On AP-Henstock Integrals of Interval-Valued Functions and Fuzzy-Number-Valued Functions ()
1. Introduction
As it is well known, the Henstock integral for a real function was first defined by Henstock [2] in 1963. The Henstock integral is a lot of powerful and easier than the Lebesgue, Wiener and Richard Phillips Feynman integrals. Furthermore, it is also equal to the Denjoy and the Perron integrals [2] [3] . In 2016, Hamid and Elmuiz [4] introduced the concept of the Henstock-Stieltjes
integrals of interval-valued functions and fuzzy-number-valued functions and discussed a number of their properties.
In this paper, we introduce the concept of the AP-Henstock integrals of interval-valued functions and fuzzy-number-valued functions and discuss some of their properties.
The paper is organized as follows. In Section 2, we have a tendency to provide the preliminary terminology used in this paper. Section 3 is dedicated to discussing the AP-Henstock integral of interval-valued functions. In Section 4, we introduce the AP- Henstock integral of fuzzy-number-valued functions. The last section provides conclusions.
2 Preliminaries
Let
be a measurable set and let
be a real number. The density of
at
is defined by
(2.1)
provided the limit exists. The point
is called a point of density of
if
. The set
represents the set of all points
such that
is a point of density of
.
A measurable set
is called an approximate neighborhood (br.ap-nbd) of
if it containing
as a point of density. We choose an ap-nbd
for each
and denote a choice on
by
. A tagged interval-point pair
is said to be
-fine if
and
.
A division
is a finite collection of interval-point pairs
, where
are non-overlapping subintervals of
. We say that
is
1) a division of
if
;
2)
-fine division of
if
and
is
-fine for all
.
Definition 2.1. [2] [3] A real-valued function
is said to be Henstock integrable to
on
if for every
, there is a function
such that for any
-fine division
of
, we have
(2.2)
where the sum
is understood to be over
and we write
, and
.
Definition 2.2. [5] A function
is AP-Henstock integrable if there exists a real number
such that for each
there is a choice
such that
(2.3)
for each
-fine division
of
.
is called AP-Henstock integral of
on
, and we write
.
Theorem 2.1. If
and
are AP-Henstock integrable on
and
almost everywhere on
, then
(2.4)
Proof. The proof is similar to the Theorem 3.6 in [3] . W
3. The AP-Henstock Integral of Interval-Valued Functions
In this section, we shall give the definition of the AP-Henstock integrals of interval-valued functions and discuss some of their properties.
Definition 3.1. [1] Let ![]()
For
, we define
iff
and
,
iff
and
, and
, where
(3.1)
and
(3.2)
Define
as the distance between intervals
and
.
Definition 3.2. [1] Let
be an interval-valued function.
, for every
there is a
such that for any
-fine division
we have
(3.3)
then
is said to be Henstock integrable over
and write
For brevity, we write ![]()
Definition 3.3. A interval-valued function
is AP-Henstock integrable to
, if for every
there exists a choice
on
such that
(3.4)
whenever
is a
-fine division of
, we write
and ![]()
Theorem 3.1. If
, then the integral value is unique.
Proof. Let integral value is not unique and let
and
. Let
be given. Then there exists a choice
on
such that
(3.5)
(3.6)
whenever
is a
-fine division of
.
Whence it follows from the Triangle Inequality that:
(3.7)
Since for
there exists a choice
on
as above so
W
Theorem 3.2. An interval-valued function
if and only if
and
(3.8)
Proof. Let
, from Definition 3.3 there is a unique interval number
with the property that for any
there exists a choice
on
such that
(3.9)
whenever
is a
-fine division of
. Since
for
we have
(3.10)
Hence
whenever
is a
-fine division of
. Thus
and
(3.11)
Conversely, let
. Then there exists
with the property that given
there exists a choice
on
such that
![]()
whenever
is a
-fine division of
. We define
then if
is a
-fine division of
, we have
(3.12)
Hence
is AP-Henstock integrable on
. W
Theorem 3.3. If
and
Then
and
(3.13)
Proof. 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
(3.14)
Hence by Theorem 3.2
and
(3.15)
W
Theorem 3.4. If
and
, then
and
(3.16)
Proof. If
and
, then by Theorem 3.2
and
. Hence
and
![]()
Similarly,
Hence by Theorem 3.2
and
(3.17)
W
Theorem 3.5. If
nearly everywhere on
and
, then
(3.18)
Proof. Let
nearly everywhere on
and
Then
and
,
nearly everywhere on
By Theorem 2.1
and
Hence
(3.19)
by Theorem 3.2. W
Theorem 3.6. Let
and
is Lebesgue integrable on
Then
(3.20)
Proof. By definition of distance,
(3.12)
W
4. The AP-Henstock Integral of Fuzzy-Number-Valued Functions
This section introduces the concept of the AP-Henstock integral of fuzzy-number- valued functions and investigates some of their properties.
Definition 4.1. [6] [7] [8] Let
be a fuzzy subset on
If for any
and
where
then
is called a fuzzy number. If
is convex, normal, upper semi-continuous and has the compact support, we say that
is a compact fuzzy number.
Let
denote the set of all fuzzy numbers.
Definition 4.2. [6] Let
, we define
iff
for all
iff
for any
iff
for any ![]()
For
is called the distance between
and ![]()
Lemma 4.1. [9] If a mapping
satisfies
when
then
(4.1)
and
(4.2)
where ![]()
Definition 4.3. [1] Let
. If the interval-valued function
is Henstock integrable on
for any
then we say that
is Henstock integrable on
and the integral value is defined by
![]()
For brevity, we write ![]()
Definition 4.4. Let
. If the interval-valued function
is AP-Henstock integrable on
for any
then
is called AP-Henstock integrable on
and the integral value is defined by
![]()
We write ![]()
Theorem 4.1.
then
and
(4.3)
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
(4.4)
From Theorem 3.2 and Lemma 4.1 we have
(4.5)
and for all
where
W
Theorem 4.2. If
and
Then
and
(4.6)
Proof. If
, then the interval-valued function
and
are AP-Henstock integrable on
for any
and
and
. From Theorem 3.3 we have
and
for any
.
Hence
and
![]()
W
Theorem 4.3. If
and
, then
and
(4.7)
Proof. If
and
, then the interval-valued function
is AP-Henstock integrable on
and
for any
and
and
. From Theorem 3.4 we have
and
for any
. Hence
and
![]()
W
Theorem 4.4. If
nearly everywhere on
and
, then
(4.8)
Proof. If
nearly everywhere on
and
, then
nearly everywhere on
for any
and
and
are AP-Henstock integrable on
for any
and
and
. From Theorem 3.5 we have
for any
. Hence
![]()
5. Conclusion
In this paper, we have a tendency to introduce the concept of the AP-Henstock integrals of interval-valued functions and fuzzy number-valued functions and investigate some properties of those integrals.