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)

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)

- (iii)

- (iv)

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

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

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

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)

- (iii)

- (iv)

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