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New Results on Multiple Solutions for
th-Order Fuzzy Differential Equations under Generalized Differentiability
Boundary Value Problems volume 2009, Article number: 395714 (2009)
Abstract
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.
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 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.
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) 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:

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 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

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
)

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 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
.
-
(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
.
-
(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 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

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
(3.11)
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

-
(ii)
If
and
is (2,2)-differentiable then
is (2,2)-differentiable and
(3.13)
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 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
,
-
(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 second-order polynomial, then for

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
4. Second-Order Fuzzy Differential Equations
In this section, we study the fuzzy initial value problem for a second-order 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 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

-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 second-order 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 ever-also
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, (2-2)-system gives

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:

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. 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.
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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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Khastan, A., Bahrami, F. & Ivaz, K. New Results on Multiple Solutions for th-Order Fuzzy Differential Equations under Generalized Differentiability.
Bound Value Probl 2009, 395714 (2009). https://doi.org/10.1155/2009/395714
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DOI: https://doi.org/10.1155/2009/395714