Research ArticleChoquet Integral of Fuzzy-Number-Valued FunctionsThe Differentiability of the Primitive with respect to FuzzyMeasures and Choquet Integral Equations

Zengtai Gong1 Li Chen12 and Gang Duan3

1 College of Mathematics and Statistics Northwest Normal University Lanzhou 730070 China2Department of Mathematics Lanzhou City University Lanzhou 730070 China3 School of Traffic and Transportation Lanzhou Jiaotong University Lanzhou 730070 China

Correspondence should be addressed to Zengtai Gong zt-gong163com

Received 25 January 2014 Accepted 22 May 2014 Published 9 June 2014

Academic Editor Marco Donatelli

Copyright copy 2014 Zengtai Gong et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper deals with the Choquet integral of fuzzy-number-valued functions based on the nonnegative real line We firstly givethe definitions and the characterizations of the Choquet integrals of interval-valued functions and fuzzy-number-valued functionsbased on the nonadditive measure Furthermore the operational schemes of above several classes of integrals on a discrete set areinvestigated which enable us to calculate Choquet integrals in some applications Secondly we give a representation of the Choquetintegral of a nonnegative continuous and increasing fuzzy-number-valued function with respect to a fuzzy measure In additionin order to solve Choquet integral equations of fuzzy-number-valued functions a concept of the Laplace transformation for thefuzzy-number-valued functions in the sense of Choquet integral is introduced For distorted Lebesgue measures it is shown thatChoquet integral equations of fuzzy-number-valued functions can be solved by the Laplace transformation Finally an example isgiven to illustrate the main results at the end of the paper

1 Introduction

The Choquet integral [1ndash4] with respect to a fuzzy measurewas proposed by Murofushi and Sugeno It was introducedby Choquet in potential theory with the concept of capacityThen it has been used for utility theory in the field ofeconomic theory [5] and has been used for image processingpattern recognition information fusion and data mining[4 6ndash8] in the context of fuzzy measure theory [9ndash13]

The development of the theory of integral equations isclosely linked to the study of mathematical physics problemsThe integral equation has the extremely widespread applica-tion in the field of engineering and mechanics and so forthThe early history of integral equation goes back to the specialintegral equation studied by several mathematicians such asLaplace Fourier Poisson Abel and Liouville in the late eigh-teenth and early nineteenth century With the developmentof computing technology the integral equation as one of theimportant foundations of engineering calculation has been

widely and effectively used Today with physical problemsbecoming more and more complex integral equation isbecoming more and more useful

Fuzzy integral and differential equations were discussedby many authors [14ndash16] which have been suggested as away of modeling uncertain and incompletely specified sys-tems Sugeno has described carefully the representation ofChoquet integral and Choquet integral equations of real-valued increasing functions and some important conclusionshave been obtained [17] Unfortunately it is not reasonableto assume that all data are real data before we elicit themfrom practical data Sometimes fuzzy data may exist such asin pharmacological financial and sociological applicationsMotivated by the above papers and related research works onthis topic the paper discusses the representation of Choquetintegral and Choquet integral equations of increasing fuzzy-number-valued functions

The rest of this study is organized as follows In Section 2we review some basic definitions of fuzzy measure and

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014 Article ID 953893 11 pageshttpdxdoiorg1011552014953893

2 Abstract and Applied Analysis

Choquet integrals of real-valued functions Section 3 givesthe definitions and the characterizations of the Choquet inte-grals of interval-valued functions and fuzzy-number-valuedfunctions based on the nonadditive Sugeno measure Fur-thermore the operational schemes of above several classes ofintegrals on a discrete sets are investigated which enable us tocalculate Choquet integrals in applications Section 4 gives arepresentation of the Choquet integral of a nonnegative con-tinuous and increasing fuzzy-number-valued function withrespect to a fuzzymeasure In Section 5 in order to solveCho-quet integral equations of fuzzy-number-valued functions aconcept of the Laplace transformation for the fuzzy-number-valued functions in the sense of Choquet integral is intro-duced For distorted Lebesguemeasures it is shown thatCho-quet integral equations of fuzzy-number-valued functionscan be solved by the Laplace transformation In addition anexample is given to illustrate themain results at the end of thepaper The paper ends with conclusions in Section 6

2 Preliminaries

In this section wewill introduce some basic definitions aboutfuzzy measures Choquet integral and fuzzy numbers

Definition 1 (see [6 7 18ndash20]) Let 119883 be a nonempty set andA a 120590-algebra on 119883 A fuzzy measure on 119883 is a set function120583 A rarr [0infin) satisfying the following conditions

(1) 120583(0) = 0(2) 119860 isin 119883 119861 isin 119883 119860 sub 119861 implies 120583(119860) le 120583(119861)(3) In 119883 if 119860

1sub 1198602sub 1198603sub sdot sdot sdot and ⋃

infin

119899=1119860119899isin 119883 then

lim119899rarrinfin

120583(119860119899) = 120583(⋃

infin

119899=1119860119899)

(4) In 119883 if 1198601sup 1198602sup 1198603sup sdot sdot sdot and ⋂

infin

119899=1119860119899isin 119883 then

lim119899rarrinfin

120583(119860119899) = 120583(⋂

infin

119899=1119860119899)

120583 is said to be lower semicontinuous if it satisfies the aboveconditions (1)ndash(3) 120583 is said to be upper semicontinuous if itsatisfies the above conditions (1) (2) and (4) 120583 is said to becontinuous if it satisfies the above conditions (1)ndash(4)

(119883A 120583) is said to be a nonadditive measure spaceOne can see that a fuzzy measure is a normal monotone

set function which vanishes at the empty set Furthermore afuzzy measure on 119883 is said to be

(i) additive if 120583(119860 cup 119861) = 120583(119860) + 120583(119861) for all disjointsubsets 119860 119861 isin 119883

(ii) subadditive if 120583(119860 cup 119861) le 120583(119860) + 120583(119861) for all disjointsubsets 119860 119861 isin 119883

(iii) superadditive if 120583(119860cup119861) ge 120583(119860)+120583(119861) for all disjointsubsets 119860 119861 isin 119883

(iv) cardinality-based if for any119860 isin 119883 120583(119860) depends onlyon the cardinality of 119860

(v) a 0 minus 1 fuzzy measure if its range is 0 1(vi) a 0minus1possibility fuzzymeasure focused on119860 denoted

by Pos119860 if Pos

119860(119861) = 1 if and only if 119860 cap 119861 = 0 and

Pos119860(119861) = 0 otherwise

(vii) a 0minus1 necessity fuzzymeasure focused on119860 denotedby Nec

119860 if Nec

119860(119861) = 1 if and only if 119861 sube 119860 and

Nec119860(119861) = 0 otherwise

Let 119891 119883 rarr (minusinfin +infin) be a measurable function withrespect toA That is 119891 satisfies the condition

119891120572= 119909 | 119891 (119909) ge 120572 isin A (1)

for any 120572 isin R

Definition 2 (see [1]) Let (119883A 120583) be a nonadditive measurespace and 119891 a measurable function on 119883 The Choquetintegral of a real-valued function 119891 119883 rarr (minusinfin +infin) isdefined as

(119888) int119883

119891119889120583 = int

0

minusinfin

[120583 (119891120572) minus 120583 (119883)] 119889120572 + int

infin

0

120583 (119891120572) 119889120572 (2)

if both of Riemann integrals exist and at least one of them hasfinite value

Let 119864 isin A Then Choquet integral of a nonnegative real-valued function 119891 119883 rarr [0 +infin) is defined as

(119888) int119864

119891119889120583 = int

infin

0

120583 (119864 cap 119891120572) 119889120572 (3)

Since 120583(119864 cap 119891120572) is nonincreasing with respect to 120572 the

Choquet integral of real-valued function 119891 with respect to 120583

exists If (119888) int119864119891119889120583 lt infin then 119891 is said to be 119862-integrable

with respect to 120583 on 119883 Choquet integral has the followingproperties [1]

(1) If 119892 le ℎ then (119888) int119864119892119889120583 le (119888) int

119864ℎ119889120583

(2) If 119860 sub 119861 then (119888) int119860119891119889120583 le (119888) int

