1. Introduction
Sugeno [1] was the first to introduce the concept of fuzzy integral. Numerical methods have been developed in recent years in order to calculate a fuzzy integral. Some numerical methods are proposed by Wu [2] [3] , Allahviranloo [4] and Fariborzi [5] [6] in order to compute fuzzy integrals by using quadrature methods and the definition of the set of levels. Wu and Gong [7] developed the Henstock integral of a fuzzy numeric valued function, then they applied the notion of differentiability of a fuzzy function. Bede and Gal in [8] have in turn applied the quadrature rule to calculate the integral of a function with fuzzy numerical value.
Some other integrals have been defined by Kumwimba et al. [9] [10] [11] . The material of this article is based on ideas developed in the article [12] to evaluate a fuzzy triple-valued function by applying the Simpson’s triple rule and introduction of the fuzzy version Henstock’s triple integral.
In Section 2, it will be a question of pinning down some basic definitions and properties of fuzzy sets and fuzzy numbers as well as some basic theorems useful for this work.
We introduce in Section 3, Simpson’s triple rule to compute a fuzzy Henstock-Kurzweil triple integral (FHTI).
At last, in order to explain an application of the proposed method, in Section 4, one triple fuzzy integral is evaluated in order to show the efficacy of the mentioned method.
2. Preliminaries
In this section, we talk about some basic definitions of fuzzy sets theory which are being used in the following.
Definition 2.1. Let
be a real set. Given a function
satisfying the properties below:
1)
is normal, i.e.
such that
,
2)
is a convex fuzzy set, i.e.,
3)
is upper semi-continuous,
4) the set
is compact, where
denotes the closure of B.
This function
is called a fuzzy number.
We denote by
the set of all fuzzy real numbers. We define
and
, for
, as the α-cut and respectively the support of a fuzzy number such as
. Moreover, we define
and
.
A triangular fuzzy number
where,
and
is defined by
and
.
For
and
, we can define the sum
and the product
by
and
,
with
the usual addition of two intervals and
the usual product between a scalar and a subset of
[13] [14] .
Definition 2.2. Let be two fuzzy numbers
and
given. The Hausdorff distance
of
and
is defined by
with
and
the Hausdorff metric. We denote
, [15] .
The following theorems will also be used.
Theorem 2.3.
1) If
, then
is a neutral element with respect to
, i.e.
.
2) With respect to
, none of
is inversible in
.
3)
such that
(or
), and any
. We have
4)
and
, we have
5)
and
, we have
6)
has the properties of a usual norm on
, i.e.
if
,
and
7)
and
[7] .
Theorem 2.4.
1)
is a complete metric space,
2)
,
3)
,
,
4)
[15] .
The concept of the Henstock integral for a fuzzy number-valued function were introduced by Wu and Gong [12] . We introduce this definition for a three-dimensional fuzzy number-valued function.
Let
and
,
and
be the partitions of the intervals
and
respectively.
Consider the points
;
;
and
.
The divisions
;
and
denoted shortly by
and
are called δ-fines if
;
and
.
Here we give our new view of fuzzy Henstock double integral on which a third integral.
Definition 2.5. The function is said to be Henstock triple integrable on
if for every
there is a function
such that for any δ-fine divisions
and R we obtain
.
Then I is called the Fuzzy Henstock Triple Integral of
and it’s denoted by (FHTI)
.
Lemma 2.6.
1) If
and
are Henstock triple integrable mappings and if
is Lebesgue integrable, then
2) Let
be a Henstock triple integrabe bounded mapping.
Then,
, the function
defined by
is Lebesgue integrable on
.
Proof (2) If
is Henstock integrable and bounded on
, then it follows that
and
are Henstock triple integrable with
. Therefore,
and
are Lebesgue measurable and uniformly bounded
, [7] . Moreover,
where the
are the rational numbers in
. According to Lebesgue’s dominated convergence theorem, it follows that
is Lebesgue integrable over
and what completes the proof.
Keeping now three integrals we reach the following definitions.
Definition 2.7. Let
be a bounded mapping. Then
the function
such that
is called the modulus of oscillation of f on
.
If
is continuous on
.
Then
is called uniform modulus of continuity of f.
We can prove the following theorem from the definition 2.7.
Theorem 2.8. The following statements, concerning the modulus of oscillation are true.
1)
,
2)
is a non-decreasing mapping in
and
,
3)
,
4)
and
,
5)
for any
.
6) If
, then
.
Proof (6) According to the hypothesis,
which is prove the relation.
We can prove similarly the other statements.
Definition 2.9. A function
is said to be
Lipschitz if for any
,
3. Triple Simpson’s Rule for the Fuzzy Henstock-Kurzweil Triple Integrals
In order to introduce triple Simpson’s rule for evaluating FHTI, firstly we prove the following theorem.
Theorem 3.1. Let
be a Henstock integrable, bounded mapping. Then, for any subdivision
,
,
and any points
,
,
we have
Proof: Since that the Henstock integral is additive related to interval [16] , hence,
Since it’s clear that
for any fuzzy constant
, we obtain
By the fourth property of the theorem 2.4, we have
Since the functions
are Lebesgue integrable for
and
from lemma 2.6 we have
From the first property of the theorem 2.8 applied to each of the above integrals we have
which completes the proof.
Corollary 3.2. Let
be a Henstock triple integrable, bounded mapping. Then,
for any
and
,
and
where
,
;
,
;
,
.
Proof It’s clear that for
and
in the theorem 3.1 the inequality stated above is obtained.
Theorem 3.3. Let
be a Lipschitz mapping with the constants
and
. Then, for any subdivision
,
and
.
;
and
; we have
Proof Similar to the proof of theorem 3.1 we have
We obtain by the definition of a Lipschitz mapping
It follows by direct computation that
Remark 3.4. If
and
, then,
where
and
. Therefore, we obtain
(1)
4. Numerical Example
Let
,
where
;
;
;
;
, and where
is a triangular fuzzy number such that
We must compute the integral
numerically.
Firstly we calculate
so
We obtain
Remark that
i.e.
is a Lipschitz mapping with
and
. We have for
:
Table 1 shows the results for different
and
and
.
In this table, the notations
and
are the approximate values of α-cut for
obtained by the triple Simpson’s rule
with
,
and
[17] .
We have
from 3.1 in this case.
5. Conclusion
We generalize the evaluating of fuzzy Henstock double integral using double Simpson’s rule [12] by introduce and evaluate Henstock’s fuzzy triple integral by applying Simpson’s triple rule. Therefore, a theorem has been demonstrated to show the upper limit of the distance between the exact and approximate values. In the following, the Monte Carlo method [3] can be used for Henstock’s fuzzy triple integral and thus compare the results of the methods with each other. We finished our paper by a numerical example of a fuzzy function in wich triple Simpson’s rule is used.