On AP Henstock Integrals of Interval Valued Functions and Fuzzy Number Valued Functions

http://www.scirp.org/journal/am ISSN Online: 2152-7393

ISSN Print: 2152-7385

On AP-Henstock Integrals of Interval-Valued

Functions and Fuzzy-Number-Valued Functions

Muawya Elsheikh Hamid

1*

, Alshaikh Hamed Elmuiz

1

, Mohammed Eldirdiri Sheima

2

1_{School of Management, Ahfad University for Women, Omdurman, Sudan} 2_{Faculty of Engineering, University of Khartoum, Khartoum, Sudan}

Abstract

In 2000, Wu and Gong [1] introduced the thought of the Henstock integrals of in-terval-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 Hens-tock [2] in 1963. The Henstock integral is a lot of powerful and easier than the Lebes-gue, 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

( )

HS 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 conclu-sions.

How to cite this paper: Hamid, M.E., El-muiz, 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

Received: October 8, 2016 Accepted: November 29, 2016 Published: December 2, 2016

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/

2 Preliminaries

Let E be a measurable set and let

c

be a real number. The density of E at

c

is defined by

(

)

(

)

0

,

lim ,

2

c h

E c h c h

d E

h µ

+ →

− +

=



_(2.1)

provided the limit exists. The point

c

is called a point of density of E if d Ec =1.

The set d

E represents the set of all points x∈E such that

x

is a point of density of

E.

A measurable set Sx⊆

[ ]

a b, is called an approximate neighborhood (br.ap-nbd) of

[ ]

,

x∈ a b if it containing

x

as a point of density. We choose an ap-nbd Sx⊆

[ ]

a b,

for each x∈ ⊆E

[ ]

a b, and denote a choice on E by S=

{

Sx:x∈E

}

. A tagged

in-terval-point pair

(

[ ]

u v, ,ξ

)

is said to be S-fine if ξ∈

[ ]

u v, and u v, ∈Sξ.

A division P is a finite collection of interval-point pairs

{

(

[

]

)

}

1

, ; n

i i i

i u v

ξ

= , where

[

]

{

,

}

1

n

i i _i

u v ₌ are non-overlapping subintervals of

[ ]

a b, . We say that

{

(

[

]

)

}

1

, ; n

i i i

i

P u v

ξ

=

=

is

1) a division of

[ ]

a b, if

[

]

[ ]

1

, ,

n

i i i

u v a b

=

=



;

2) S -fine division of

[ ]

a b, if

ξ

_i∈

[

u v_i, _i

]

and

(

[

u vi, i

]

,ξi

)

is S -fine for all

1, 2, ,

i=  n.

Definition 2.1. [2][3] A real-valued function f :

[ ]

a b, →R is said to be Henstock

integrable to A on

[ ]

a b, if for every ε> 0, there is a function δ

( )

t > 0 such that

for any δ-fine division P=

{

[

u vi, i

]

;ξi

}

_in₌₁ of

[ ]

a b, , we have

( )(

)

1

,

n

i i i

i

f ξ v u A ε

=

− − <

∑

(2.2)

where the sum

∑

is understood to be over P and we write

( ) ( )

d

b

a

H

∫

f t t=A, and

[ ]

,

f∈H a b .

Definition 2.2. [5] A function f :

[ ]

a b, →R is AP-Henstock integrable if there

ex-ists a real number A∈R such that for each ε> 0 there is a choice S such that

( )(

)

1 n

i i i

i

f ξ v u A ε

=

− − <

∑

(2.3)

for each S-fine division

{

(

[

]

)

}

1

, ; n

i i i _i

P= u v

ξ

₌ of

[ ]

a b, . A is called AP-Henstock

integral of f on

[ ]

a b, , and we write =

(

)

b

a A APH

∫

f .

Theorem 2.1. If f and g are AP-Henstock integrable on

[ ]

a b, and f ≤g

almost everywhere on

[ ]

a b, , then

(

)

(

)

.

b b

a a

Proof. The proof is similar to the Theorem 3.6 in [3]. 

