theorem 3.13.
Let and be secondorder differentiable functions ( is generalized differentiable as in Definition 3.7).
(i)If and is (1,1)differentiable then is (1,1)differentiable and
(ii) If and is (2,2)differentiable then is (2,2)differentiable and
Proof.
We prove (i), and the proof of another case is similar. If and is (1)differentiable, then by Theorem 3.12 we have
Now by differentiation as first case in Definition 3.2, since is (1)differentiable and then we conclude the result.
Remark 3.14.
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 secondorder differentiability as follows.
Theorem 3.15.
Let be twice differentiable on , and define by , for all .
(i)If and then is (1,1)differentiable and its second derivative, is ,
(ii) If and then is (1,2)differentiable with ,
(iii) If and then is (2,1)differentiable with ,
(iv) If and then is (2,2)differentiable with .
Proof.
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.
Example 3.16.
If is a fuzzy number and where
is crisp secondorder polynomial, then for
we have the following
(i)for: and then by (iv), is (22)differentiable and its second derivative, is ,
(ii)for: and then by (ii), is (12)differentiable with ,
(iii)for: and then by (iii), is (21)differentiable and ,
(iv)for: and then by (i), is (11)differentiable and ,
(v)for: we have , then by [9, Theorem 10] we have , again by applying this theorem, we get
4. SecondOrder Fuzzy Differential Equations
In this section, we study the fuzzy initial value problem for a secondorder linear fuzzy differential equation:
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).
Definition 4.1.
Let be a fuzzy function and One says is an solution for problem (4.1) on , if exist on and .
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 secondorder linear ordinary differential equations hereafter, called corresponding system for problem (4.1).
Therefore, four ODEs systems are possible for problem (4.1), as follows:
system
system
system
system
Theorem 4.2.
Let and be an solution for problem (4.1) on . Then and solve the associated systems.
Proof.
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.
Theorem 4.3.
Let and and solve the system on for every . Let . If has valid level sets on and exists, then is an solution for the fuzzy initial value problem (4.1).
Proof.
Since is ()differentiable fuzzy function, by Theorems 3.6 and 3.9 we can compute and according to . Due to the fact that solve system, from Definition 4.1, it comes that is an solution for (4.1).
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 [5] we can construct the solution of the fuzzy initial value problem (4.1).
Remark 4.4.
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.
Example 4.5.
Let us consider the following secondorder fuzzy initial value problem
where are the triangular fuzzy number having level sets
If is (1,1)solution for the problem, then
and they satisfy (1,1)system associated with (4.1). On the other hand, by ordinary differential theory, the corresponding (1,1)system has only the following solution:
We see that are valid level sets for and
By Theorem 3.15, is (1,1)differentiable for . Therefore, defines a (1,1)solution for .
For (1,2)solution, we get the following solutions for (1,2)system:
where has valid level sets for How everalso where is (1,2)differentiable. Then gives us a (1,2)solution on .
(2,1)system yields
where has valid level sets for We can see is a (2,1)solution on
Finally, (22)system gives
where has valid level sets for all and defines a (2,2)solution on .
Then we have an example of a secondorder fuzzy initial value problem with four different solutions.
Example 4.6.
Consider the fuzzy initial value problem:
where is the fuzzy number having level sets and
To find (1,1)solution, we have
where has valid level sets for and . From Theorem 3.15, is (1,2)differentiable on , then by Remark 3.8, is not differentiable on . Hence, no (1,1)solution exists for .
For (1,2)solutions we deduce
we see that has valid level sets and is (1,1)differentiable for . Since the (1,2)system has only the above solution, then (1,2)solution does not exist.
For (2,1)solutions we get
we see that the fuzzy function has valid level sets for and define a (2,1)solution for the problem on
Finally, to find (2,2)solution, we find
that has valid level sets for and is (2,2)differentiable on .
We then have a linear fuzzy differential equation with initial condition and two solutions.
5. HigherOrder Fuzzy Differential Equations
Selecting different types of derivatives, we get several solutions to fuzzy initial value problem for secondorder fuzzy differential equations. Theorem 4.2 has a crucial role in our strategy. To extend the results to thorder 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 secondorder fuzzy differential equation to thorder. Under generalized derivatives, we would expect at most solutions for an thorder fuzzy differential equation by choosing the different types of derivatives.
Acknowledgments
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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