R E S E A R C H
Open Access
Some properties of the interval-valued
g
-integrals and a standard interval-valued
g
-convolution
Lee-Chae Jang
**Correspondence:
General Education Institute, Konkuk University, Chungju, 138-701, Korea
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 andg-integral, we define the interval-valuedg-integral represented by its
interval-valued generatorgand a standard interval-valuedg-convolution by means of the corresponding interval-valuedg-integral. We also investigate some
characterizations of the interval-valuedg-integral and a standard interval-valued
g-convolution.
MSC: 28E10; 28E20; 03E72; 26E50; 11B68
Keywords: fuzzy measure;g-integral; interval-representable pseudo-multiplication; interval-valued function; interval-valued idempotent;g-convolution
1 Introduction
Benvenuti and Mesiar [], Daraby [], Deschrijver [], Grbicet al.[], Klementet al.[], Mesiaret al.[], Stajner-Papuga et al.[], Sugeno [], Sugeno and Murofushi [], Wu
et al.[, ] have been studying pseudo-multiplications and various pseudo-integrals of measurable functions. Markova and Stupnanova [], Maslov and Samborskij [], and Pap and Stajner [] 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 [, , –] 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 [] 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-valuedg-integral represented by its interval-valued generatorgand a standard interval-valuedg-convolution by means of the corresponding interval-valuedg-integral, and to investigate an interval-valued idempo-tent function with respect to a standard interval-valuedg-convolution.
This paper is organized in five sections. In Section , we list definitions and some prop-erties 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 , we define an interval-representable pseudo-addition, an interval-representable pseudo-multiplication, the interval-valuedg-integral represented by its interval-valued generatorg, and inves-tigate some characterizations of the interval-valuedg-integral. In Section , we define a standard interval-valuedg-convolution by means of the corresponding interval-valuedg -integral and investigate some basic characterizations of them. In Section , we give a brief summary of results and some conclusions.
2 Definitions and preliminaries
LetXbe a set,Abe aσ-algebra ofX, andF(X) be a set of all measurable functionsf :
X−→[,∞). We introduce a pseudo-addition and a pseudo-multiplication (see [–, , , , , , ]).
Definition .([]) () A binary operation⊕: [,∞]−→[,∞] is called a pseudo-addition if it satisfies the following axioms:
(i) x⊕y=y⊕xfor allx,y∈[,∞],
(ii) x≤y⇒x⊕z≤y⊕zfor allx,y,z∈[,∞],
(iii) (x⊕y)⊕z=x⊕(y⊕z)for allx,y,z∈[,∞],
(iv) ∃∈[,∞]such thatx⊕=xfor allx∈[,∞],
(v) xn−→x,yn−→y⇒xn⊕yn−→x⊕y.
() A binary operation: [,∞]−→[,∞] is called a pseudo-multiplication with
re-spect to⊕if it satisfies the following axioms:
(i) xy=yxfor allx,y∈[,∞],
(ii) x(yz) =x(yz)for allx,y,z∈[,∞],
(iii) ∃∈[,∞]such thatx=xfor allx∈[,∞],
(iv) (xy)⊕z= (xy)⊕(xz)for allx,y,z∈[,∞],
(v) x=for allx∈[,∞],
(vi) x≤y⇒xz≤yzfor allx,y,z∈[,∞].
Remark .([, , ]) Ifg: [,∞]−→[,∞] is a generating function for a semigroup ([,∞],⊕,), then the pseudo-operations are of the following forms:
x⊕y=g–g(x) +g(y) ()
and
xy=g–g(x)g(y). ()
Definition .([]) A set functionμ:A−→[,∞] is called aσ–⊕-measure if it satisfies the following axioms:
(i) μ(∅) = , (ii) μ(∞i=A)i=
∞
i=μ(Ai)for any sequence{Ai}of pairwise disjoint sets fromA,
where∞i=xi=limn→∞
LetF(X) be the set of all measurable functionsf :X−→[,∞). We introduce theg -integral with respect to a fuzzy measure induced by a pseudo-addition⊕and a pseudo-multiplicationin Remark ..
Definition .([]) () Letg: [,∞]−→[,∞] be a continuous strictly monotone in-creasing surjection function such thatg() = andf ∈F(X). Theg-integral off onAis defined by
⊕
A
fdμ=g–
A
gf(x)dx, ()
wheredxis related to the Lebesgue measure and the integral on the right-hand side is the Lebesgue integral.
()f is said to be integrable ifA⊕fdμ∈[,∞).