3. The AP-Henstock Integral of Interval-Valued Functions

In this section, we shall give the definition of the AP-Henstock integrals of inter-val-valued functions and discuss some of their properties.

Definition 3.1. [1] Let

{

, : I is the closed bounded interval on the real line

}

.

R

I = I = I− I+_ R

For A B, ∈IR , we define A≤B iff A B −_≤ − _and

A+≤B+ , A+ =B C iff C−=A−+B− and C+ =A++B+, and A B⋅ =

{

a b a⋅ : ∈A b, ∈B

}

, where

(

A B⋅

)

− =min

{

A−⋅B−,A−⋅B+,A+⋅B−,A+⋅B+

}

(3.1)

and

(

A B⋅

)

+=max

{

A−⋅B−,A−⋅B+,A+⋅B−,A+⋅B+

}

. (3.2)

Define d

(

A B,

)

=max

(

A−−B− ,A+−B+

)

as the distance between intervals A

and B.

Definition 3.2. [1] Let F:

[ ]

a b, →I_R be an interval-valued function. I₀∈I_R, for

every ε >0 there is a δ

( )

t >0 such that for any δ-fine division

{

[

]

}

1

, , n

i i i _i

P= u v ξ ₌

we have

( )(

)

0 1

d , ,

n

i i i

i

F ξ v u I ε

=

 ₋ _<

 



∑

 (3.3)

then F t

( )

is said to be Henstock integrable over

[ ]

a b, and write

( ) ( )

d 0. b

a

IH

∫

F t t=I For brevity, we write F t

( )

∈IH a b

[ ]

, .

Definition 3.3. A interval-valued function F:

[ ]

a b, →I_R is AP-Henstock

integra-ble to I0∈IR, if for every ε> 0 there exists a choice S on

[ ]

a b, such that

( )(

)

0 1

d , ,

n

i i i

i

F ξ v u I ε

=

 ₋ _<

 



∑

 (3.4)

whenever

{

(

[

]

)

}

1

, ; n

i i i

i

P u v

ξ

=

= is a S-fine division of

[ ]

a b, , we write

(

)

0

b

a

APIH

∫

F =I and F∈APIH a b

[ ]

, .

Theorem 3.1. If F∈APIH a b

[ ]

, , then the integral value is unique.

Proof. Let integral value is not unique and let 1

(

)

b

a

B = APIH

∫

F and

(

)

2

b

a

B = APIH

∫

F. Let ε >0 be given. Then there exists a choice S on

[ ]

a b, such

that

( )(

)

1 1

d , ,

2

n

i i i

i

F ξ v u B ε

=

 ₋ _<

 

( )(

)

2 1

d ,

2

n

i i i

i

F ξ v u B ε

=

 ₋ _<

 



∑



(3.6)

whenever

{

(

[

i, i

]

; i

)

}

n₁ i

P u v

ξ

=

= _{is a} S-fine division of

[ ]

a b, .

Whence it follows from the Triangle Inequality that:

(

1 2

)

( )(

)

1

( )(

)

2

1 1

d , d , d , .

2 2

n n

i i i i i i

i i

B B F ξ v u B F ξ v u B ε ε ε

= =

   

= _ − _+ _ − _< + =



∑

 

∑

 (3.7)

Since for ∀ >ε 0, there exists a choice S on

[ ]

a b, as above so B1 =B2. 

Theorem 3.2. An interval-valued function F∈APIH a b

[ ]

, if and only if

[ ]

, ,

F− F+∈APH a b and

(

)

(

)

,

(

)

.

b b b

a a a

APIH F=  APH F− APH F+_

 

∫

∫

∫

(3.8)

Proof. Let F∈APIH a b

[ ]

, , from Definition 3.3 there is a unique interval number

0 0, 0

I = I− I+_ with the property that for any ε> 0 there exists a choice S on

[ ]

a b,

such that

( )(

)

0 1

d , ,

n

i i i

i

F ξ v u I ε

=

 ₋ _<

 



∑



(3.9)

whenever

{

(

[

i, i

]

; i

)