119861119891119889120583

(3) Let 120583 be lower semicontinuous If 119891119899

uarr 119891 ae in 119864then (119888) int

119864119891119899119889120583 uarr (119888) int

119864119891119889120583

(4) Let 120583 be upper semicontinuous If 119891119899

darr 119891 ae in 119864and there exists a 119862-integrable function 119892 such that1198911le 119892 then (119888) int

119864119891119899119889120583 darr (119888) int

119864119891119889120583

119868(119877+) = 119903 [119903 119903] sub 119877

+ denotes the set of all interval

numbers on 119877+ where 119877

+= [0 +infin) With the definition

from Wu et al [21] interval numbers should satisfy thefollowing basic operations

(1) 119903 lowast 119901 = [119903 lowast 119901 119903 lowast 119901](lowast denotes + or and)

(2) 119896 sdot 119903 = [119896119903 119896119903] (119896 isin R+)(3) 119903 le 119901 hArr 119903 le 119901 119903 le 119901

(4) 119889(119903 119901) = max|119903 minus 119901| |119903 minus 119901|(5) If 119889(119903

119899 119903) rarr 0 then 119903

119899rarr 119903

A fuzzy subset of 119877 is a function 119906 119877 rarr [0 1]For each fuzzy set 119906 defined as above we denote by 119906

120582=

119909 isin 119877 119906(119909) ge 120582 for any 120582 isin [0 1] its 120582-level set Bysupp119906 we denote the support of 119906 that is the set 119909 isin 119877

119906(119909) gt 0 By 1199060we denote the closure of supp119906 that is

1199060

= 119909 isin 119877 119906(119909) gt 0 Let 119864 be the collection of all fuzzysets of 119877 We call 119906 isin 119864 a fuzzy number if it satisfies thefollowing conditions [22]

Abstract and Applied Analysis 3

(1) 119906 is normal that is there exists 1199090

isin 119877 such that119906(1199090) = 1

(2) 119906 is fuzzy convex that is 119906(120582119909 + (1 minus 120582)119910) ge

min119906(119909) 119906(119910) for any 119909 119910 isin 119877 0 le 120582 le 1(3) 119906 is upper semicontinuous that is 119906(119909

0) ge

lim119896rarrinfin

119906(119909119896) for any 119909

119896isin 119877 (119896 = 0 1 2 ) 119909

119896rarr

1199090

(4) 1199060= 119909 isin 119877 119906(119909) gt 0 is compact

We denote the collection of all fuzzy numbers by 119886 is said to be a nonnegative fuzzy number if supp 119886 =

119909 isin 119877 | 119886(119909) gt 0 sub 119877+We can define the semiorder and the

distance in space [22] Let 119886 isin Then 119886 le if 119886120582

le 119887120582

and 119886120582

le 119887120582for 120582 isin (0 1] 119886 + = 119888 if 119886

120582+ 119887120582

= 119888120582for

120582 isin (0 1] 119863(119886 ) = sup120582isin[01]

119863(119886120582 119887120582) is the distance of 119886

and Let 119886119899sub If 119863(119886

119899 119886) rarr 0 then 119886

119899rarr 119886

119877+ denotes the collection of all nonnegative fuzzy num-bers

Lemma 3 (see [22]) Let 119886 isin Then

(1) 119886120582is a nonempty bounded and closed interval for each

120582 isin [0 1](2) if 0 le 120582

1le 1205822le 1 then 119886

1205822

sub 1198861205821

(3) if 120582119899ge 0 and 120582

119899rarr 120582 (120582 isin (01]) then⋂

infin

119899=1119886120582119899

= 119886120582

Conversely there exists a 119887120582

sub 119877 for each 120582 isin (0 1]which satisfies the conditions (1)ndash(3) then there exists a uniquefuzzy number 119886 isin such that 119887

120582= 119886120582 120582 isin (0 1] and

1198860= ⋃120582isin(01]

119886120582sub 1198870

3 Choquet Integrals of Interval-ValuedFunctions and Fuzzy-Number-ValuedFunctions Based on Nonadditive Measures

In this section we will give the definitions and the characteri-zations of the Choquet integrals of interval-valued functionsand fuzzy-number-valued functions based on the nonaddi-tive Sugeno measure Furthermore the operational schemesof above several classes of integrals on a discrete sets areinvestigated which enable us to calculate Choquet integralsin applications

We first introduce the concept of the Choquet integralsfor the interval-valued functions as follows

Definition 4 (see [23]) An interval-valued function 119865 119883 rarr

119868(119877+) is said to be measurable if both 119865(119909) and 119865(119909) are

measurable functions where 119865(119909) = [119865(119909) 119865(119909)] 119865(119909) is theleft end point of interval 119865(119909) and 119865(119909) is the right end pointof interval 119865(119909)

Interval-valued function 119865 119883 rarr 119868(119877+) is 119862-integrally

bounded if there exists a Choquet integrable function ℎ

119883 rarr 119877+ such that || le ℎ(119905) for every selection isin 119865

We denote 1198750(119883) = 119864 | 119864 sub 119883 and 119864 = 0 Let (119883A 120583)

be a measure space 119860 119861 sub A and let 119865 119860 rarr 1198750(119861) be a

measurable set valued mapping 119891 is said to be a measurable

selection of 119865 if there exists a measurable mapping 119891 119860 rarr

119861 such that 119891(119909) isin 119865(119909) for every 119909 isin 119860

Definition 5 Let (119883A 120583) be a nonadditive measure spaceAssume that 119865 119883 rarr 119868(119877

+) is measurable 119862-integrally

bounded interval-valued function and 119864 isin A 119865 is said tobe 119862-integrable if

(119888) int119864

119865119889120583 = (119888) int119864

119892119889120583 | 119892 isin 119878119865(119909)

(4)

is a closed interval on 119868(119877+) where

119878119865(119909)

= 119892 | 119892 119883 997888rarr 119877+ is a measurable selection of 119865 (119909)

(5)

Theorem 6 Let (119883A 120583) be a nonadditive measure spaceSuppose that 120583 is a fuzzymeasure119864 isin A and119865 is nonnegativemeasurable 119862-integrally bounded interval-valued functionthen 119865 is 119862-integrable on 119864 and

(119888) int119864

119865119889120583 = [(119888) int119864

119865119889120583 (119888) int119864

119865119889120583] (6)

Proof Since 119865 is nonnegative measurable on 119864 119865 and 119865 aremeasurable on119864Thus 119865 and 119865 are twomeasurable selectionsof 119865 On the other hand 119865 is 119862-integrally bounded we have

119865 le ℎ 119865 le ℎ (7)

By the properties of Choquet integral of real valued functionswe know that 119865 and 119865 are 119862-integrable on 119864 Let 119872 isin

(119888) int119864119865119889120583 That is to say there exists a measurable selection

119892 of 119865 such that (119888) int119864119892119889120583 = 119872 We can prove that 119872 isin

[(119888) int119864119865119889120583 (119888) int

119864119865119889120583] Indeed notice that 119865 le 119892 le 119865 and

by the properties of Choquet integral of real-valued functionswe have

(119888) int119864

119865119889120583 le (119888) int119864

119892119889120583 le (119888) int119864

119865119889120583 (8)

This follows that

119872 isin [(119888) int119864

119865119889120583 (119888) int119864

119865119889120583] (9)

Thus

(119888) int119864

119892119889120583 | 119892 isin 119878119865(119909)

sub [(119888) int119864

119865119889120583 (119888) int119864

119865119889120583] (10)

Conversely we can show that

[(119888) int119864

119865119889120583 (119888) int119864

119865119889120583] sub (119888) int119864

119865119889120583 (11)

For measurable selections 119901(119909)119898(119909) isin 119865(119909) we get that

119905119901 (119909) + (1 minus 119905)119898 (119909) isin 119865 (119909) (12)

4 Abstract and Applied Analysis

is measurable selection of 119865 for any 0 le 119905 le 1 Therefore

119905 sdot (119888) int119864

119901 (119909) 119889120583 + (1 minus 119905) sdot (119888) int119864

119898(119909) 119889120583

= (119888) int119864

[119905119901 (119909) + (1 minus 119905)119898 (119909)] 119889120583

isin (119888) int119864

119892119889120583 | 119892 isin 119878119865(119909)

(13)

It implies that (119888) int119864119865119889120583 = (119888) int

119864119892119889120583 | 119892 isin 119878

119865(119909) is a convex

set On the other hand since 119865 and 119865 are two measurableselections of 119865 and

