Open Access

Some properties of the interval-valued g ¯ -integrals and a standard interval-valued g ¯ -convolution

Journal of Inequalities and Applications20142014:88

https://doi.org/10.1186/1029-242X-2014-88

Received: 27 October 2013

Accepted: 30 January 2014

Published: 20 February 2014

Abstract

Pap and Stajner (Fuzzy Sets Syst. 102:393-415, 1999) investigated a generalized pseudo-convolution of functions based on pseudo-operations. Jang (Fuzzy Sets Syst. 222:45-57, 2013) studied the interval-valued generalized fuzzy integral by using an interval-representable pseudo-multiplication.

In this paper, by using the concepts of interval-representable pseudo-multiplication and g-integral, we define the interval-valued g ¯ -integral represented by its interval-valued generator g ¯ and a standard interval-valued g ¯ -convolution by means of the corresponding interval-valued g ¯ -integral. We also investigate some characterizations of the interval-valued g ¯ -integral and a standard interval-valued g ¯ -convolution.

MSC:28E10, 28E20, 03E72, 26E50, 11B68.

Keywords

fuzzy measureg-integralinterval-representable pseudo-multiplicationinterval-valued functioninterval-valued idempotentg-convolution

1 Introduction

Benvenuti and Mesiar [1], Daraby [2], Deschrijver [3], Grbic et al. [4], Klement et al. [5], Mesiar et al. [6], Stajner-Papuga et al. [7], Sugeno [8], Sugeno and Murofushi [9], Wu et al. [10, 11] have been studying pseudo-multiplications and various pseudo-integrals of measurable functions. Markova and Stupnanova [12], Maslov and Samborskij [13], and Pap and Stajner [14] introduced a general notion of pseudo-convolution of functions based on pseudo-mathematical operations and investigated the idempotent with respect to a pseudo-convolution.

Many researchers [3, 4, 1529] have studied the pseudo-integral of measurable multi-valued function, for example, the Aumann integral, the fuzzy integral, and the Choquet integral of measurable interval-valued functions, in many different mathematical theories and their applications.

Recently, Jang [26] defined the interval-valued generalized fuzzy integral by using an interval-representable pseudo-multiplication and investigated their characterizations. The purpose of this study is to define the interval-valued g ¯ -integral represented by its interval-valued generator g ¯ and a standard interval-valued g ¯ -convolution by means of the corresponding interval-valued g ¯ -integral, and to investigate an interval-valued idempotent function with respect to a standard interval-valued g ¯ -convolution.

This paper is organized in five sections. In Section 2, we list definitions and some properties of a pseudo-addition, a pseudo-multiplication, a g-integral, and a g-convolution of functions by means of the corresponding g-integral. In Section 3, we define an interval-representable pseudo-addition, an interval-representable pseudo-multiplication, the interval-valued g ¯ -integral represented by its interval-valued generator g ¯ , and investigate some characterizations of the interval-valued g ¯ -integral. In Section 4, we define a standard interval-valued g ¯ -convolution by means of the corresponding interval-valued g ¯ -integral and investigate some basic characterizations of them. In Section 5, we give a brief summary of results and some conclusions.

2 Definitions and preliminaries

Let X be a set, A be a σ-algebra of X, and F ( X ) be a set of all measurable functions f : X [ 0 , ) . We introduce a pseudo-addition and a pseudo-multiplication (see [14, 6, 7, 12, 14, 26, 30]).

Definition 2.1 ([12])
  1. (1)

    A binary operation : [ 0 , ] 2 [ 0 , ] is called a pseudo-addition if it satisfies the following axioms:

     
  2. (i)

    x y = y x for all x , y [ 0 , ] ,

     
  3. (ii)

    x y x z y z for all x , y , z [ 0 , ] ,

     
  4. (iii)

    ( x y ) z = x ( y z ) for all x , y , z [ 0 , ] ,

     
  5. (iv)

    0 [ 0 , ] such that x 0 = x for all x [ 0 , ] ,

     
  6. (v)

    x n x , y n y x n y n x y .