}

n₁ i

P u v

ξ

=

= _{is a} S-fine division of

[ ]

a b, . Since vi− ≥ui 0 for

1≤ ≤i n, we have

( )(

)

( )(

)

( )(

)

( )(

)

( )(

)

0 1 0 0 1 1 0 0 1 1 d ,

max , .

max , | .

n

i i i

i

n n

i i i i i i

i i

n n

i i i i i i

i i

F v u I

F v u I F v u I

F v u I F v u I

ξ ε

ξ ξ ε

ξ ξ ε

= − + − + = = − − + + = =

 ₋ _<

 

 

_{   } 

 

= _ − _ − _ − _ − <

_{   } 

 

 

= _ − − − − _<

 

∑

∑

∑

∑

∑

(3.10)

Hence

( )(

)

0

1

,

n

i i i

i

F− ξ v u I− ε

=

− − <

∑

( )(

)

0

1 n

i i i

i

F+ ξ v u I+ ε

=

− − <

∑

whenever

[

]

(

)

{

, ;

}

1

n

i i i

i

P u v

ξ

=

= _{is a} S-fine division of

[ ]

a b, . Thus F−,F+∈APH a b

[ ]

, and

(

)

(

)

,

(

)

.

b b b

a a a

APIH F=  APH F− APH F+_

 

∫

∫

∫

(3.11)

Conversely, let F−,F+∈APH a b

[ ]

, . Then there exists H H₁, ₂∈R with the

prop-erty that given ε> 0 there exists a choice S on

[ ]

a b, such that

( )(

)

1

( )(

)

2

1 1

,

n n

i i i i i i

i i

F− ξ v u H ε F+ ξ v u H ε

= =

− − < − − <

∑

∑

whenever

{

(

[

i, i

]

; i

)

}

n₁ i

P u v

ξ

=

= _{is a} S-fine division of

[ ]

a b, . We define

[

]

0 1, 2 ,

I = H H then if

{

[

]

}

1

, , n

i i i _i

( )(

)

0 1

d , .

n

i i i

i

F ξ v u I ε

=

 ₋ _<

 



∑

 (3.12)

Hence F:

[ ]

a b, →I_R is AP-Henstock integrable on

[ ]

a b, . 

Theorem 3.3. If F G, ∈APIH a b

[ ]

, and β γ, ∈R. Then βF+γG∈APIH a b

[ ]

,

and

(

) (

)

(

)

(

)

.

b b b

a a a

APIH

∫

βF+γG =β APIH

∫

F+γ APIH

∫

G (3.13)

Proof. If F G, ∈APIH a b

[ ]

, , then

F

−

,

F G G

+

,

−

,

+

∈

APH a b

[ ]

,

by Theorem 3.2.

Hence

β

F

−

+

γ

G

−

,

β

F

−

+

γ

G

+

,

β

F

+

+

γ

G

−

,

β

F

+

+

γ

G

+

∈

APH a b

[ ]

,

.

(1) If β >0 and γ >0, then

(

) (

) (

)

(

)

(

)

(

)

(

)

(

)

(

)

(

)

.

b b

a a

b b

a a

b b

a a

b b

a a

APH F G APH F G

APH F APH G

APIH F APIH G

APIH F APIH G

β γ β γ

β γ

β γ

β γ

− − −

− −

− −

−

+ = +

= +

   

=   +  

   

 

= + 

 

∫

∫

∫

∫

∫

∫

∫

∫

(2) If β <0 and γ <0, then

(

) (

) (

)

(

)

(

)

(

)

(

)

(

)

(

)

(

)

.

b b

a a

b b

a a

b b

a a

b b

a a

APH F G APH F G

APH F APH G

APIH F APIH G

APIH F APIH G

β γ β γ

β γ

β γ

β γ

− _{+ +}

+ +

+ +

−

+ = +

= +

   

=   +  

   

 

= + 

 

∫

∫

∫

∫

∫

∫

∫

∫

(3) If β >0 and γ <0, (or β <0 and γ <0,), then

(

) (

) (

)

(

)

(

)

(

)

(

)

(

)

(

)

(

)

( )

.