LetF∗(X) be the set of all integrable functions. Then we obtain some basic properties of theg-integral with respect to a fuzzy measure.
Theorem . ()If A∈A,f,h∈F∗(X)and f ≤h,then we have
⊕
A
fdμ≤
⊕
A
hdμ. ()
()Let g: [,∞]−→[,∞]be a continuous strictly monotone increasing surjection func-tion such that g() = ,⊕,are the same pseudo-operations as in Remark..If A∈A, f,h∈F∗(X),then we have
⊕
A
(f ⊕h)dμ=
⊕
A
fdμ⊕
⊕
A
hdμ. ()
() Let g : [,∞]−→[,∞]be a continuous strictly monotone increasing surjection function such that g() = ,⊕,are the same pseudo-operations as in Remark.,and u⊗v=g–(g(u)g(v))for u,v∈[,∞).If A∈A,c∈[,∞),h∈F∗(X),then we have
⊕
A
(c⊗h)dμ=c⊗
⊕
A
hdμ. ()
Proof () Note that iff,h∈F∗(X) andf≤h, then
gf(x)≤gh(x) and g–f(x)≤g–h(x). ()
By Definition .(), (), and the monotonicity of the Lebesgue integral,
⊕
A
fdμ=g–
A
gf(x)dx
≤g–
A
gh(x)dx
=
⊕
A
() By Definition .() and the additivity of the Lebesgue integral,
⊕
A
(f ⊕h)dμ=g–
A
gg–gf(x)+gh(x)dx
=g–
A
gf(x)+gh(x)dx
=g–
A
gf(x)dx+
A
gh(x)dx
=g–gg–
A
gf(x)dx+gg–
A
gh(x)dx
=g–
g
⊕
A
f dμ+g
⊕
A
hdμ
=
⊕
A
f dμ⊕
⊕
A
hdμ. ()
() By Definition .() and the linearity of the Lebesgue integral,
⊕
A
(c⊗h)dμ=g–
A
gg–g(c)g(h)dx
=g–
A
g(c)g(h)dx
=g–g(c)
A
g(h)dx
=g–g(c)gg–
A
g(h)dx
=g–g(c)g
⊕
A
hdμ
=c⊗
⊕
A
hdμ. ()
By using theg-integral, we define theg-convolution of functions by means of the
corre-spondingg-integral (see [, –]).
Definition .([]) Letgbe the same function as in Theorem ., let⊕,be the same pseudo-operations as in Remark .,u⊗v=g–(g(u)g(v)) foru,v∈[,∞), andf,h∈F∗(X).
Theg-convolution off andhby means of theg-integral is defined by
(f ∗h)(t) =
⊕
[,t]
f(t–u)⊗h(u)dμ(u) ()
for allt∈[,∞).
Finally, we introduce the following basic characterizations of theg-convolution in [].
we have
(f ∗h)(t) =g–
t
gf(t–u)gh(u)du ()
for all t∈[,∞).
Theorem .([]) If g is the same function as in Theorem.,⊕,are the same pseudo-operations as in Remark.,u⊗v=g–(g(u)g(v))for u,v∈[,∞),and f,h,k∈F∗(X),then
we have
f ∗h=h∗f ()
and
(f ∗h)∗k=f ∗(h∗k). ()
3 The interval-valuedg-integrals
In this section, we consider the intervals, a standard interval-valued pseudo-addition, and a standard interval-valued pseudo-multiplication. LetI(Y) be the set of all closed intervals (for short, intervals) inYas follows:
I(Y) =a= [al,ar]|al,ar∈Yandal≤ar
, ()
whereYis [,∞) or [,∞]. For anya∈Y, we definea= [a,a]. Obviously,a∈I(Y) (see [, –]).
Definition .([]) Ifa= [al,ar],b= [bl,br],an= [anl,anr],aα= [aαl,aαr]∈I(Y) for all
n∈Nandα∈[,∞), andk∈[,∞), then we define arithmetic, maximum, minimum, order, inclusion, superior, and inferior operations as follows:
() a+b= [al+bl,ar+br],
() ka= [kal,kar],
() ab= [albl,arbr],
() a∨b= [al∨bl,ar∨br],
() a∧b= [al∧bl,ar∧br],
() a≤bif and only ifal≤blandar≤br,
() a<bif and only ifal≤blandal=bl,
() a⊂bif and only ifbl≤alandar≤br,
() supnan= [supnanl,supnanr],
() infnan= [infnanl,infnanr],
() supαaα= [supαaαl,supαaαr], and
() infαaα= [infαaαl,infαaαr].