     
  1. (2)

    A binary operation : [ 0 , ] 2 [ 0 , ] is called a pseudo-multiplication with respect to if it satisfies the following axioms:

     
  2. (i)

    x y = y x for all x , y [ 0 , ] ,

     
  3. (ii)

    x ( y z ) = x ( y z ) for all x , y , z [ 0 , ] ,

     
  4. (iii)

    1 [ 0 , ] such that x 1 = x for all x [ 0 , ] ,

     
  5. (iv)

    ( x y ) z = ( x y ) ( x z ) for all x , y , z [ 0 , ] ,

     
  6. (v)

    x 0 = 0 for all x [ 0 , ] ,

     
  7. (vi)

    x y x z y z for all x , y , z [ 0 , ] .

     

Remark 2.1 ([6, 12, 14])

If g : [ 0 , ] [ 0 , ] is a generating function for a semigroup ( [ 0 , ] , , ) , then the pseudo-operations are of the following forms:
x y = g 1 ( g ( x ) + g ( y ) )
(1)
and
x y = g 1 ( g ( x ) g ( y ) ) .
(2)

Definition 2.2 ([6])

A set function μ : A [ 0 , ] is called a σ -measure if it satisfies the following axioms:
  1. (i)

    μ ( ) = 0 ,

     
  2. (ii)

    μ ( i = 1 A ) i = i = 1 μ ( A i ) for any sequence { A i } of pairwise disjoint sets from A , where i = 1 x i = lim n i = 1 n x i .

     

Let F ( X ) be the set of all measurable functions f : X [ 0 , ) . We introduce the g-integral with respect to a fuzzy measure induced by a pseudo-addition and a pseudo-multiplication in Remark 2.1.

Definition 2.3 ([6])
  1. (1)
    Let g : [ 0 , ] [ 0 , ] be a continuous strictly monotone increasing surjection function such that g ( 0 ) = 0 and f F ( X ) . The g-integral of f on A is defined by
    A f d μ = g 1 A g ( f ( x ) ) d x ,
    (3)
     
where dx is related to the Lebesgue measure and the integral on the right-hand side is the Lebesgue integral.
  1. (2)

    f is said to be integrable if A f d μ [ 0 , ) .

     

Let F ( X ) be the set of all integrable functions. Then we obtain some basic properties of the g-integral with respect to a fuzzy measure.

Theorem 2.2 (1) If A A , f , h F ( X ) and f h , then we have
A f d μ A h d μ .
(4)
  1. (2)
    Let g : [ 0 , ] [ 0 , ] be a continuous strictly monotone increasing surjection function such that g ( 0 ) = 0 , , are the same pseudo-operations as in Remark  2.1. If A A , f , h F ( X ) , then we have
    A ( f h ) d μ = A f d μ A h d μ .
    (5)
     
  2. (3)
    Let g : [ 0 , ] [ 0 , ] be a continuous strictly monotone increasing surjection function such that g ( 0 ) = 0 , , are the same pseudo-operations as in Remark  2.1, and u v = g 1 ( g ( u ) g ( v ) ) for u , v [ 0 , ) . If A A , c [ 0 , ) , h F ( X ) , then we have
    A ( c h ) d μ = c A h d μ .
    (6)
     
Proof (1) Note that if f , h F ( X ) and f h , then
g ( f ( x ) ) g ( h ( x ) ) and g 1 ( f ( x ) ) g 1 ( h ( x ) ) .
(7)
By Definition 2.3(1), (7), and the monotonicity of the Lebesgue integral,
A f d μ = g 1 A g ( f ( x ) ) d x g 1 A g ( h ( x ) ) d x = A h d μ .
(8)
  1. (2)
    By Definition 2.3(1) and the additivity of the Lebesgue integral,
    A ( f h ) d μ = g 1 A g ( g 1 ( g ( f ( x ) ) + g ( h ( x ) ) ) ) d x = g 1 A ( g ( f ( x ) ) + g ( h ( x ) ) ) d x = g 1 [ A g ( f ( x ) ) d x + A g ( h ( x ) ) d x ] = g 1 g g 1 A g ( f ( x ) ) d x + g g 1 A g ( h ( x ) ) d x = g 1 ( g A f d μ + g A h d μ ) = A f d μ A h d μ .
    (9)
     
  2. (3)
    By Definition 2.3(1) and the linearity of the Lebesgue integral,
    A ( c h ) d μ = g 1 A g ( g 1 ( g ( c ) g ( h ) ) ) d x = g 1 ( A g ( c ) g ( h ) d x ) = g 1 g ( c ) A g ( h ) d x = g 1 g ( c ) g g 1 ( A g ( h ) d x ) = g 1 g ( c ) g ( A h d μ ) = c A h d μ .
    (10)
     