(119888) int119864

119865119889120583 isin (119888) int119864

119865119889120583 (119888) int119864

119865119889120583 isin (119888) int119864

119865119889120583 (14)

we have

[(119888) int119864

119865119889120583 (119888) int119864

119865119889120583] sub (119888) int119864

119892119889120583 | 119892 isin 119878119865(119909)

(15)

Hence

(119888) int119864

119865119889120583 = [(119888) int119864

119865119889119892120583 (119888) int119864

119865119889120583] (16)

That is 119865 is 119862-integrable

From the above theorem we know that interval-valuedfunction 119865 is 119862-integrable on 119864 if (119888) int

119864119865119889120583 and (119888) int

119864119865119889120583

exist and are boundedNext we will introduce the concept of the Choquet

integrals for the Fuzzy-number-valued functions as followsFuzzy-number-valued function 119865 119883 rarr

+ on119883 is saidto be measurable if 119865

120582and 119865

120582are measurable functions with

respect to 119909 isin 119883 for any 120582 isin [0 1]Fuzzy-number-valued function 119865 119883 rarr

+ is said tobe 119862-integrally bounded if there exists a Choquet integrablefunction ℎ 119883 rarr 119877

+ such that || le ℎ(119905) for every selection isin [119865(119905)]

0

Definition 7 Let (119883A 120583) be a nonadditive measure spaceAssume that 119864 isin A 119865 119883 rarr

+ is measurable and 119862-integrally bounded function 119865 is said to be 119865119862-integrable if

[(119888) int119864

119865119889120583]

120582

= (119888) int119864

119892119889120583 | 119892 isin 119878119865120582

0 le 120582 le 1 (17)

determines a unique fuzzy number 119886 isin + which is denoted

by (119888) int119864119865119889120583 = 119886 where 119878

119865120582

= 119892 119883 rarr 119877+ 119892 isin 119865

120582is a

measurable selection of 119865120582

Theorem 8 Let (119883A 120583) be a nonadditive measurable spaceAssume 120583 is a continuous fuzzy measure 119864 isin A 119865 119883 rarr

+

ismeasurable and119862-integrally bounded function then119865 is119865119862-integrable on 119864 if and only if 119865

120582and 119865

120582are 119862-integrable on 119864

and

[(119888) int119864

119865119889120583]

120582

= [(119888) int119864

119865120582119889120583 (119888) int

119864

119865120582119889120583] (18)

for any 120582 isin [0 1]

Proof For the necessity since 119865 is measurable and 119862-integrally bounded function 119865

120582and 119865

120582are measurable for

every 120582 isin [0 1] and there exists a Choquet integrable func-tion ℎ(119909) such that 0 le 119865

120582le ℎ(119909) 0 le 119865

120582le ℎ(119909) and then

(119888) int119864

119865120582119889120583 le (119888) int

119864

119865120582119889120583 le (119888) int

119864

ℎ (119909) 119889120583 lt infin (19)

That is 119865120582 119865120582are 119862-integrable and

[(119888) int119864

119865119889120583]

120582

= [(119888) int119864

119865120582119889120583 (119888) int

119864

119865120582119889120583] (20)

for any 120582 isin [0 1]For the sufficiency let 119865 be a Fuzzy-number-valued func-

tion Note that

[(119888) int119864

119865119889120583]

120582

= [(119888) int119864

119865120582119889120583 (119888) int

119864

119865120582119889120583] (21)

we need only to prove that the interval family

[(119888) int119864

119865119889120583]

120582

= [(119888) int119864

119865120582119889120583 (119888) int

119864

119865120582119889120583] 120582 isin [0 1]

(22)

determines a unique fuzzy number Indeed the interval fam-ily

[(119888) int119864

119865120582119889120583 (119888) int

119864

119865120582119889120583] (23)

satisfies the conditions of Lemma 3

(1) 119865 is a measurable fuzzy-number-valued function foreach 120582 isin [0 1] we have 119865

120582(119909) le 119865

120582(119909) and therefore

(119888) int119864

119865120582119889120583 le (119888) int

119864

119865120582119889120583 (24)

(2) Since 1198651205822

sub 1198651205821

for 0 le 1205821le 1205822le 1 that is

1198651205821

(119909) le 1198651205822

(119909) 1198651205821(119909) ge 119865

1205822(119909) (25)

we have

(119888) int119864

1198651205821

119889120583 le (119888) int119864

1198651205822

119889120583

(119888) int119864

1198651205821

119889120583 ge (119888) int119864

1198651205822

119889120583

[(119888) int119864

1198651205821

119889120583 (119888) int119864

1198651205821

119889120583]

sup [(119888) int119864

1198651205822

119889120583 (119888) int119864

1198651205822

119889120583]

(26)

(3) For each 120582119899uarr 120582 isin (0 1] ⋂infin

119899=1119865120582119899

(119909) = 119865120582(119909) that is

lim119899rarrinfin

119865120582119899

= 119865120582 lim119899rarrinfin

119865120582119899

= 119865120582 It is easy to see

Abstract and Applied Analysis 5

that 1198651205821

le 119865120582119899

le 119865120582119899

le 1198651205821

1198651205821

1198651205821

are integrableand by the continuity of 120583 then

infin

⋂

119899=1

[(119888) int119864

119865120582119899

119889120583 (119888) int119864

119865120582119899

119889120583]

= [(119888) int119864

119865120582119889120583 (119888) int

119864

119865120582119889120583]

(27)

In conclusion there exists a unique fuzzy number 119886 isin +

such that

119886120582= [(119888) int

119864

119865120582119889120583 (119888) int

119864

119865120582119889120583] (28)

Furthermore we get that 119865 is 119865119862-integrable on 119864 and

[(119888) int119864

119865119889120583]

120582

= [(119888) int119864

119865120582119889120583 (119888) int

119864

119865120582119889120583] (29)

In the last part of the section we will investigate theoperational schemes of above several classes of integrals ona discrete set

Let119883 = 1199091 1199092 119909

119899 be a discrete setThenwewill give

a new scheme to calculate the value of the Choquet integral

Theorem 9 (see [8]) Let 119891 be a real-valued function on 119883 =

1199091 1199092 119909

119899 Then Choquet integral of 119891 with respect to a

fuzzy measure 120583 on 119883 is given by

(119888) int119883

119891119889120583 =

119899

sum

119894=1

[119891 (1199091015840

119894) minus 119891 (119909

1015840

119894minus1)] 120583 (119883

1015840

119894) (30)

or equivalently by

(119888) int119883

119891119889120583 =

119899

sum

119894=1

[120583 (1198831015840

119894) minus 120583 (119883

1015840

119894+1)] 119891 (119909

1015840

119894) (31)

where 1199091015840

1 1199091015840

2 119909

1015840

119899is a permutation of 119909

1 1199092 119909

119899such

that 119891(1199091015840

0) le 119891(119909

1015840

1) le sdot sdot sdot le 119891(119909

1015840

119899) 119891(119909

1015840

0) = 0 119883

1015840

119894=

1199091015840

119894 1199091015840

119894+1 119909

1015840

119899 119894 = 1 2 119899 and 119883

1015840

119899+1= 0

Theorem 10 Let 119865 be an interval-valued function on 119883 =

1199091 1199092 119909

119899 Then Choquet integral of 119865 with respect to a

fuzzy measure 120583 on 119883 is given by

(119888) int119883

119865119889120583 =

119899

sum

119894=1

[120583 (1198831015840

119894) minus 120583 (119883

1015840

119894+1)] 119865 (119909

1015840

119894) (32)

where 1199091015840

1 1199091015840

2 119909

1015840

119899is a permutation of 119909

1 1199092 119909

119899such

that 119865(1199091015840

0) le 119865(119909

1015840

1) le sdot sdot sdot le 119865(119909

1015840

119899) 119865(119909

1015840

0) = [0 0] 1198831015840

119894=

1199091015840

119894 1199091015840

119894+1 119909

1015840

119899 119894 = 1 2 119899 and 119883

1015840

119899+1= 0

Proof Since 119865 is an interval-valued function on119883 in view ofTheorem 6 we have

(119888) int119883

119865119889120583 = [(119888) int119883

119865119889120583 (119888) int119883

119865119889120583] (33)

Note that 119865 and 119865 are real-valued function on119883 respectivelyByTheorem 9 we get