Definition .([]) () A binary operation:I([,∞])−→I([,∞]) is called a
stan-dard interval-valued pseudo-addition if there exist pseudo-additions⊕land⊕rsuch that
x⊕ly≤x⊕ryfor allx,y∈[,∞], and such that for alla= [al,ar],b= [bl,br]∈I([,∞]),
ab= [al⊕lbl,ar⊕rbr]. ()
Then⊕land⊕rare called the representants of
() A binary operation:I([,∞])−→I([,∞]) is called a standard interval-valued
pseudo-multiplication if there exist pseudo-multiplicationslandrsuch thatxly≤
xryfor allx,y∈[,∞], and such that for alla= [al,ar],b= [bl,br]∈I([,∞]),
ab= [allbl,arrbr]. ()
Thenlandrare called the representants of
.
Theorem . If two pseudo-additions⊕land⊕rare representants of a standard
interval-valued pseudo-addition,two pseudo-multiplicationslandrare representants of a
standard interval-valued pseudo-multiplication,then we have
() xy=yxfor allx,y∈I([,∞]),
() (xy)z=x(yz)for allx,y,z∈I([,∞]), () xy=yxfor allx,y∈I([,∞]),
() (xy)z=x(yz)for allx,y,z∈I([,∞]), () x(yz) = (xy)(xz)for allx,y,z∈I([,∞]).
Proof () By the commutativity of⊕land⊕r, for anyx,y∈I([,∞]), we have
xy= [xl⊕lyl,xr⊕ryr]
= [yl⊕lxl,yr⊕rxr]
=yx. ()
() By the associativity of⊕land⊕r, for anyx,y,z∈I([,∞]), we have
xy z= [xl⊕lyl,xr⊕ryr]
[zl,zr]
=(xl⊕lyl)⊕lzl, (xr⊕ryr)⊕rzr
=xl⊕l(yl⊕lzl),xr⊕r(yr⊕rzr)
= [xl,xr]
[yl⊕lxl,yr⊕rxr]
=x yz. ()
() By the commutativity oflandr, for anyx,y∈I([,∞]), we have
xy= [xllyl,xrryr]
= [yllxl,yrrxr]
=yx. ()
() By the associativity oflandrin Definition .()(ii), for anyx,y,z∈I([,∞]), we
have
(xy)z= [xllyl,xrryr]
[zl,zr]
=(xllyl)lzl, (xrryr)rzr
=xll(yllzl),xrr(yrrzr)
= [xl,xr]
I[yllxl,yrrxr]
=x(yz). ()
() By the distributivity of⊕sandsfors=l,rin Definition .()(iv), for anyx,y,z∈
I([,∞]), we have
x yz= [xl,xr]
[yl⊕lzl,yr⊕rzr]
=xll(yl⊕lzl),xrr(yr⊕rzr)
=(xllyl)⊕l(xllzl), (xrryr)⊕r(xrrzr)
= [xllyl,xrryr]
[xllzl,xrrzr]
= (xy)(xz). ()
By using a standard interval-valued pseudo-addition and a standard interval-valued pseudo-multiplication, we define the interval-valuedg-integral represented by its
interval-valued generatorg.
Definition . Let Xbe a set, two pseudo-additions⊕l and⊕r be representants of a
standard interval-valued pseudo-addition, and two pseudo-multiplicationslandr
be representants of a standard interval-valued pseudo-multiplication.
() An interval-valued functionf :X→I([,∞))\ {∅}is said to be measurable if for any open setO⊂[,∞),
f–(O) =x∈X|f(x)∩O=∅∈A. () () Let gsbe a continuous strictly increasing surjective function fors=l,rsuch that
gl≤gr,g= [gl,gr], andgs() = fors=l,r. The interval-valuedg-integral with respect to a
fuzzy measureμof a measurable interval-valued functionf = [fl,fr] is defined by
A
f dμ=
⊕l
A
flldμ,
⊕r
A
frrdμ ()
for allA∈A.
()f is said to be integrable onA∈Aif
A
f dμ∈I[,∞]. ()
LetIF(X) be the set of all measurable interval-valued functions andIF∗(X) be the set of all integrable interval-valued functions. Then, by Definition ., we directly obtain the following theorem.