By using the g-integral, we define the g-convolution of functions by means of the corresponding g-integral (see [2, 1214]). □

Definition 2.4 ([14])

Let g be the same function as in Theorem 2.2, let , be the same pseudo-operations as in Remark 2.1, u v = g 1 ( g ( u ) g ( v ) ) for u , v [ 0 , ) , and f , h F ( X ) . The g-convolution of f and h by means of the g-integral is defined by
( f h ) ( t ) = [ 0 , t ] [ f ( t u ) h ( u ) ] d μ ( u )
(11)

for all t [ 0 , ) .

Finally, we introduce the following basic characterizations of the g-convolution in [14].

Theorem 2.3 ([14])

If g is the same function as in Theorem  2.2, , are the same pseudo-operations as in Remark  2.1, u v = g 1 ( g ( u ) g ( v ) ) for u , v [ 0 , ) , and f , h F ( X ) , then we have
( f h ) ( t ) = g 1 0 t g ( f ( t u ) ) g ( h ( u ) ) d u
(12)

for all t [ 0 , ) .

Theorem 2.4 ([14])

If g is the same function as in Theorem  2.2, , are the same pseudo-operations as in Remark  2.1, u v = g 1 ( g ( u ) g ( v ) ) for u , v [ 0 , ) , and f , h , k F ( X ) , then we have
f h = h f
(13)
and
( f h ) k = f ( h k ) .
(14)

3 The interval-valued g ¯ -integrals

In this section, we consider the intervals, a standard interval-valued pseudo-addition, and a standard interval-valued pseudo-multiplication. Let I ( Y ) be the set of all closed intervals (for short, intervals) in Y as follows:
I ( Y ) = { a ¯ = [ a l , a r ] a l , a r Y  and  a l a r } ,
(15)

where Y is [ 0 , ) or [ 0 , ] . For any a Y , we define a = [ a , a ] . Obviously, a I ( Y ) (see [1, 2129]).

Definition 3.1 ([26])

If a ¯ = [ a l , a r ] , b ¯ = [ b l , b r ] , a ¯ n = [ a n l , a n r ] , a ¯ α = [ a α l , a α r ] I ( Y ) for all n N and α [ 0 , ) , and k [ 0 , ) , then we define arithmetic, maximum, minimum, order, inclusion, superior, and inferior operations as follows:
  1. (1)

    a ¯ + b ¯ = [ a l + b l , a r + b r ] ,

     
  2. (2)

    k a ¯ = [ k a l , k a r ] ,

     
  3. (3)

    a ¯ b ¯ = [ a l b l , a r b r ] ,

     
  4. (4)

    a ¯ b ¯ = [ a l b l , a r b r ] ,

     
  5. (5)

    a ¯ b ¯ = [ a l b l , a r b r ] ,

     
  6. (6)

    a ¯ b ¯ if and only if a l b l and a r b r ,

     
  7. (7)

    a ¯ < b ¯ if and only if a l b l and a l b l ,

     
  8. (8)

    a ¯ b ¯ if and only if b l a l and a r b r ,

     
  9. (9)

    sup n a ¯ n = [ sup n a n l , sup n a n r ] ,

     
  10. (10)

    inf n a ¯ n = [ inf n a n l , inf n a n r ] ,

     
  11. (11)

    sup α a ¯ α = [ sup α a α l , sup α a α r ] , and

     
  12. (12)

    inf α a ¯ α = [ inf α a α l , inf α a α r ] .

     
Definition 3.2 ([26])
  1. (1)
    A binary operation : I ( [ 0 , ] ) 2 I ( [ 0 , ] ) is called a standard interval-valued pseudo-addition if there exist pseudo-additions l and r such that x l y x r y for all x , y [ 0 , ] , and such that for all a ¯ = [ a l , a r ] , b ¯ = [ b l , b r ] I ( [ 0 , ] ) ,
    a ¯ b ¯ = [ a l l b l , a r r b r ] .
    (16)
     
Then l and r are called the representants of .
  1. (2)
    A binary operation : I ( [ 0 , ] ) 2 I ( [ 0 , ] ) is called a standard interval-valued pseudo-multiplication if there exist pseudo-multiplications l and r such that x l y x r y for all x , y [ 0 , ] , and such that for all a ¯ = [ a l , a r ] , b ¯ = [ b l , b r ] I ( [ 0 , ] ) ,
    a ¯ b ¯ = [ a l l b l , a r r b r ] .
    (17)
     

Then l and r are called the representants of .