(119888) int119883

119865119889120583 = [

119899

sum

119894=1

[120583 (1198831015840

119894) minus 120583 (119883

1015840

119894+1)] 119865 (119909

1015840

119894)

119899

sum

119894=1

[120583 (1198831015840

119894) minus 120583 (119883

1015840

119894+1)] 119865 (119909

1015840

119894)]

=

119899

sum

119894=1

[120583 (1198831015840

119894) minus 120583 (119883

1015840

119894+1)] 119865 (119909

1015840

119894)

(34)

where 1199091015840

1 1199091015840

2 119909

1015840

119899is a permutation of 119909

1 1199092 119909

119899such

that 119865(1199091015840

0) le 119865(119909

1015840

1) le sdot sdot sdot le 119865(119909

1015840

119899) 119865(119909

1015840

0) = [0 0] 1198831015840

119894=

1199091015840

119894 1199091015840

119894+1 119909

1015840

119899 119894 = 1 2 119899 and 119883

1015840

119899+1= 0

Theorem 11 Let119865 be a fuzzy-number-valued function on119883 =

1199091 1199092 119909

119899 Then Choquet integral of 119865 with respect to a

fuzzy measure 120583 on 119883 is given by

(119888) int119883

119865119889120583 =

119899

sum

119894=1

[120583 (1198831015840

119894) minus 120583 (119883

1015840

119894+1)] 119865 (119909

1015840

119894) (35)

where 1199091015840

1 1199091015840

2 119909

1015840

119899is a permutation of 119909

1 1199092 119909

119899such

that 119865(1199091015840

0) le 119865(119909

1015840

1) le sdot sdot sdot le 119865(119909

1015840

119899) 119865(119909

1015840

0) = 0 119883

1015840

119894=

1199091015840

119894 1199091015840

119894+1 119909

1015840

119899 119894 = 1 2 119899 and 119883

1015840

119899+1= 0

Proof Since 119865 is a fuzzy-number-valued on 119883 in view ofTheorem 8 we have

[(119888) int119883

119865119889120583]

120582

= [(119888) int119883

119865120582119889120583 (119888) int

119883

119865120582119889120583] (36)

for any 120582 isin [0 1] By the semiorder in space (ie let 119886 isin

Then 119886 le if 119886120582le 119887120582and 119886

120582le 119887120582) andTheorem 10 there

is a sequence 1199091015840

1 1199091015840

2 119909

1015840

119899on 119883 such that 119865(119909

1015840

0) le 119865(119909

1015840

1) le

sdot sdot sdot le 119865(1199091015840

119899) 119865(119909

1015840

0) = 0 1198831015840

119894= 1199091015840

119894 1199091015840

119894+1 119909

1015840

119899 119894 = 1 2 119899

1198831015840

119899+1= 0 and 119909

1015840

1 1199091015840

2 119909

1015840

119899is a permutation of 119909

1 1199092 119909

119899

Consequently

[(119888) int119883

119865119889120583]

120582

= [

119899

sum

119894=1

[120583 (1198831015840

119894) minus 120583 (119883

1015840

119894+1)] 119865120582(1199091015840

119894)

119899

sum

119894=1

[120583 (1198831015840

119894) minus 120583 (119883

1015840

119894+1)] 119865120582(1199091015840

119894)]

(37)

for any 120582 isin [0 1] Therefore

(119888) int119883

119865119889120583 =

119899

sum

119894=1

[120583 (1198831015840

119894) minus 120583 (119883

1015840

119894+1)] 119865 (119909

1015840

119894) (38)

4 The Representation of Choquet Integral ofFuzzy-Number-Valued Functions

Sugeno has described carefully the representation of Cho-quet integral of real-valued increasing functions and some

6 Abstract and Applied Analysis

important conclusions have been obtained [17] Motivated bythis we will discuss the representation of Choquet integral offuzzy-number-valued functions in this section

Definition 12 Fuzzy-number-valued function 119865 119883 rarr +

is said to be continuous on 119883 if for every 1199090isin 119883 there are

lim119909rarr119909

0

119865120582(119909) = 119865

120582(1199090)

lim119909rarr119909

0

119865120582(119909) = 119865

120582(1199090)

(39)

where 120582 isin [0 1]

Definition 13 Fuzzy-number-valued function 119865 119883 rarr +

is said to be increasing on 119883 if for every 1199091

le 1199092there are

119865120582(1199091) le 119865120582(1199092) and 119865

120582(1199091) le 119865120582(1199092) where 119909

1 1199092isin 119883 and

120582 isin [0 1]LetF+ be a class ofmeasurable nonnegative continuous

and increasing fuzzy-number-valued functionsLet ] be a Lebesgue measure for [119886 119887] sub [0infin) ][119886 119887] =

119887 minus 119886

Definition 14 (see [17]) Let 119898 119877+

rarr 119877+ be a continuous

and increasing function and 119898(0) = 0 A fuzzy measure 120583119898

a distorted Lebesgue measure is defined by 120583119898(sdot) = 119898(](sdot))

Definition 15 (see [24]) A fuzzy-number-valued function 119865

[119886 119887] rarr is differentiable at 119909 isin [119886 119887] if there exist fuzzynumber-valued functions 1198651015840(119909) such that

[1198651015840(119909)]120582= [1198651015840

120582(119909) 119865

1015840

120582(119909)] (40)

for every 120582 isin [0 1] where 1198651015840(119909) is the fuzzy derivative of

119865(119909)Note that 120583

119898is induced from the Lebesguemeasure ] by a

monotone transformation where 120583119898([119886 119887]) = 119898(]([119886 119887])) =

119898(119887 minus 119886) Apparently it loses additivity unless 119898(119905) = 119905 butreserves monotonicity In what follows we assume that 119898(119905)

is differentiableIn this section we consider the calculation of Choquet

integrals Let 120583 be a general fuzzy measure and consider120583([120591 119905]) for a closed interval [120591 119905] then 120583([120591 119905]) is decreasingfor 120591 and increasing for 119905 Throughout the paper we assumethat the functions 119898(119905) 119865(119905) and 119866(119905) are continuouslydifferentiable We also assume that 120583([120591 119905]) is continuouslydifferentiable with respect to 120591 on [0 119905] for every 119905 gt 0 Inaddition we require the regularity condition that 120583(119905) = 0

holds for every 119905 ge 0 We write 1205831015840([120591 119905]) = (120597120597120591)120583([120591 119905])

where we note that 1205831015840([120591 119905]) le 0 for 120591 le 119905 If 120583 = 120583119898then

1205831015840([120591 119905]) = minus119898

1015840(119905 minus 120591) where 119898

1015840(119905) = 119889119898(119905)119889119905 First we

consider a case that 119865(119905) is strictly increasingFuzzy-number-valued function 119865 119883 rarr is said to

be 119871-integrally bounded if there exists a Lebesgue integrablefunction ℎ 119883 rarr 119877

+ such that || le ℎ(119905) for every section isin [119865(119905)]

0

Definition 16 (see [22]) Let fuzzy-number-valued functions119865 [119886 +infin) rarr be measurable and 119871-integrally bounded119865 is called Kaleva-integrable if

[int

+infin

119886

119865(119905)119889119905]

120582

= (119871) int

+infin

119886

119892 (119905) 119889119905 | 119892 (119905) isin 119878[119865(119905)]

120582

0 le 120582 le 1

(41)

determines a unique fuzzy number 119886 isin which is denotedby (119870) int

+infin

119886119865(119905) 119889120583 = 119886 where 119878

[119865(119905)]120582

= 119892(119905) isin [119865(119905)]120582is a

measurable selection

Lemma 17 (see [22]) Let 119865 [119886 +infin) rarr be measurableand C-integrally bounded fuzzy-number-valued function then119865 is Kaleva integrable on [119886 +infin) if and only if 119865

120582(119905) and119865

120582(119905)

are 119871-integrable on [119886 +infin) and

[(119870)int

119905

0

119865(120591)119889120591]

120582

= [(119871) int

119905

0

119865120582(120591) 119889120591 (119871) int

119905

0

119865120582(120591) 119889120591]

(42)

for any 120582 isin [0 1]

Theorem 18 Let 119865(119905) isin F+ Then minus1205831015840([120591 119905])119865(120591) is Kaleva

integrable on [0 119905] and

[(119870)int

119905

0

[minus1205831015840([120591 119905])]119865(120591)119889120591]