b b

a a

b b

a a

b b

a a

b b

a a

APH F G APH F G

APH F APH G

APIH F APIH G

APIH F APIH G

β γ β γ

β γ

β γ

β γ

− − +

− +

− +

−

+ = +

= +

   

= _{ } + _{ }

   

 

=_ + _

 

∫

∫

∫

∫

∫

∫

∫

∫

(

) (

)

(

)

(

)

.

b b b

a a a

APH βF γG β APIH F γ APIH G

+

+  

+ = + 

 

∫

∫

∫

(3.14)

Hence by Theorem 3.2 βF+γG∈APIH a b

[ ]

, and

(

) (

)

(

)

(

)

.

b b b

a a a

APIH

∫

βF+γG =β APIH

∫

F+γ APIH

∫

G (3.15)



Theorem 3.4. If F∈APIH a c

[ ]

, and F∈APIH c b

[ ]

, , then F∈APIH a b

[ ]

, and

(

)

(

)

(

)

.

b c b

a a c

APIH

∫

F= APIH

∫

F+ APIH

∫

F

(3.16)

Proof. If F∈APIH a c

[ ]

, and F∈APIH c b

[ ]

, , then by Theorem 3.2

[ ]

,

,

F

−

F

+

∈

APH a c

and

F

−

,

F

+

∈

APH c b

[ ]

,

. Hence

F

−

,

F

+

∈

APH a b

[ ]

,

and

(

)

(

)

(

)

(

)

(

)

.

b c b

a a c

c b

a c

APH F APH F APH F

APIH F APIH F

− − −

−

= +

 

=_ + _

 

∫

∫

∫

∫

∫

Similarly,

(

)

(

)

(

)

.

b c b

a a c

APH F APIH F APIH F

+

+ ₌ ₊ 

 

 

∫

∫

∫

Hence by Theorem 3.2

[ ]

,

F∈APIH a b and

(

)

(

)

(

)

.

b c b

a a c

APIH

∫

F= APIH

∫

F+ APIH

∫

F

(3.17)



Theorem 3.5. If F≤G nearly everywhere on

[ ]

a b, and F G, ∈APIH a b

[ ]

, , then

(

)

(

)

.

b b

a a

APIH

∫

F≤ APIH

∫

G (3.18)

Proof. Let F≤G nearly everywhere on

[ ]

a b, and F G, ∈APIH a b

[ ]

, Then

[ ]

, , , ,

F− F G G+ − +∈APH a b and F−≤G−, F+≤G+ nearly everywhere on

[ ]

a b,

By Theorem 2.1

(

)

b

(

)

b

a a

APH

∫

F−≤ APH

∫

G− and

(

)

(

)

.

b b

a a

APH

∫

F+≤ APH

∫

G+ Hence

(

)

(

)

,

b b

a a

APIH

∫

F≤ APIH

∫

G (3.19)

by Theorem 3.2. 

Theorem 3.6. Let F G, ∈APIH a b

[ ]

, and d

(

F G,

)

is Lebesgue integrable on

[ ]

a b, . Then

(

) (

)

( ) (

)

d , d , .

b b b

a a a

APIH F APIH G L F G

 

≤

 



∫

∫



∫

(3.20)

(

) (

)

(

)

(

)

(

)

(

)

(

)

(

)

(

)

(

)

( )

( )

( )

(

)

( ) (

)

d ,

max ,

max ,

max ,

max ,

d , .

b b

a a

b b b b

a a a a

b b

a a

b b

a a

b

a b

a

APIH F APIH G

APIH F APIH G APIH F APIH G

APH F G APH F G

L F G L F G

L F G F G

L F G

− − + +

− − + +

− − + +

− − + +

 

 

 

_{       } 

 

= _{ } −_{   } −_{ }

_{       } 

 

 

= _ − − _

 

 

≤ _ − − _

 

≤ − −

=

∫

∫

∫

∫

∫

∫

∫

∫

∫

∫

∫

∫

(3.12)



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

A



∈

F R

( )

be a fuzzy subset on R. If for any

[ ]

0,1 ,

λ∈ A_λ = A_λ−,A_λ+_ and A₁≠φ, where Aλ =

{

t A t: 

( )

≥λ

}

, then A is called

a fuzzy number. If A is convex, normal, upper semi-continuous and has the compact

support, we say that A is a compact fuzzy number.