Theorem . If gsis a continuous strictly increasing surjective function for s=l,r such that
gl≤gr,g= [gl,gr],and gs() = for s=l,r,two pseudo-additions⊕land⊕rare
landrare representants of a standard interval-valued pseudo-multiplication
,then we have
A
f dμ=
gl–
A
gl
fl(x)
dx,gr–
A
gr
fr(x)
dx . ()
Proof By Definition .(),
⊕s
A
fssdμ=gs–
A
gs
fs(x)
dx ()
fors=l,r. By () and Definition ., we have
A
f dμ=
⊕l
A
flldμ,
⊕r
A
frrdμ
=
gl–
A
gl
fl(x)
dx,gr–
A
gr
fr(x)
dx . ()
By the definition of the interval-valuedg-integral, we directly obtain the following basic
properties.
Theorem . Let gsbe a continuous strictly increasing surjective function for s=l,r such
that gl≤gr,g= [gl,gr],and gs() = for s=l,r,two pseudo-additions⊕land⊕rbe
repre-sentants of a standard interval-valued pseudo-addition,two pseudo-multiplicationsl
andrbe representants of a standard interval-valued pseudo-multiplication
,and two pseudo-multiplications⊗land⊗rbe representants of a standard interval-valued
pseudo-multiplication.
() IfA∈Aandf,h∈IF∗(X)andf ≤h,then we have
A
f dμ≤
A
hdμ. ()
() IfA∈Aandf,h∈IF∗(X),then we have
A
f h dμ=
A
f dμ
A
hdμ. ()
() IfA∈Aandc= [cl,cr]∈I([,∞)),h∈IF∗(X),then we have
A
ch dμ=c
A
hdμ. ()
Proof () Note that iff,h∈IF∗(X) andf≤h, then
fs≤hs ()
fors=l,r. Sinceglandgrare strictly monotone increasing,
fors=l,r. By () and Theorem .(),
⊕s
A
fssdμ≤
⊕s
A
hssdμ ()
fors=l,r. By () and Theorem .,
A
f dμ=
⊕l
A
flldμ,
⊕r
A
frrdμ
≤
⊕l
A
hlldμ,
⊕r
A
hrrdμ =
A
hdμ. ()
() Note that iff,h∈IF∗(X), then
f h= [fl⊕lhl,fr⊕rhr]. ()
By Theorem .(),
⊕s
A
(fs⊕shs)sdμ=
⊕s
A
fssdμ⊕s
⊕s
A
hssdμ ()
fors=l,r. By () and Theorem .,
A
f h dμ
=
⊕l
A
(fl⊕lhl)ldμ,
⊕r
A
(fr⊕rhr)rdμ
=
⊕l
A
flldμ⊕l
⊕l
A
hlldμ,
⊕r
A
frrdμ⊕r
⊕r
A
hrrdμ
=
⊕l
A
flldμ,
⊕r
A
frrdμ
I
⊕l
A
hlldμ,
⊕r
A
hrrdμ
=
A
f dμ
A
hdμ. ()
() Note that iff ∈IF∗(X) andc∈I([,∞)), then
cf = [cl⊗lfl,cr⊗rfr]. ()
By Theorem .(),
⊕s
A
(cs⊗sfs)sdμ=cs⊗s
⊕s
A
fssdμ ()
fors=l,r. By () and Definition .(),
A
cf dμ
=
⊕l
A
(cl⊗lfl)ldμ,
⊕r
A
=
cl⊗l
⊕l
A
flldμ,cr⊗r
⊕r
A
frrdμ
=c
A
f dμ. ()
4 An interval-valuedg-convolution
In this section, by using the interval-valuedg-integral, we define the interval-valuedg -convolution of interval-valued functions inIF∗(X).
Definition . If g, , , and satisfy the hypotheses of Theorem ., then the interval-valuedg-convolution is defined by
(f h)(t) =
[,t]
f(t–u)h(u) dμ(u) ()
for allt∈[,∞).
From Definition ., we directly obtain some characterization of an interval-valuedg -convolution by means of the interval-valuedg-integrals.
Theorem . If g,,,andsatisfy the hypotheses of Theorem.,then we have f h= [fl∗lhl,fr∗rhr], ()
where(fs∗shs)(t) =
⊕s
[,t]fs(t–u)sdμfor s=l,r.
Proof By Definition ., we have
(fs∗shs)(t) =
⊕s
[,t]
fs(t–u)⊗shs(u)sdμ(u) ()
fors=l,r. By Theorem . and (),
(f h)(t) =
[,t]
f(t–u)h(u) dμ(u)
=
[,t]
fl(t–u)⊗lhl(u),fr(t–u)⊗rhr(u) dμ(u)
=
⊕l
[,t]
fl(t–u)⊗lhl(u)ldμ,
⊕r
[,t]
fr(t–u)⊗rhr(u)rdμ
=(fl∗lhl)(t), (fr∗rhr)(t)
. ()
From Theorem ., we investigate the commutativity and the associativity of a standard interval-valuedg-convolution.