Theorem 3.1 If two pseudo-additions l and r are representants of a standard interval-valued pseudo-addition , two pseudo-multiplications l and r are representants of a standard interval-valued pseudo-multiplication , then we have
  1. (1)

    x ¯ y ¯ = y ¯ x ¯ for all x ¯ , y ¯ I ( [ 0 , ] ) ,

     
  2. (2)

    ( x ¯ y ¯ ) z ¯ = x ¯ ( y ¯ z ¯ ) for all x ¯ , y ¯ , z ¯ I ( [ 0 , ] ) ,

     
  3. (3)

    x ¯ y ¯ = y ¯ x ¯ for all x ¯ , y ¯ I ( [ 0 , ] ) ,

     
  4. (4)

    ( x ¯ y ¯ ) z ¯ = x ¯ ( y ¯ z ¯ ) for all x ¯ , y ¯ , z ¯ I ( [ 0 , ] ) ,

     
  5. (5)

    x ¯ ( y ¯ z ¯ ) = ( x ¯ y ¯ ) ( x ¯ z ¯ ) for all x ¯ , y ¯ , z ¯ I ( [ 0 , ] ) .

     
Proof (1) By the commutativity of l and r , for any x ¯ , y ¯ I ( [ 0 , ] ) , we have
x ¯ y ¯ = [ x l l y l , x r r y r ] = [ y l l x l , y r r x r ] = y ¯ x ¯ .
(18)
  1. (2)
    By the associativity of l and r , for any x ¯ , y ¯ , z ¯ I ( [ 0 , ] ) , we have
    ( x ¯ y ¯ ) z ¯ = [ x l l y l , x r r y r ] [ z l , z r ] = [ ( x l l y l ) l z l , ( x r r y r ) r z r ] = [ x l l ( y l l z l ) , x r r ( y r r z r ) ] = [ x l , x r ] [ y l l x l , y r r x r ] = x ¯ ( y ¯ z ¯ ) .
    (19)
     
  2. (3)
    By the commutativity of l and r , for any x ¯ , y ¯ I ( [ 0 , ] ) , we have
    x ¯ y ¯ = [ x l l y l , x r r y r ] = [ y l l x l , y r r x r ] = y ¯ x ¯ .
    (20)
     
  3. (4)
    By the associativity of l and r in Definition 2.1(2)(ii), for any x ¯ , y ¯ , z ¯ I ( [ 0 , ] ) , we have
    ( x ¯ y ¯ ) z ¯ = [ x l l y l , x r r y r ] [ z l , z r ] = [ ( x l l y l ) l z l , ( x r r y r ) r z r ] = [ x l l ( y l l z l ) , x r r ( y r r z r ) ] = [ x l , x r ] I [ y l l x l , y r r x r ] = x ¯ ( y ¯ z ¯ ) .
    (21)
     
  4. (5)
    By the distributivity of s and s for s = l , r in Definition 2.1(2)(iv), for any x ¯ , y ¯ , z ¯ I ( [ 0 , ] ) , we have
    x ¯ ( y ¯ z ¯ ) = [ x l , x r ] [ y l l z l , y r r z r ] = [ x l l ( y l l z l ) , x r r ( y r r z r ) ] = [ ( x l l y l ) l ( x l l z l ) , ( x r r y r ) r ( x r r z r ) ] = [ x l l y l , x r r y r ] [ x l l z l , x r r z r ] = ( x ¯ y ¯ ) ( x ¯ z ¯ ) .
    (22)
     

By using a standard interval-valued pseudo-addition and a standard interval-valued pseudo-multiplication, we define the interval-valued g ¯ -integral represented by its interval-valued generator g ¯ . □

Definition 3.3 Let X be a set, two pseudo-additions l and r be representants of a standard interval-valued pseudo-addition , and two pseudo-multiplications l and r be representants of a standard interval-valued pseudo-multiplication .
  1. (1)
    An interval-valued function f ¯ : X I ( [ 0 , ) ) { } is said to be measurable if for any open set O [ 0 , ) ,
    f ¯ 1 ( O ) = { x X f ¯ ( x ) O } A .
    (23)
     