Let R denote the set of all fuzzy numbers.

Definition 4.2. [6] Let A B , ∈R, we define A≤B iff Aλ≤Bλ for all λ∈

(

0,1 ,

]

A+ =B C iff A_λ +B_λ =C_λ for any λ∈

(

0,1 ,

]

A B ⋅ =D iff A_λ⋅B_λ =D_λ for any

(

0,1 .

]

λ∈

For A B , ∈RC,

( )

[ ]0,1

(

)

, sup d ,

D A B A B_{λ λ}

λ∈

=

  _{is called the distance between} A and

.

B

Lemma 4.1. [9] If a mapping H: 0,1

[ ]

→I_R, λ→H

( )

λ =

[

m n_λ, _λ

]

, satisfies

1, 1 2, 2

m_λ n_λ m_λ n_λ

 ⊃ 

    when λ1<λ2, then

(0,1]

( )

:

A H R

λ

λ

λ

∈

= ∈



_

 _(4.1)

and

( )

1

,

n n

Aλ H λ

∞

=

=

_

(4.2)

where 1

₍

1

₎

.

1

n

n λ = − ₊ λ

 

 

Definition 4.3. [1] Let F:

[ ]

a b, →R. If the interval-valued function

( )

( )

,

( )

F t_λ = F_λ− t F_λ+ t _ is Henstock integrable on

[ ]

a b, for any λ∈

(

0,1 ,

]

then we

(

) ( )

( ]

( )

( )

( ]

( )

( )

0,1

0,1

d : d

d , d .

b b

a a

b b

a a

FH F t t IH F t t

H F t H F t

λ λ

λ λ

λ

λ

λ ∈

− +

∈

=

 

= _{ }

 

∫

∫

∫

∫



_



For brevity, we write

F t



( )

∈

FH a b

[ ]

,

.

Definition 4.4. Let

F



:

[ ]

a b

,

→

R



. If the interval-valued function

( )

( )

,

( )

)

F tλ = Fλ− t Fλ+ t  is AP-Henstock integrable on

[ ]

a b, for any λ∈

(

0,1 ,

]

then

F t



( )

is called AP-Henstock integrable on

[ ]

a b, and the integral value is

de-fined by

(

) ( )

( ]

(

)

( )

( ]

(

)

(

)

0,1

0,1

d : d

, d .

b b

a a

b b

a a

APFH F t t APIH F t t

APH F dt APH F t

λ λ

λ λ

λ

λ

λ ∈

− +

∈

=

 

= _{ }

 

∫

∫

∫

∫



_



We write

F t



( )

∈

APFH a b

[ ]

,

.

Theorem 4.1.

F



∈

APFH a b

[ ]

,

,

then

(

) ( )

d

b

a

APFH

∫

F t t∈R and

(

) ( )

(

)

( )

1

d d ,

b b

n n

a a

APFH F t t APIH F_λ t t

λ

∞

=

 

=

 



∫

 



∫

(4.3)

where 1

₍

1

₎

.

1

n

n λ = − λ

+

 

 

Proof. Let H: 0,1

(

]

→I_R, be defined by

( ) (

)

( ) (

d ,

)

( )

d .

b b

a a

H λ APH Fλ− t t APH Fλ+ t t

 

=  



∫

∫



Since

F

_λ−

( )

t

and

F

_λ+

( )

t

are increasing and decreasing on λ respectively, there-fore, when 0<λ1≤λ2 ≤1, we have Fλ1

( )

t Fλ2

( )

t ,

− _≤ −

( )

( )

1 2 ,

Fλ+ t ≥Fλ+ t on

[ ]

a b, .