Theorem . If g, ,,andsatisfy the hypotheses of Theorem.and f,h,and k∈IF∗(X),then we have
() fh=hf,
Proof Letf = [fl,fr],h= [hl,hr],k= [kl,kr]∈IF∗(X). By (), we have
fl∗lhl=hl∗lfl and fr∗rhr=hr∗rfr. ()
By Theorem . and (), we have
f h= [fl∗lhl,fr∗rhr]
= [hl∗lfl,hr∗rfr]
=hf. ()
By (), we have
(fl∗lhl)∗lkl=fl∗l(hl∗lkl) and (fr∗rhr)∗rkr=fr∗r(hr∗rkr). ()
By Theorem . and (),
(f h)k=(fl∗lhl)∗lkl, (fr∗rhr)∗rkr
=fl∗l(hl∗lkl),fr∗r(hr∗rkr)
=f(hk). ()
Finally, we illustrate the following examples which are related with the interval-valued
g-integral and the interval-valuedg-convolution as follows.
Example . We give three examples of the interval-valuedg-integral.
() Ifgl(x) =gr(x) =xfor allx∈[,∞] are the generators ofl,r,⊕l, and⊕r, and
f(x) = [e–x,e–x] for allx∈[,∞), andA= [,t] for allt∈[,∞), then we have
A
f dμ=
gl–
t
gl
fl(x)
dx,gr–
t
gr
fr(x)
dx
=
t
e
–xdx,
t
e–xdx
=
–e–t, –e–t . ()
() Ifgl(x) = x,gr(x) =xfor allx∈[,∞] are the generators ofl,r, andgl(x) =gr(x) =
xfor allx∈[,∞] are the generators of⊕l,⊕r, andf(x) = [e –x
,e
–x] for allx∈[,∞), and
A= [,t] for allt∈[,∞), then we have
A
f dμ=
gl–
t
gl
fl(x)
dx,gr–
t
gr
fr(x)
dx
=
t
e
–xdx,
t
e–xdx
=
() Ifgl(x) =x,gr(x) = xfor allx∈[,∞] are the generators ofl,r, andgl(x) =
gr(x) =xfor allx∈[,∞] are the generators of⊕l,⊕r, andf(x) = [e –x
,e–x] for allx∈[,∞),
andA= [,t] for allt∈[,∞), then we have
A
f dμ=
gl–
t
gl
fl(x)
dx,gr–
t
gr
fr(x)
dx
=
t
e
–xdx,
t
e–xdx
=
–e–t,
–e–t . ()
Example . We give an example of the interval-valuedg-convolution.
Ifgl(x) =x,gr(x) = xfor allx∈[,∞] are the generators ofl,r, andgl(x) =gr(x) =x
for allx∈[,∞] are the generators of⊕l,⊕r,⊗l,⊗r, andf(x) = [e –x
,e
–x] for allx∈[,∞),
h(x) = [x,x] for allx∈[,∞), andA= [,t] for allt∈[,∞), then we have
(f h)(t) =
A
f(t–u)h(u) dμ(u)
=
t
e–(t–u)e–udu,
t
e
–(x–u)e–udu
=
t
e
–t,t
e
–t . ()
5 Conclusions
In this paper, we have considered the g-integral represented by its generating g, the pseudo-addition, the pseudo-multiplication (see Definition .). This study was to de-fine theg-convolution by means of theg-integral (see Definition .) and to investigate some characterizations of theg-integral and the commutativity and the associativity of theg-convolution (see Theorems ., ., and .).
We also defined the interval-valuedg-integral represented by its interval-valued gener-atorg. By using general notions of an interval-representable pseudo-multiplication (see Definition .), we defined an interval-valuedg-integral (see Definition .) and investi-gated some basic characterizations of them (see Theorems ., .).
From Definitions ., ., and Theorems ., ., we defined a standard interval-valued
g-convolution (see Definition .). We also investigated some characterizations of a stan-dard interval-valuedg-convolution of valued functions by means of the interval-valuedg-integral including commutativity and associativity of an interval-representable convolution (see Theorems ., .).
Competing interests
The author declares that they have no competing interests.
Acknowledgements
This paper was supported by Konkuk University in 2014.
Received: 27 October 2013 Accepted: 30 January 2014 Published:20 Feb 2014
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10.1186/1029-242X-2014-88
Cite this article as:Jang:Some properties of the interval-valuedg-integrals and a standard interval-valued