120582

= [(119871) int

119905

0

[minus1205831015840([120591 119905])] 119865

120582(120591) 119889120591

(119871) int

119905

0

[minus1205831015840([120591 119905])] 119865

120582(120591) 119889120591]

(43)

Proof Since119865(119905) isin F+ for every120582 isin [0 1]we get that119865120582and

119865120582are measurable nonnegative continuous and increasing

real-valued functions on [0 119905] It follows that 119865120582and 119865

120582are

119871-integrable on [0 119905] 120583([120591 119905]) is 119871-integrable on [0 119905] as itis continuously differentiable with respect to 120591 on [0 119905] forevery 119905 gt 0 In view of Lemma 17 [minus120583

1015840([120591 119905])]119865(120591) is Kaleva

integrable on [0 119905] and

[(119870)int

119905

0

[minus1205831015840([120591 119905])] 119865 (120591) 119889120591]

120582

= [(119871) int

119905

0

[minus1205831015840([120591 119905])] 119865

120582(120591) 119889120591

(119871) int

119905

0

[minus1205831015840([120591 119905])] 119865

120582(120591) 119889120591]

(44)

Lemma 19 (see [17]) Let 119891(119905) be a real strictly increasingfunction Then the Choquet integral of 119891 with respect to 120583 on[0 119905] is represented as

(119888) int[0119905]

119891 (120591) 119889120583 (120591) = minusint

119905

0

1205831015840([120591 119905]) 119891 (120591) 119889120591 (45)

Abstract and Applied Analysis 7

In particular for 120583 = 120583119898

(119888) int[0119905]

119891 (120591) 119889120583119898

(120591) = int

119905

0

1198981015840(119905 minus 120591) 119891 (120591) 119889120591 (46)

Lemma 20 (see [17]) Let 119891(119905) be a constant real-valuedfunction 119891(119905) = 119862 forall119905 isin 119877

+= (0 +infin) Then

(119888) int[0119905]

119891 (120591) 119889120583 (120591) = minusint

119905

0

1205831015840([120591 119905]) 119862119889120591 (47)

In particular for 120583 = 120583119898

(119888) int[0119905]

119891 (120591) 119889120583119898

(120591) = 119862119898 (119905) (48)

Lemma 21 (see [17]) Let 119891(119905) be a measurable nonnegativecontinuous and increasing real-valued function Then theChoquet integral of 119891 with respect to 120583 on [0 119905] is representedas

(119888) int[0119905]

119891 (120591) 119889120583 (120591) = int

infin

0

120583 (120591 | 119891 (120591) ge 119903 cap [0 119905]) 119889119903

= minusint

119905

0

1205831015840([120591 119905]) 119891 (120591) 119889120591

(49)

In particular for 120583 = 120583119898

(119888) int[0119905]

119891 (120591) 119889120583119898

(120591) = int

infin

0

120583 (120591 | 119891 (120591) ge 119903 cap [0 119905]) 119889119903

= int

119905

0

1198981015840(119905 minus 120591) 119891 (120591) 119889120591

(50)

Theorem 22 Let 119865(119905) be a strictly increasing fuzzy-number-valued functionThen the Choquet integral of 119865with respect to120583 on [0 119905] is represented as

(119888) int[0119905]

119865 (120591) 119889120583 (120591) = (119870)int

119905

0

[minus1205831015840([120591 119905])] 119865 (120591) 119889120591 (51)

In particular for 120583 = 120583119898

(119888) int[0119905]

119865 (120591) 119889120583119898

(120591) = (119870)int

119905

0

1198981015840(119905 minus 120591) 119865 (120591) 119889120591 (52)

Proof ByTheorem 8 we have

[(119888) int[0119905]

119865 (120591) 119889120583 (120591)]

120582

= [(119888) int[0119905]

119865120582(120591) 119889120583 (120591) (119888) int

[0119905]

119865120582(120591) 119889120583 (120591)]

(53)

for any 120582 isin [0 1] Since 119865(119905) is strictly increasing 119865120582(119905) and

119865120582(119905) are strictly increasing real-valued functions respec-

tively In view of Lemma 19 we have

(119888) int[0119905]

119865120582(120591) 119889120583 (120591) = minusint

119905

0

1205831015840([120591 119905]) 119865

120582(120591) 119889120591

(119888) int[0119905]

119865120582(120591) 119889120583 (120591) = minusint

119905

0

1205831015840([120591 119905]) 119865

120582(120591) 119889120591

(54)

It follows that

[(119888) int[0119905]

119865(120591)119889120583(120591)]

120582

= [minusint

119905

0

1205831015840([120591 119905]) 119865

120582(120591) 119889120591 minusint

119905

0

1205831015840([120591 119905]) 119865120582 (120591) 119889120591]

(55)

On the other hand

int

119905

0

[minus1205831015840([120591 119905])] 119865120582 (120591) 119889120591

= int

119905

0

[minus1205831015840([120591 119905])] [119865

120582(120591) 119865

120582(120591)] 119889120591

= [minusint

119905

0

1205831015840([120591 119905]) 119865

120582(120591) 119889120591 minusint

119905

0

1205831015840([120591 119905]) 119865120582 (120591) 119889120591]

(56)

Therefore

[(119888) int[0119905]

119865(120591)119889120583(120591)]

120582

= int

119905

0

[minus1205831015840([120591 119905])] 119865120582 (120591) 119889120591 (57)

By the arbitrary of 120582 we have

(119888) int[0119905]

119865 (120591) 119889120583 (120591) = (119870)int

119905

0

[minus1205831015840([120591 119905])] 119865 (120591) 119889120591 (58)

For 120583 = 120583119898 we note that minus120583

1015840([120591 119905]) = 119898

1015840(119905 minus 120591) hence

(119888) int[0119905]

119865 (120591) 119889120583119898

(120591) = (119870)int

119905

0

1198981015840(119905 minus 120591) 119865 (120591) 119889120591 (59)

Theorem 23 Let 119865(119905) be a constant fuzzy-valued functionthat is 119865(119905) = 119862 forall119905 isin 119877

+= (0 +infin) Then

(119888) int[0119905]

119865 (120591) 119889120583 (120591)

= (119870)int

119905

0

[minus1205831015840([120591 119905])] 119862119889120591 = 119862120583 ([0 119905])

(60)

In particular for 120583 = 120583119898

(119888) int[0119905]

119865 (120591) 119889120583119898

(120591) = 119862119898 (119905) (61)

Proof Note that119862 = [119862 119862] where119862 and119862 are constant real-valued functions respectively In view of Lemma 20 we have

(119888) int[0119905]

119862119889120583 (120591) = minusint

119905

0

1205831015840([120591 119905]) 119862119889120591

(119888) int[0119905]

119862119889120583 (120591) = minusint

119905

0

1205831015840([120591 119905]) 119862119889120591

(62)

8 Abstract and Applied Analysis

The rest of the proof follows in exactly the same way as thatof Theorem 22 Hence

(119888) int[0119905]

119865 (120591) 119889120583 (120591)

= (119870)int

119905

0

[minus1205831015840([120591 119905])] 119862119889120591 = 119862120583 ([0 119905])

(63)

For 120583 = 120583119898 we obtain 120583

1015840([120591 119905]) = minus119898

1015840(119905 minus 120591) It follows that

(119888) int[0119905]

119862119889120583 (120591) = (119870)int

119905

0

[minus1205831015840([120591 119905])] 119862119889120591

= int

119905

0

1198981015840(119905 minus 120591) 119862119889120591

= (119870)119862int

119905

0

1198981015840(119905 minus 120591) 119889120591 = 119862119898 (119905)

(64)

where 119898(0) = 0

Theorem 24 Let 119865(119905) isin F+ Then the Choquet integral of 119865with respect to 120583 on [0 119905] is represented as

(119888) int[0119905]

119865 (120591) 119889120583 (120591) = (119870)int

119905

0

[minus1205831015840([120591 119905])] 119865 (120591) 119889120591 (65)

In particular for 120583 = 120583119898

(119888) int[0119905]

119865 (120591) 119889120583119898

(120591) = (119870)int

119905

0

1198981015840(119905 minus 120591) 119865 (120591) 119889120591 (66)

