On Henstock-Stieltjes Integrals of Interval-Valued Functions and Fuzzy-Number-Valued Functions ()
Received 2 March 2016; accepted 24 April 2016; published 28 April 2016
![](//html.scirp.org/file/11-1720539x5.png)
1. Introduction
As it is well known, the Henstock (H) integral for a real function was first defined by Henstock [1] in 1963. The Henstock (H) integral is a lot powerful and easier than the Lebesgue, Wiener and Richard Phillips Feynman integrals. Furthermore, it is also equal to the Denjoy and the Perron integrals [1] [2] . In 2000, Congxin Wu and Zengtai Gong [3] introduced the notion of the Henstock (H) integrals of interval-valued functions and fuzzy- number-valued functions and obtained a number of their properties. In 2016, Yoon [4] introduced the interval- valued Henstock-Stieltjes integral on time scales and investigated some properties of these integrals. In 1998, Lim et al. [5] introduced the notion of the Henstock-Stieltjes (HS) integral of real-valued function which was a generalization of the Henstock (H) integral and obtained its properties.
In this paper, we tend to introduce the notion of the Henstock-Stieltjes (HS) integrals of interval-valued functions and fuzzy-number-valued functions and discuss some of their properties.
The paper is organized as follows. In Section two, we tend to give the preliminary terminology used in the present paper. Section three is dedicated to discussing the Henstock-Stieltjes (HS) integral of interval-valued functions. In Section four, we tend to introduce the Henstock-Stieltjes (HS) integral of fuzzy-number-valued functions. The last section provides conclusions.
2. Preliminaries
Definition 2.1 [1] [2] Let
be a positive real-valued function.
is called a d- fine division, if the subsequent conditions are satisfied:
1)
,
2) ![](//html.scirp.org/file/11-1720539x10.png)
For brevity, we write
, wherever
denotes a typical interval in P and
is that the associated point of
.
Definition 2.2 [1] [2] A real-valued function
is called Henstock (H) integrable to A on
if for each
, there exists a function
such that for any d-fine division
of
, we have
(1)
where the sum
is understood to be over P, we write
, and
.
Definition 2.3 [5] Let
be an increasing function. A real-valued function
is Henstock-Stieltjes (HS) integrable to
with respect to
on
if for each
, there exists a function
, such that for any d-fine division
we have
(2)
We write
, and
.
Lemma 2.1 [5] Let
be an increasing function and let f, g are Henstock-Stieltjes (HS) integrable with respect to
on
. If
and
almost everywhere on
, then
(3)
3. The Henstock-Stieltjes (HS) Integrals of Interval-Valued Functions
Definition 3.1 [3] Let
.
For
, we define
if and only if
and
,
if and only if
and
, and
, wherever
and ![]()
Define
as the distance between intervals A and B.
Definition 3.2 [3] Let
be an interval-valued function.
, for each
there exists a
such that for any d-fine division
we have
(4)
then
is called the Henstock (H) integrable over
and write
. Also, we write
.
Definition 3.3 Let
be an increasing function. An interval-valued function
is Henstock-Stieltjes (HS) integrable to
with respect to
on
, if for each
there exists a
such that for any d- fine division
, we have
(5)
We write
and ![]()
Theorem 3.1 Let
be an increasing function. If
, then there exists a unique integral value.
Proof Let the integral value is not unique and let
and
. If
is given. Then there exists a
such that for any d- fine division
, we have
(6)
(7)
![]()
Since for all
there exists a
as above then
![]()
Theorem 3.2 Let
be an increasing function. Then an interval-valued function
iff
and
(8)
Proof If
, by Definition 3.3 there exists a unique interval number
with the
property, for any
there exists a
such that for any d- fine division
, we have
(9)
that is
(10)
Since
for
we have
(11)
(12)
Therefore, by Definition 2.3 we can obtain
and
(13)
(14)
Conversely, let
, then there exists a unique
with the property, given ![]()
there exists a
such that for any
-fine division
, we have
(15)
It is similar to find
such that for any
-fine division
, we have
(16)
If
, then
We define
and
then for any d- fine division
, we have
(17)
Hence
is Henstock-Stieltjes (HS) integrable with respect to
on
. ![]()
Theorem 3.3 If
and
Then
i)
and
(18)
ii) Let
almost everywhere on
. Then
(19)
Proof i) If
, then
by Theorem 3.2. Hence ![]()
1) If
and
then
![]()
2) If
and
then
![]()
3) If
and
(or
and
), then
![]()
Similarly, for four cases above we have
(20)
Hence by Theorem 3.2
and
(21)
ii) The proof is similar to Theorem 2.8 in [5] . ![]()
Theorem 3.4 Let
and let
Then
and
(22)
Proof If
and
, then by Theorem 3.2
and
. Hence
and
![]()
Similarly,
Hence by Theorem 3.2
and
(23)
![]()
Theorem 3.5 Let
be an increasing function such that
If
nearly everywhere on
and
, then
(24)
Proof Let
nearly everywhere on
and
Then
and
,
nearly everywhere on
. By Lemma 2.1
and
Hence
(25)
by Theorem 3.2. ![]()
Theorem 3.6 Let
and
is Lebesgue-Stieltjes (LS) integrable on
. Then
(26)
Proof By definition of distance,
(27)
![]()
4. The Henstock-Stieltjes (HS) Integral of Fuzzy-Number-Valued Functions
Definition 4.1 [6] - [8] If
is a fuzzy subset on
. If for any
and
wherever
then
is called a fuzzy number. If
satisfy the following conditions: 1) convex, 2) normal, 3) upper semi-continuous, 4) has the compact support, then
is called a compact fuzzy number.
Let
denote the set of all fuzzy numbers and
denote the set of all compact fuzzy numbers.
Definition 4.2 [6] Let
, we define
if and only if
for all
if and only if
for any
if and only if
for any ![]()
For
is called the distance between
and ![]()
Lemma 4.1 [9] If a mapping
satisfies
when
then
(28)
and
(29)
where ![]()
Definition 4.3 [3] Let
and let the interval-valued function
is Henstock (H) integrable on
for any
then
is called Henstock (H) integrable on
and the integral value is defined by
![]()
We write ![]()
Definition 4.4 Let
be an increasing function and let
. If the interval-valued function
is Henstock-Stieltjes (HS) integrable with respect to
on
for any
then
is called Henstock-Stieltjes (HS) integrable with respect to
on
and the integral value is defined by
![]()
We write ![]()
Theorem 4.1
then
and
(30)
where ![]()
Proof Let
be defined by ![]()
Since
and
are increasing and decreasing on
respectively, therefore, when
we have
on
. From Theorem 3.5 we have
(31)
From Theorem 3.2 and Lemma 4.1 we have
(32)
and
wherever
![]()
Using Theorem 4.1 and the properties of
integral, we are able to get the properties of
integral, for example, 1) the linear, 2) monotone, 3) interval additive properties of
integral.
5. Conclusion
In this paper, we proposed the definition of the Henstock-Stieltjes (HS) integrals of interval-valued functions and fuzzy-number-valued functions and investigated some properties of those integrals.
NOTES
![]()
*Corresponding author.