Applied Mathematics
Vol.07 No.18(2016), Article ID:72490,11 pages
10.4236/am.2016.718180
On AP-Henstock Integrals of Interval-Valued Functions and Fuzzy-Number-Valued Functions
Muawya Elsheikh Hamid1*, Alshaikh Hamed Elmuiz1, Mohammed Eldirdiri Sheima2
1School of Management, Ahfad University for Women, Omdurman, Sudan
2Faculty of Engineering, University of Khartoum, Khartoum, Sudan
Copyright © 2016 by authors and Scientific Research Publishing Inc.
This work is licensed under the Creative Commons Attribution International License (CC BY 4.0).
http://creativecommons.org/licenses/by/4.0/
Received: October 8, 2016; Accepted: November 29, 2016; Published: December 2, 2016
ABSTRACT
In 2000, Wu and Gong [1] introduced the thought of the Henstock integrals of interval-valued functions and fuzzy-number-valued functions and obtained a number of their properties. The aim of this paper is to introduce the thought of the AP- Henstock integrals of interval-valued functions and fuzzy-number-valued functions which are extensions of [1] and investigate a number of their properties.
Keywords:
Fuzzy Numbers, AP-Henstock Integrals of Interval-Valued Functions, AP-Henstock Integrals of 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.
Cite this paper
Hamid, M.E., Elmuiz, A.H. and Sheima, M.E. (2016) On AP-Henstock Integrals of Interval-Valued Functions and Fuzzy-Number-Valued Func- tions. Applied Mathematics, 7, 2285-2295. http://dx.doi.org/10.4236/am.2016.718180
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