Proof The theorem has been proved when 119865(119905) is a strictlyincreasing or constant fuzzy-number-valued function byTheorem 22 or Theorem 23 Without loss of generality weconsider a continuous and increasing function such that

119865 (119905) =

1198651(119905) 0 le 119905 lt 119905

1

1198652(119905) 119905

1le 119905 lt 119905

2

1198653(119905) 119905

2le 119905 lt infin

(67)

where 1198651(119905) and 119865

3(119905) are strictly increasing 119865

2(119905) is constant

and 1198651(1199051) = 1198652(119905) = 119865

3(1199052)

(i) For 0 le 119905 lt 1199051

Note that 1198651(119905) is strictly increasing on [0 119905

1) from

Theorem 22 we obtain

(119888) int[0119905]

119865 (120591) 119889120583 (120591) = (119870)int

119905

0

[minus1205831015840([120591 119905])] 1198651 (120591) 119889120591 (68)

(ii) For 1199051le 119905 lt 119905

2

In view of the construction of 119865(119905) on [0 119905] we have

(119888) int[0119905]

119865 (120591) 119889120583 (120591) = (119888) int[01199051]

1198651(120591) 119889120583 (120591)

+ (119888) int[1199051119905]

1198652(120591) 119889120583 (120591)

(69)

Since 1198651(119905) is strictly increasing on [0 119905

1) and 119865

2(119905) is

constant on [1199051 1199052) by Theorems 22 and 23

(119888) int[0119905]

119865 (120591) 119889120583 (120591) = (119870)int

1199051

0

[minus1205831015840([120591 119905])] 119865

1(120591) 119889120591

+ (119870)int

119905

1199051

[minus1205831015840([120591 119905])] 119865

2(120591) 119889120591

= (119870)int

119905

0

[minus1205831015840([120591 119905])] 119865 (120591) 119889120591

(70)

(iii) For 1199052le 119905 lt infin

Since1198651(119905) is strictly increasing on [0 119905

1) 1198652(119905) is constant

on [1199051 1199052) and 119865

3(119905) is strictly increasing on [119905

1infin) by

Theorems 22 and 23 we have

(119888) int[0119905]

119865 (120591) 119889120583 (120591)

= (119888) int[01199051]

1198651(120591) 119889120583 (120591) + (119888) int

[11990511199052]

1198652(120591) 119889120583 (120591)

+ (119888) int[1199052119905]

1198653(120591) 119889120583 (120591)

= (119870)int

1199051

0

[minus1205831015840([120591 119905])] 119865

1(120591) 119889120591

+ (119870)int

1199052

1199051

[minus1205831015840([120591 119905])] 119865

2(120591) 119889120591

+ (119870)int

119905

1199052

[minus1205831015840([120591 119905])] 119865

3(120591) 119889120591

= (119870)int

119905

0

[minus1205831015840([120591 119905])] 119865 (120591) 119889120591

(71)

For 120583 = 120583119898 we obtain that

(119888) int[0119905]

119865 (120591) 119889120583119898

(120591) = (119870)int

119905

0

1198981015840(119905 minus 120591) 119865 (120591) 119889120591 (72)

Remark 25 Note that Theorem 24 holds only for a continu-ous and increasing 119865 but in the case 120583

119898= ] it holds for any

119865

Remark 26 Let us consider the representation of theChoquetintegral for a continuous case shown in Theorem 24 inrelation with a discrete case Now [0 119905] is transformed intothe discrete set 119909

1 1199092 119909

119899 Since119865 is an increasing fuzzy-

number-valued function we have 119909119894= 1199091015840

119894 119894 = 1 2 119899 in

Theorem 11 For the sake of simplicity let 0 = 1199090lt 1199091lt sdot sdot sdot lt

Abstract and Applied Analysis 9

119909119899

= 119905 119865(1199090) = 0 and [119909

119894 119905] = 119909

119894 119909119894+1

119909119899 It follows

that

(119888) int[0119905]

119865 (120591) 119889120583 (120591)

=

119899

sum

119894=1

[120583 ([119909119894 119905]) minus 120583 ([119909

119894+1 119905])] 119865 (119909

1015840

119894)

=

119899

sum

119894=1

minus [120583 ([119909119894 119905]) minus 120583 ([119909

119894+1 119905])]

Δ119909119894

119865 (1199091015840

119894) Δ119909119894

= (119870)int

119905

0

[minus1205831015840([120591 119905])] 119865 (120591) 119889120591

(73)

5 Choquet Integral Equations ofFuzzy-Number-Valued Functions

In this section let us consider a Choquet integral equationbased onTheorem 24 as shown below Given continuous andincreasing fuzzy-number-valued functions 119866(119905) with 119866(0) =

0 let us find continuous and increasing fuzzy-number-valuedfunctions 119865(119905) such that

119866 (119905) = (119888) int[0119905]

119865 (120591) 119889120583119898

(119905) (74)

which is expressed as

119866 (119905) = (119870)int

119905

0

1198981015840(119905 minus 120591) 119865 (120591) 119889120591 (75)

Definition 27 Let 119891(119905) be a real-valued function defined on[0 +infin) The following

F (119904) = int

+infin

0

119890minus119904119905

119891 (119905) 119889119905 (76)

is said to be the Laplace transformation of 119891(119905) if the infiniteintegralint+infin

0119890minus119904119905

119891(119905)119889119905 is convergentwith respect to the valueof parameter 119904 We denote its Laplace transformation asF(119904) = 119871[119891(119905)] and the inverse Laplace transformation as119891(119905) = 119871

minus1[F(119904)]

Definition 28 Let119865(119909 119910) be a fuzzy-number-valued functiondefined on [0 +infin) times [0 +infin) (119870) int

infin

0119865(119909 119910)119889119909 is said to be

convergent with respect to the parameter 119910 if for every fixed119910 isin [0 +infin) (119870) int

+infin

0119865(119909 119910)119889119909 exists

Definition 29 Let 119865(119905) be a fuzzy-number-valued functiondefined on [0 +infin) The following

F (119904) = (119870)int[0+infin)

119890minus119904119905

119865 (119905) 119889119905 (77)

is said to be the Laplace transformation of 119865(119905) if the Kalevaintegral intinfin

0119890minus119904119905

119865(119905)119889119905 is convergent with respect to the valueof parameter 119904 We denote its Laplace transformation asF(119904) = 119871[119865(119905)] and the inverse Laplace transformation as119865(119905) = 119871

minus1[F(119904)]

Remark 30 In view of Theorem 8 we have

F120582(119904) = [int

+infin

0

119890minus119904119905

119865120582(119905) 119889119905 int

+infin

0

119890minus119904119905

119865120582(119905) 119889119905] (78)

for all 120582 isin [0 1]

Theorem 31 Let 119866(119905) = (119888) int[0119905]

119865(120591)119889120583119898(119905) Then

G (119904) = 119904M (119904) F (119904)

119866 (119904) = 119871minus1

[119904M (119904) F (119904)]

(79)

where G(119904) = 119871[119866(119905)]M(s) = 119871[119898(119905)] and F(119904) = 119871[119865(119905)]

Proof Since 119866(119905) = (119888) int[0119905]

119865(120591)119889120583119898(119905) = (119870) int

119905

01198981015840(119905 minus

120591)119865(120591)119889120591 it follows easily from the condition that

G120582(119904) = [int

+infin

0

119890minus119904119905

119866120582(119905) 119889119905 int

+infin

0

119890minus119904119905

119866120582(119905) 119889119905]

= [int

+infin

0

119890minus119904119905

int

119905

0

1198981015840(119905 minus 120591) 119865

120582(120591) 119889120591 119889119905

int

+infin

0

119890minus119904119905

int

119905

0

1198981015840(119905 minus 120591) 119865

120582(120591) 119889120591 119889119905]

(80)

for all 120582 isin [0 1]Notice that

int

+infin

0

119890minus119904119905

int

119905

0

1198981015840(119905 minus 120591) 119865

120582(120591) 119889120591 119889119905

= int

+infin

0

int

+infin

120591

119890minus119904119905

1198981015840(119905 minus 120591) 119865

120582(120591) 119889119905 119889120591

= int

+infin

0

119865120582(120591) int

+infin

120591

119890minus119904119905

1198981015840(119905 minus 120591) 119889119905 119889120591

= int

+infin

0

119865120582(120591) [119890

minus119904119905119898(119905 minus 120591)