  2. (2)
    Let g s be a continuous strictly increasing surjective function for s = l , r such that g l g r , g ¯ = [ g l , g r ] , and g s ( 0 ) = 0 for s = l , r . The interval-valued g ¯ -integral with respect to a fuzzy measure μ of a measurable interval-valued function f ¯ = [ f l , f r ] is defined by
    A f ¯ d μ = [ A l f l l d μ , A r f r r d μ ]
    (24)
     
for all A A .
  1. (3)
    f ¯ is said to be integrable on A A if
    A f ¯ d μ I ( [ 0 , ] ) .
    (25)
     

Let IF ( X ) be the set of all measurable interval-valued functions and IF ( X ) be the set of all integrable interval-valued functions. Then, by Definition 3.3, we directly obtain the following theorem.

Theorem 3.2 If g s is a continuous strictly increasing surjective function for s = l , r such that g l g r , g ¯ = [ g l , g r ] , and g s ( 0 ) = 0 for s = l , r , two pseudo-additions l and r are representants of a standard interval-valued pseudo-addition , and two pseudo-multiplications l and r are representants of a standard interval-valued pseudo-multiplication , then we have
A f ¯ d μ = [ g l 1 A g l ( f l ( x ) ) d x , g r 1 A g r ( f r ( x ) ) d x ] .
(26)
Proof By Definition 2.3(1),
A s f s s d μ = g s 1 A g s ( f s ( x ) ) d x
(27)
for s = l , r . By (27) and Definition 3.3, we have
A f ¯ d μ = [ A l f l l d μ , A r f r r d μ ] = [ g l 1 A g l ( f l ( x ) ) d x , g r 1 A g r ( f r ( x ) ) d x ] .
(28)

By the definition of the interval-valued g ¯ -integral, we directly obtain the following basic properties. □

Theorem 3.3 Let g s be a continuous strictly increasing surjective function for s = l , r such that g l g r , g ¯ = [ g l , g r ] , and g s ( 0 ) = 0 for s = l , r , two pseudo-additions l and r be representants of a standard interval-valued pseudo-addition , two pseudo-multiplications l and r be representants of a standard interval-valued pseudo-multiplication , and two pseudo-multiplications l and r be representants of a standard interval-valued pseudo-multiplication .
  1. (1)
    If A A and f ¯ , h ¯ IF ( X ) and f ¯ h ¯ , then we have
    A f ¯ d μ A h ¯ d μ .
    (29)
     
  2. (2)
    If A A and f ¯ , h ¯ IF ( X ) , then we have
    A ( f ¯ h ¯ ) d μ = A f ¯ d μ A h ¯ d μ .
    (30)
     
  3. (3)
    If A A and c ¯ = [ c l , c r ] I ( [ 0 , ) ) , h ¯ IF ( X ) , then we have
    A ( c ¯ h ¯ ) d μ = c ¯ A h ¯ d μ .
    (31)
     
Proof (1) Note that if f ¯ , h ¯ IF ( X ) and f ¯ h ¯ , then
f s h s
(32)
for s = l , r . Since g l and g r are strictly monotone increasing,
g s f s g s h s
(33)
for s = l , r . By (33) and Theorem 2.2(1),
A s f s s d μ A s h s s d μ
(34)
for s = l , r . By (34) and Theorem 3.2,
A f ¯ d μ = [ A l f l l d μ , A r f r r d μ ] [ A l h l l d μ , A r h r r d μ ] = A h ¯ d μ .
(35)
  1. (2)
    Note that if f ¯ , h ¯ IF ( X ) , then
    f ¯ h ¯ = [ f l l h l , f r r h r ] .
    (36)
     
By Theorem 2.2(2),
A s ( f s s h s ) s d μ = A s f s s d μ s A s h s s d μ
(37)
for s = l , r . By (37) and Theorem 3.2,
A ( f ¯ h ¯ ) d μ = [ A l ( f l l h l ) l d μ , A r ( f r r h r ) r d μ ] = [ A l f l l d μ l A l h l l d μ , A r f r r d μ r A r h r r d μ ] = [ A l f l l d μ , A r f r r d μ ] I [ A l h l l d μ , A r h r r d μ ] = A f ¯ d μ A h ¯ d μ .
(38)
  1. (3)
    Note that if f ¯ IF ( X ) and c ¯ I ( [ 0 , ) ) , then
    c ¯ f ¯ = [ c l l f l , c r r f r ] .
    (39)
     