From Theorem 3.5 we have

(

)

₁

( ) (

)

( )

(

)

₂

( ) (

)

₂

( )

1

d , d d , d .

b b b b

a a a a

APH F_λ− t t APH F_λ+ t t APH F_λ− t t APH F_λ+ t t

   

⊃

   



∫

∫

 

∫

∫

 (4.4)

From Theorem 3.2 and Lemma 4.1 we have

(

) ( )

(

)

(

)

(0,1]

d : d , d

b b b

a a a

APFH F t t APH F_λ t APH F_λ t R

λ

λ − +

∈

 

= _{ }∈

 

∫



_

∫

∫



_(4.5)

and for all λ∈

(

0,1 ,

]

(

) ( )

(

)

( )

1

d d ,

b b

n n

a a

APFH F t t APIH F_λ t t

λ

∞

=

 

=

 



∫

 



∫

where

(

1

)

1 .

1

n

n λ = −_ _λ

+

 

   Theorem 4.2. If F G, ∈APFH a b

[ ]

, and

β γ

, ∈R. Then

β

F+

γ

G∈APFH a b

[ ]

,

and

(

)

(

)

d

(

) ( )

d

(

) ( )

d .

b b b

a a a

APFH

∫

βF+γG t=β APFH

∫

F t t+γ APFH

∫

G t t (4.6)

( )

( )

( ) ,

F t_λ = F_λ− t F_λ+ t _ and G_λ

( )

t = G_λ−

( )

t G, _λ+

( )

t _ are AP-Henstock integrable on

[ ]

a b, for any λ∈

(

0,1

]

and

(

) ( )

(0,1]

(

) ( )

d d

b b

a a

APFH F t t APIH F t_λ t

λ

λ

∈

=

∫



_

∫

_and

(

) ( )

(0,1]

(

)

( )

d d

b b

a a

APFH G t t APIH G_λ t t

λ

λ

∈

=

∫



_

∫

_{. From Theorem 3.3 we have}

[ ]

,

F_λ G_λ APIH a b

β +γ ∈ _and

(

) (

)

d

(

)

d

(

)

d

b b b

a a a

APIH

∫

βF_λ+γG_λ t=β APIH

∫

F t_λ +γ APIH

∫

G t_λ for any λ∈

(

0,1

]

.

Hence

β

F



+

γ

G



∈

APFH a b

[ ]

,

and

(

)

(

)

( ]

(

) (

)

( ]

(

)

(

)

( ]

(

)

(

)

(

) ( )

(

) ( )

0,1 0,1 0,1 (0,1] d d d d d d

d d .

b b a a b b a a b b a a b b a a

APFH F G t APIH F G t

APIH F t APIH G t

APIH F t APIH G t

APFH F t t APFH G t t

λ λ λ λ λ λ λ λ λ λ

β γ λ β γ

λ β γ

β λ γ λ

β γ ∈ ∈ ∈ ∈ + = +   = _ + _   = + = +

∫

∫

∫

∫

∫

∫

∫

∫

   











Theorem 4.3. If

F



∈

APFH a c

[ ]

,

and

F



∈

APFH c b

[ ]

,

, then

F



∈

APFH a b

[ ]

,

and

(

) ( )

d

(

) ( )

d

(

) ( )

d .

b c b

a a c

APFH

∫

F t t= APFH

∫

F t t+ APFH

∫

F t t (4.7)

Proof. If

F



∈

APFH a c

[ ]

,

and

F



∈

APFH c b

[ ]

,

, then the interval-valued function

( )

( )

,

( )

F t_λ = F_λ− t F_λ+ t _ is AP-Henstock integrable on

[ ]

a c, and

[ ]

c b, for any

(

0,1

]

λ∈ _and

(

) ( )

(0,1]

(

) ( )

d d

c c

a a

APFH F t t APIH F t_λ t

λ

λ

∈

=

∫



_

∫

_and

(

) ( )

(0,1]

(

)

( )

d d

b b

c c

APFH F t t APIH F t_λ t

λ

λ

∈

=

∫



_

∫

_{. From Theorem 3.4 we have}

[ ]

,

Fλ∈APIH a b and

(

)

d

(

)

d

(

)

d

b c b

a a c

APIH

∫

F tλ = APIH

∫

F t+ APIH

∫

F tλ for any

(

0,1

]