10038161003816100381610038161003816

+infin

120591minus int

+infin

120591

119898(119905 minus 120591) 119889119890minus119904119905

] 119889120591

= int

+infin

0

119865120582(120591) [minusint

+infin

120591

119898(119905 minus 120591) (minus119904) 119890minus119904119905

119889119905] 119889120591

= 119904int

+infin

0

119865120582(120591) [int

+infin

0

119898(119905 minus 120591) (minus119904) 119890minus119904(119905minus120591)

119890minus119904120591

119889 (119905 minus 120591)] 119889120591

= 119904int

+infin

0

119865120582(120591) [119890

minus119904120591int

+infin

0

119890minus119904(119905minus120591)

119898(119905 minus 120591) 119889 (119905 minus 120591)] 119889120591

= 119904int

+infin

0

119865120582(120591) 119890minus119904120591

M (119904) 119889120591

= 119904M (119904) int

+infin

0

119890minus119904120591

119865120582(120591) 119889120591 = 119904M (119904)F

120582(119904)

(81)

Following the same argument we can prove that

int

+infin

0

119890minus119904119905

int

119905

0

1198981015840(119905 minus 120591) 119865

120582(120591) 119889120591 119889119905 = 119904M (119904)F

120582(119904) (82)

10 Abstract and Applied Analysis

Hence

G (119904) = 119904M (119904) F (119904)

119866 (119904) = 119871minus1

[119904M (119904) F (119904)]

(83)

Theorem 32 For continuous and increasing fuzzy-valuedfunctions 119866(119905) with 119866(0) = 0

F (119904) =G (119904)

119904M (119904) 119865 (119904) = 119871

minus[

G (119904)

119904M (119904)] (84)

where G(119904) = 119871[119866(119905)]M(119904) = 119871[119898(119905)] and F(119904) = 119871[119865(119905)]

Proof It follows easily fromTheorem 31 that

G (119904) = 119904M (119904) F (119904)

119866 (119904) = 119871minus1

[119904M (119904) F (119904)]

(85)

It is obvious that

F (119904) =G (119904)

119904M (119904)

119865 (119904) = 119871minus1

[G (119904)

119904M (119904)]

(86)

where G(119904) = 119871[119866(119905)]M(119904) = 119871[119898(119905)] and F(119904) = 119871[119865(119905)]The Dirac delta function is defined by the properties

120575 (119905) = 0 for 119905 = 0

undefined at 119905 = 0(87)

and int+infin

minusinfin120575(119905)119889119905 = 1 that is the function has unit area

The Laplace transform of the Dirac Delta function 120575(119905) isshown as follows

119871 [120575 (119905)] = int

+infin

0

119890minus119904119905

120575 (119905) 119889119905 = 1 (88)

So

119871minus1

[1] = 120575 (119905) (89)

Example 33 Let 119898(119905) = 119905 and let the membership functionof 119866(119905) be

119898119866(119905)

(119909) = 119909 119909 isin [0 1]

0 119909 notin [0 1] 119905 le 1 (90)

Then by Theorem 32 we can solve the Choquet integralequation

119866 (119905) = (119888) int[0119905]

119865 (120591) 119889120583119898

(119905) (91)

Indeed let 119898(119905) = 119905 thenM(119904) = 11199042 The 120582-cut of 119866(119905) is

represented by interval 119866120582(119905) = 119909 | 119898

119866(119905)(119909) ge 120582 = [120582 1]

that is 119866120582(119905) = 120582 119866

120582(119905) = 1 In view of Remark 30 we have

G120582(119904) = [int

+infin

0119890minus119904119905

119866120582(119905)119889119905 int

+infin

0119890minus119904119905

119866120582(119905)119889119905] = [120582119878 1119878]

Then

F120582(119904) =

G120582(119904)

119904M (119904)= 120582

F120582(119904) =

G120582(119904)

119904M (119904)= 1

(92)

Furthermore

119865120582(119905) = 119871

minus1(F120582(119904)) = 120582120575 (119905)

119865120582(119905) = 119871

minus1(F120582(119904)) = 120575 (119905)

(93)

Therefore for every 120582 isin [0 1]

119865120582(119905) = [119865

120582(119905) 119865120582(119905)] = [120582120575 (119905) 120575 (119905)] (94)

Then we have the membership function of 119865(119905) as

119898119865(119905)

(119909) =

119909

120575 (119905) 119909 isin [0 120575 (119905)] 119905 = 0

0 otherwise(95)

6 Conclusions and Remarks

In this paper we have considered the Choquet integrals offuzzy-number-valued functions based on the nonnegativereal lineWe have discussed the Choquet integrals of interval-valued functions and fuzzy-number-valued functions basedon nonadditive Sugeno measures and showed the repre-sentation theorem of them respectively We also gave theoperational schemes of above several classes of integralson discrete sets Then we have given a representation ofthe Choquet integral of a nonnegative continuous andincreasing fuzzy-number-valued function with respect to afuzzymeasure In addition in order to solve Choquet integralequations of fuzzy-number-valued function a concept of theLaplace transformation for calculation has been introducedFor distorted Lebesgue measures it was shown that Choquetintegral equations of fuzzy-number-valued function could besolved by the Laplace transformation Finally we have givenan example to illustrate themain result at the end of the paper

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The work is supported by the Natural Scientific Funds ofChina (61262022) the Natural Scientific Fund of GansuProvince of China (1208RJZA251) Lanzhou Jiaotong Univer-sity Young Scientific Research Fund Project (2011020) theFundamental Research Funds for Gansu Province Univer-sities (no 213060) and the Scientific Research Project ofNorthwestNormalUniversity (noNWNU-KJCXGC-03-61)

Abstract and Applied Analysis 11

References

[1] G Choquet ldquoTheory of capacitiesrdquoAnnales de lrsquoInstitut Fouriervol 5 pp 131ndash295 1954

[2] D Denneberg Non-Additive Measure and Integral KluwerAcademic Publishers Boston Mass USA 1994

[3] T Murofushi M Sugeno and M Machida ldquoNon-monotonicfuzzy measures and the Choquet integralrdquo Fuzzy Sets andSystems vol 64 no 1 pp 73ndash86 1994

[4] Z Y Wang K-S Leung M-L Wong and J Fang ldquoA new typeof nonlinear integrals and the computational algorithmrdquo FuzzySets and Systems vol 112 no 2 pp 223ndash231 2000

[5] D Schmeidler ldquoSubjective probability and expected utilitywithout additivityrdquo Econometrica vol 57 no 3 pp 571ndash5871989

[6] M Grabisch ldquo119896-order additive discrete fuzzy measures andtheir representationrdquo Fuzzy Sets and Systems vol 92 no 2 pp167ndash189 1997

[7] MGrabisch I Kojadinovic and PMeyer ldquoA review ofmethodsfor capacity identification in Choquet integral based multi-attribute utility theory applications of the Kappalab R packagerdquoEuropean Journal of Operational Research vol 186 no 2 pp766ndash785 2008

[8] K Xu Z Wang P Heng and K Leung ldquoClassification bynonlinear integral projectionsrdquo IEEE Transactions on FuzzySystems vol 11 no 2 pp 187ndash201 2003

[9] M J Bolanos L M de Campos Ibanez and A GonzalezMunoz ldquoConvergence properties of the monotone expectationand its application to the extension of fuzzy measuresrdquo FuzzySets and Systems vol 33 no 2 pp 201ndash212 1989

[10] T Murofushi and M Sugeno ldquoA theory of fuzzy measuresrepresentations the Choquet integral and null setsrdquo Journal ofMathematical Analysis and Applications vol 159 no 2 pp 532ndash549 1991

[11] Z Wang ldquoConvergence theorems for sequences of Choquetintegralsrdquo International Journal of General Systems vol 26 no1-2 pp 133ndash143 1997

[12] YOuyang andHZhang ldquoOn the space ofmeasurable functionsand its topology determined by the Choquet integralrdquo Interna-tional Journal of Approximate Reasoning vol 52 no 9 pp 1355ndash1362 2011

[13] J Kawabe ldquoThe Choquet integral representability of comono-tonically additive functionals in locally compact spacesrdquo Inter-national Journal of Approximate Reasoning vol 54 no 3 pp418ndash426 2013