By Theorem 2.2(3),
A s ( c s s f s ) s d μ = c s s A s f s s d μ
(40)
for s = l , r . By (40) and Definition 3.3(2),
A ( c ¯ f ¯ ) d μ = [ A l ( c l l f l ) l d μ , A r ( c r r f r ) r d μ ] = [ c l l A l f l l d μ , c r r A r f r r d μ ] = c ¯ A f ¯ d μ .
(41)

 □

4 An interval-valued g ¯ -convolution

In this section, by using the interval-valued g ¯ -integral, we define the interval-valued g ¯ -convolution of interval-valued functions in IF ( X ) .

Definition 4.1 If g ¯ , , , and satisfy the hypotheses of Theorem 3.2, then the interval-valued g ¯ -convolution is defined by
( f ¯ h ¯ ) ( t ) = [ 0 , t ] [ f ¯ ( t u ) h ¯ ( u ) ] d μ ( u )
(42)

for all t [ 0 , ) .

From Definition 4.1, we directly obtain some characterization of an interval-valued g ¯ -convolution by means of the interval-valued g ¯ -integrals.

Theorem 4.1 If g ¯ , , , and satisfy the hypotheses of Theorem  3.2, then we have
f ¯ h ¯ = [ f l l h l , f r r h r ] ,
(43)

where ( f s s h s ) ( t ) = [ 0 , t ] s f s ( t u ) s d μ for s = l , r .

Proof By Definition 2.4, we have
( f s s h s ) ( t ) = [ 0 , t ] s f s ( t u ) s h s ( u ) s d μ ( u )
(44)
for s = l , r . By Theorem 3.2 and (44),
( f ¯ h ¯ ) ( t ) = [ 0 , t ] ( f ¯ ( t u ) h ¯ ( u ) ) d μ ( u ) = [ 0 , t ] [ f l ( t u ) l h l ( u ) , f r ( t u ) r h r ( u ) ] d μ ( u ) = [ [ 0 , t ] l f l ( t u ) l h l ( u ) l d μ , [ 0 , t ] r f r ( t u ) r h r ( u ) r d μ ] = [ ( f l l h l ) ( t ) , ( f r r h r ) ( t ) ] .
(45)

 □

From Theorem 4.1, we investigate the commutativity and the associativity of a standard interval-valued g ¯ -convolution.

Theorem 4.2 If g ¯ , , , and satisfy the hypotheses of Theorem  3.2 and f ¯ , h ¯ , and k ¯ IF ( X ) , then we have
  1. (1)

    f ¯ h ¯ = h ¯ f ¯ ,

     
  2. (2)

    ( f ¯ h ¯ ) k ¯ = f ¯ ( h ¯ k ¯ ) .

     
Proof Let f ¯ = [ f l , f r ] , h ¯ = [ h l , h r ] , k ¯ = [ k l , k r ] IF ( X ) . By (16), we have
f l l h l = h l l f l and f r r h r = h r r f r .
(46)
By Theorem 4.1 and (46), we have
f ¯ h ¯ = [ f l l h l , f r r h r ] = [ h l l f l , h r r f r ] = h ¯ f ¯ .
(47)
By (17), we have
( f l l h l ) l k l = f l l ( h l l k l ) and ( f r r h r ) r k r = f r r ( h r r k r ) .
(48)
By Theorem 4.1 and (48),
( f ¯ h ¯ ) k ¯ = [ ( f l l h l ) l k l , ( f r r h r ) r k r ] = [ f l l ( h l l k l ) , f r r ( h r r k r ) ] = f ¯ ( h ¯ k ¯ ) .
(49)

 □

Finally, we illustrate the following examples which are related with the interval-valued g ¯ -integral and the interval-valued g ¯ -convolution as follows.

