New Results on Multiple Solutions for th-Order Fuzzy Differential Equations under Generalized Differentiability

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 set-valued 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 fuzzy-number-valued function.

The strongly generalized differentiability was introduced in [8] 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 [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 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, [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

(2.1)

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) left-continuous function on and right-continuous for ,

(ii) is a bounded monotonic decreasing (nonincreasing) left-continuous function on and right-continuous 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:

(2.2)

by

(2.3)

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.

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 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

Definition 3.1.

Let be a fuzzy function. One says, is differentiable at if there exists an element such that the limits

(3.1)

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 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 [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 fuzzy-number-valued 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 )

(3.2)

is (2)-differentiable if for all sufficiently near to , there exist and the limits (in the metric )

(3.3)

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 H-derivative introduced in [4], 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 .

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 .

1. (ii)

   If is (2)-differentiable, then and are differentiable functions and .

    

Proof.

See [11].

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.

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 .

1. (ii)

   If is (2)-differentiable, then and are differentiable functions and .

    
2. (iii)

   If is (1)-differentiable, then and are differentiable functions and .

    
3. (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

(3.4)

and multiplying by we have

(3.5)

Similarly, we obtain

(3.6)

Passing to the limit, we have

(3.7)

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.

Lemma 3.10.

If are th-order 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 second-order 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 second-order generalized differentiable and

(3.8)

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

(3.9)

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

(3.10)

Proof.

See [10].

Theorem 3.13.

Let and be second-order differentiable functions ( is generalized differentiable as in Definition 3.7).

(i)If and is (1,1)-differentiable then is (1,1)-differentiable and

(3.12)

Proof.

We prove (i), and the proof of another case is similar. If and is (1)-differentiable, then by Theorem 3.12 we have

(3.14)

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 second-order 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 ,

1. (ii)

   If and then is (1,2)-differentiable with ,

    
2. (iii)

   If and then is (2,1)-differentiable with ,

    
3. (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

(3.15)

is crisp second-order polynomial, then for

(3.16)

we have the following

(i)for : and then by (iv), is (2-2)-differentiable and its second derivative, is ,

(ii)for : and then by (ii), is (1-2)-differentiable with ,

(iii)for : and then by (iii), is (2-1)-differentiable and ,

(iv)for : and then by (i), is (1-1)-differentiable and ,

(v)for : we have , then by [9, Theorem 10] we have , again by applying this theorem, we get

In this section, we study the fuzzy initial value problem for a second-order linear fuzzy differential equation:

(4.1)

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 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:

-system

(4.2)

-system

(4.3)

-system

(4.4)

-system

(4.5)

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 second-order fuzzy initial value problem

(4.6)

where are the triangular fuzzy number having -level sets

If is (1,1)-solution for the problem, then

(4.7)

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:

(4.8)

We see that are valid level sets for and

(4.9)

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:

(4.10)

where has valid level sets for How ever-also where is (1,2)-differentiable. Then gives us a (1,2)-solution on .

(2,1)-system yields

(4.11)

where has valid level sets for We can see is a (2,1)-solution on

Finally, (2-2)-system gives

(4.12)

where has valid level sets for all and defines a (2,2)-solution on .

Then we have an example of a second-order fuzzy initial value problem with four different solutions.

Example 4.6.

Consider the fuzzy initial value problem:

(4.13)

where is the fuzzy number having -level sets and

To find (1,1)-solution, we have

(4.14)

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

(4.15)

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

(4.16)

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

(4.17)

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.

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.