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

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.

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)

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

  3. (ii)

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

  4. (iii)

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

  5. (iv)

    0[0,] such that x0=x for all x[0,],

  6. (v)

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

  1. (2)

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

  2. (i)

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

  3. (ii)

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

  4. (iii)

    1[0,] such that x1=x for all x[0,],

  5. (iv)

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

  6. (v)

    x0=0 for all x[0,],

  7. (vi)

    xyxzyz 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:

xy= g 1 ( g ( x ) + g ( y ) )
(1)

and

xy= 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 fF(X). The g-integral of f on A is defined by

    A fdμ= g 1 A g ( f ( x ) ) dx,
    (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 fdμ[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 AA, f,h F (X) and fh, then we have

A fdμ A hdμ.
(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 AA, f,h F (X), then we have

    A (fh)dμ= A fdμ A hdμ.
    (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 uv= g 1 (g(u)g(v)) for u,v[0,). If AA, c[0,), h F (X), then we have

    A (ch)dμ=c A hdμ.
    (6)

Proof (1) Note that if f,h F (X) and fh, 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, uv= 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

(fh)(t)= [ 0 , t ] [ f ( t u ) h ( u ) ] dμ(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, uv= g 1 (g(u)g(v)) for u,v[0,), and f,h F (X), then we have

(fh)(t)= g 1 0 t g ( f ( t u ) ) g ( h ( u ) ) du
(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, uv= g 1 (g(u)g(v)) for u,v[0,), and f,h,k F (X), then we have

fh=hf
(13)

and

(fh)k=f(hk).
(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 aY, we define a=[a,a]. Obviously, aI(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 nN 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 yx 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 yx 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 ¯ :XI([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 ¯ dμ= [ A l f l l d μ , A r f r r d μ ]
    (24)

for all AA.

  1. (3)

    f ¯ is said to be integrable on AA if

    A f ¯ dμI ( [ 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 ¯ dμ= [ 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 dμ= g s 1 A g s ( f s ( x ) ) dx
(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 AA and f ¯ , h ¯ IF (X) and f ¯ h ¯ , then we have

    A f ¯ dμ A h ¯ dμ.
    (29)
  2. (2)

    If AA and f ¯ , h ¯ IF (X), then we have

    A ( f ¯ h ¯ ) dμ= A f ¯ dμ A h ¯ dμ.
    (30)
  3. (3)

    If AA and c ¯ =[ c l , c r ]I([0,)), h ¯ IF (X), then we have

    A ( c ¯ h ¯ ) dμ= c ¯ A h ¯ dμ.
    (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 dμ A s h s s dμ
(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 dμ= A s f s s dμ s A s h s s dμ
(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 dμ= c s s A s f s s dμ
(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 ) ] dμ(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 (tu) s dμ for s=l,r.

Proof By Definition 2.4, we have

( f s s h s )(t)= [ 0 , t ] s f s (tu) s h s (u) s dμ(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.

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Acknowledgements

This paper was supported by Konkuk University in 2014.

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Correspondence to Lee-Chae Jang.

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Jang, LC. Some properties of the interval-valued g ¯ -integrals and a standard interval-valued g ¯ -convolution. J Inequal Appl 2014, 88 (2014). https://doi.org/10.1186/1029-242X-2014-88

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Keywords

  • fuzzy measure
  • g-integral
  • interval-representable pseudo-multiplication
  • interval-valued function
  • interval-valued idempotent
  • g-convolution