Example 4.1 We give three examples of the interval-valued g ¯ -integral.
  1. (1)
    If g l ( x ) = g r ( x ) = x for all x [ 0 , ] are the generators of l , r , l , and r , and f ¯ ( x ) = [ e x 2 , e x ] for all x [ 0 , ) , and A = [ 0 , t ] for all t [ 0 , ) , then we have
    A f ¯ d μ = [ g l 1 0 t g l ( f l ( x ) ) d x , g r 1 0 t g r ( f r ( x ) ) d x ] = [ 0 t 1 2 e x d x , 0 t e x d x ] = [ 1 2 ( 1 e t ) , ( 1 e t ) ] .
    (50)
     
  2. (2)
    If g l ( x ) = 1 2 x , g r ( x ) = x for all x [ 0 , ] are the generators of l , r , and g l ( x ) = g r ( x ) = x for all x [ 0 , ] are the generators of l , r , and f ¯ ( x ) = [ e x 2 , e x ] for all x [ 0 , ) , and A = [ 0 , t ] for all t [ 0 , ) , then we have
    A f ¯ d μ = [ g l 1 0 t g l ( f l ( x ) ) d x , g r 1 0 t g r ( f r ( x ) ) d x ] = [ 2 0 t 1 4 e x d x , 0 t e x d x ] = [ 1 2 ( 1 e t ) , ( 1 e t ) ] .
    (51)
     
  3. (3)
    If g l ( x ) = x 2 , g r ( x ) = 3 x 2 for all x [ 0 , ] are the generators of l , r , and g l ( x ) = g r ( x ) = x for all x [ 0 , ] are the generators of l , r , and f ¯ ( x ) = [ e x 2 , e x ] for all x [ 0 , ) , and A = [ 0 , t ] for all t [ 0 , ) , then we have
    A f ¯ d μ = [ g l 1 0 t g l ( f l ( x ) ) d x , g r 1 0 t g r ( f r ( x ) ) d x ] = [ 0 t 1 2 e 2 x d x , 1 3 0 t 3 e 2 x d x ] = [ 1 4 ( 1 e 2 t ) , 1 2 ( 1 e t ) ] .
    (52)
     

Example 4.2 We give an example of the interval-valued g ¯ -convolution.

If g l ( x ) = x 2 , g r ( x ) = 3 x 2 for all x [ 0 , ] are the generators of l , r , and g l ( x ) = g r ( x ) = x for all x [ 0 , ] are the generators of l , r , l , r , and f ¯ ( x ) = [ e x 2 , e x ] for all x [ 0 , ) , h ¯ ( x ) = [ 1 2 x , x ] for all x [ 0 , ) , and A = [ 0 , t ] for all t [ 0 , ) , then we have
( f ¯ h ¯ ) ( t ) = A [ f ¯ ( t u ) h ¯ ( u ) ] d μ ( u ) = [ 1 2 0 t e 2 ( t u ) e 2 u d u , 3 0 t 1 2 e 2 ( x u ) e 2 u d u ] = [ t 4 e 2 t , 3 t 2 e 2 t ] .
(53)

5 Conclusions

In this paper, we have considered the g-integral represented by its generating g, the pseudo-addition, the pseudo-multiplication (see Definition 2.3). This study was to define the g-convolution by means of the g-integral (see Definition 2.4) and to investigate some characterizations of the g-integral and the commutativity and the associativity of the g-convolution (see Theorems 2.2, 2.3, and 2.4).

We also defined the interval-valued g ¯ -integral represented by its interval-valued generator g ¯ . By using general notions of an interval-representable pseudo-multiplication (see Definition 3.2), we defined an interval-valued g ¯ -integral (see Definition 3.3) and investigated some basic characterizations of them (see Theorems 3.2, 3.3).

From Definitions 2.3, 2.4, and Theorems 2.2, 2.3, we defined a standard interval-valued g ¯ -convolution (see Definition 4.1). We also investigated some characterizations of a standard interval-valued g ¯ -convolution of interval-valued functions by means of the interval-valued g ¯ -integral including commutativity and associativity of an interval-representable convolution (see Theorems 4.1, 4.2).

In the future, we can study various inequalities of the interval-valued g ¯ -integral and expect that the standard interval-valued g ¯ -convolutions are used (i) to generalize the g-Laplace transform, Hamilton-Jacobi equation on the space of functions, such as in nonlinearity and optimization and such as in information theory (see [1, 14, 29]); (ii) to generalize the Stolasky-type inequality for the pseudo-integral of functions such as in economics, finance, decision making (see [2, 30]), etc.

Declarations

Acknowledgements

This paper was supported by Konkuk University in 2014.

Authors’ Affiliations

(1)
General Education Institute, Konkuk University

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© Jang; licensee Springer. 2014

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