λ∈ . Hence F∈APFH a b

[ ]

, and

(

) ( )

( ]

(

)

( )

( ]

(

)

( )

(

)

( )

( ]

(

)

( )

(

)

( )

(

) ( )

(

) ( )

0,1 0,1 0,1 (0,1] d d d d d d

d d .

b b a a c b a c c b a c c b a c

APFH F t t APIH F t t

APIH F t t APIH F t t

APIH F t t APIH F t t

APFH F t t APFH F t t

Theorem 4.4. If

F t



( )

≤

G t



( )

nearly everywhere on

[ ]

a b, and

[ ]

,

,

F G





∈

APFH a b

, then

(

) ( )

d

(

) ( )

d .

b b

a a

APFH

∫

F t t≤ APFH

∫

G t t (4.8)

Proof. If

F t



( )

≤

G t



( )

nearly everywhere on

[ ]

a b, and

F G



,



∈

APFH a b

[ ]

,

, then

( )

( )

F_λ t ≤G_λ t nearly everywhere on

[ ]

a b, for any λ∈

(

0,1

]

and F_λ

( )

t and

( )

G_λ t are AP-Henstock integrable on

[ ]

a b, for any λ∈

(

0,1

]

and

(

) ( )

(0,1]

(

) ( )

d d

b b

a a

APFH F t t APIH F tλ t

λ

λ

∈

=

∫



_

∫

_and

(

) ( )

(0,1]

(

)

( )

d d

b b

a a

APFH G t t APIH Gλ t t

λ

λ

∈

=

∫



_

∫

_{. From Theorem 3.5 we have}

(

)

( )

d

(

)

( )

d

b b

a a

APIH

∫

F t_λ t≤ APIH

∫

G_λ t t for any λ∈

(

0,1

]

. Hence

(

) ( )

( ]

(

)

( )

( ]

(

)

( )

(

) ( )

0,1

0,1

d d

d

d .

b b

a a

b

a

b

a

APFH F t t APIH F t t

APIH G t t

APFH G t t

λ λ

λ λ

λ

λ

∈

∈

=

≤

=

∫

∫

∫

∫











5. Conclusion

In this paper, we have a tendency to introduce the concept of the AP-Henstock inte-grals of interval-valued functions and fuzzy number-valued functions and investigate some properties of those integrals.

References

[1] Wu, C.X. and Gong, Z. (2000) On Henstock Integrals of Interval-Valued Functions and Fuzzy-Valued Functions. Fuzzy Sets and Systems, 115, 377-391.

https://doi.org/10.1016/S0165-0114(98)00277-2

[2] Henstock, R. (1963) Theory of Integration. Butterworth, London.

[3] Peng-Yee, L. (1989) Lanzhou Lectures on Henstock Integration. World Scientific, Singa-pore.

[4] Hamid, M.E. and Elmuiz, E.H. (2016) On Henstock-Stieljes Integrals of Interval-Valued Functions and Fuzzy-Valued Functions. Journal of Applied Mathematics and Physics, 4, 779-786. https://doi.org/10.4236/jamp.2016.44088

[5] Park, J.M., Park, C.G., Kim, J.B., Lee, D.H. and Lee, W.Y. (2004) The Integrals of s-Perron, sap-Perron and ap-McShane. Czechoslovak Mathematical Journal, 54, 545-557.

https://doi.org/10.1007/s10587-004-6407-7

[6] Nanda, S. (1989) On Integration of Fuzzy Mappings. Fuzzy Sets and Systems, 32, 95-101. https://doi.org/10.1016/0165-0114(89)90090-0

Sets and Systems, 44, 33-38. https://doi.org/10.1016/0165-0114(91)90030-T

[8] Wu, C.X. and Ma, M. (1992) Embedding Problem of Fuzzy Number Spaces: Part II. Fuzzy Sets and Systems, 45, 189-202. https://doi.org/10.1016/0165-0114(92)90118-N

[9] Cheng, Z.L. and Demou, W. (1983) Extension of the Integral of Interval-Valued Function and the Integral of Fuzzy-Valued Function. Fuzzy Math, 3, 45-52.

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