[14] S Abbasbandy E Babolian and M Alavi ldquoNumerical methodfor solving linear Fredholm fuzzy integral equations of thesecond kindrdquo Chaos Solitons amp Fractals vol 31 no 1 pp 138ndash146 2007

[15] O Kaleva ldquoFuzzy differential equationsrdquo Fuzzy Sets and Sys-tems vol 24 no 3 pp 301ndash317 1987

[16] J Mordeson and W Newman ldquoFuzzy integral equationsrdquoInformation Sciences vol 87 no 4 pp 215ndash229 1995

[17] M Sugeno ldquoA note on derivatives of functions with respect tofuzzy measuresrdquo Fuzzy Sets and Systems vol 222 pp 1ndash17 2013

[18] I Kojadinovic J-L Marichal and M Roubens ldquoAn axiomaticapproach to the definition of the entropy of a discrete Choquetcapacityrdquo Information Sciences vol 172 no 1-2 pp 131ndash1532005

[19] E Pap Null-Additive Set Functions vol 337 of Mathematicsand Its Applications KluwerAcademic Publishers LondonUK1994

[20] M Sugeno Theory of fuzzy integrals and its applications [PhDdissertation] Tokyo Institute of Technology Tokyo Japan 1974

[21] C X Wu D L Zhang and C M Guo ldquoFuzzy number fuzzymeasures and fuzzy integralsrdquo Fuzzy Sets and Systems vol 98no 3 pp 355ndash360 1998

[22] C X Wu and M Ma The Foundament of Fuzzy AnalysisNational Defense Press Beijing China 1991 (Chinese)

[23] R Yang Z Y Wang P A Heng and K S Leung ldquoFuzzynumbers and fuzzification of the Choquet integralrdquo Fuzzy Setsand Systems vol 153 no 1 pp 95ndash113 2005

[24] S Seikkala ldquoOn the fuzzy initial value problemrdquo Fuzzy Sets andSystems vol 24 no 3 pp 319ndash330 1987

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

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Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Abstract and Applied Analysis

Choquet integrals of real-valued functions Section 3 givesthe definitions and the characterizations of the Choquet inte-grals of interval-valued functions and fuzzy-number-valuedfunctions based on the nonadditive Sugeno measure Fur-thermore the operational schemes of above several classes ofintegrals on a discrete sets are investigated which enable us tocalculate Choquet integrals in applications Section 4 gives arepresentation of the Choquet integral of a nonnegative con-tinuous and increasing fuzzy-number-valued function withrespect to a fuzzymeasure In Section 5 in order to solveCho-quet integral equations of fuzzy-number-valued functions aconcept of the Laplace transformation for the fuzzy-number-valued functions in the sense of Choquet integral is intro-duced For distorted Lebesguemeasures it is shown thatCho-quet integral equations of fuzzy-number-valued functionscan be solved by the Laplace transformation In addition anexample is given to illustrate themain results at the end of thepaper The paper ends with conclusions in Section 6

2 Preliminaries

In this section wewill introduce some basic definitions aboutfuzzy measures Choquet integral and fuzzy numbers

Definition 1 (see [6 7 18ndash20]) Let 119883 be a nonempty set andA a 120590-algebra on 119883 A fuzzy measure on 119883 is a set function120583 A rarr [0infin) satisfying the following conditions

(1) 120583(0) = 0(2) 119860 isin 119883 119861 isin 119883 119860 sub 119861 implies 120583(119860) le 120583(119861)(3) In 119883 if 119860

1sub 1198602sub 1198603sub sdot sdot sdot and ⋃

infin

119899=1119860119899isin 119883 then

lim119899rarrinfin

120583(119860119899) = 120583(⋃

infin

119899=1119860119899)

(4) In 119883 if 1198601sup 1198602sup 1198603sup sdot sdot sdot and ⋂

infin

119899=1119860119899isin 119883 then

lim119899rarrinfin

120583(119860119899) = 120583(⋂

infin

119899=1119860119899)

120583 is said to be lower semicontinuous if it satisfies the aboveconditions (1)ndash(3) 120583 is said to be upper semicontinuous if itsatisfies the above conditions (1) (2) and (4) 120583 is said to becontinuous if it satisfies the above conditions (1)ndash(4)

(119883A 120583) is said to be a nonadditive measure spaceOne can see that a fuzzy measure is a normal monotone

set function which vanishes at the empty set Furthermore afuzzy measure on 119883 is said to be

(i) additive if 120583(119860 cup 119861) = 120583(119860) + 120583(119861) for all disjointsubsets 119860 119861 isin 119883

(ii) subadditive if 120583(119860 cup 119861) le 120583(119860) + 120583(119861) for all disjointsubsets 119860 119861 isin 119883

(iii) superadditive if 120583(119860cup119861) ge 120583(119860)+120583(119861) for all disjointsubsets 119860 119861 isin 119883

(iv) cardinality-based if for any119860 isin 119883 120583(119860) depends onlyon the cardinality of 119860

(v) a 0 minus 1 fuzzy measure if its range is 0 1(vi) a 0minus1possibility fuzzymeasure focused on119860 denoted

by Pos119860 if Pos

119860(119861) = 1 if and only if 119860 cap 119861 = 0 and

Pos119860(119861) = 0 otherwise

(vii) a 0minus1 necessity fuzzymeasure focused on119860 denotedby Nec

119860 if Nec

119860(119861) = 1 if and only if 119861 sube 119860 and

Nec119860(119861) = 0 otherwise

Let 119891 119883 rarr (minusinfin +infin) be a measurable function withrespect toA That is 119891 satisfies the condition

119891120572= 119909 | 119891 (119909) ge 120572 isin A (1)

for any 120572 isin R

Definition 2 (see [1]) Let (119883A 120583) be a nonadditive measurespace and 119891 a measurable function on 119883 The Choquetintegral of a real-valued function 119891 119883 rarr (minusinfin +infin) isdefined as

(119888) int119883

119891119889120583 = int

0

minusinfin

[120583 (119891120572) minus 120583 (119883)] 119889120572 + int

infin

0

120583 (119891120572) 119889120572 (2)

if both of Riemann integrals exist and at least one of them hasfinite value

Let 119864 isin A Then Choquet integral of a nonnegative real-valued function 119891 119883 rarr [0 +infin) is defined as

(119888) int119864

119891119889120583 = int

infin

0

120583 (119864 cap 119891120572) 119889120572 (3)

Since 120583(119864 cap 119891120572) is nonincreasing with respect to 120572 the

Choquet integral of real-valued function 119891 with respect to 120583

exists If (119888) int119864119891119889120583 lt infin then 119891 is said to be 119862-integrable

with respect to 120583 on 119883 Choquet integral has the followingproperties [1]

(1) If 119892 le ℎ then (119888) int119864119892119889120583 le (119888) int

119864ℎ119889120583

(2) If 119860 sub 119861 then (119888) int119860119891119889120583 le (119888) int

119861119891119889120583

(3) Let 120583 be lower semicontinuous If 119891119899

uarr 119891 ae in 119864then (119888) int

119864119891119899119889120583 uarr (119888) int

119864119891119889120583

(4) Let 120583 be upper semicontinuous If 119891119899

darr 119891 ae in 119864and there exists a 119862-integrable function 119892 such that1198911le 119892 then (119888) int

119864119891119899119889120583 darr (119888) int

119864119891119889120583

119868(119877+) = 119903 [119903 119903] sub 119877

+ denotes the set of all interval

numbers on 119877+ where 119877

+= [0 +infin) With the definition

from Wu et al [21] interval numbers should satisfy thefollowing basic operations

(1) 119903 lowast 119901 = [119903 lowast 119901 119903 lowast 119901](lowast denotes + or and)

(2) 119896 sdot 119903 = [119896119903 119896119903] (119896 isin R+)(3) 119903 le 119901 hArr 119903 le 119901 119903 le 119901

(4) 119889(119903 119901) = max|119903 minus 119901| |119903 minus 119901|(5) If 119889(119903

119899 119903) rarr 0 then 119903

119899rarr 119903

A fuzzy subset of 119877 is a function 119906 119877 rarr [0 1]For each fuzzy set 119906 defined as above we denote by 119906

120582=

119909 isin 119877 119906(119909) ge 120582 for any 120582 isin [0 1] its 120582-level set Bysupp119906 we denote the support of 119906 that is the set 119909 isin 119877

119906(119909) gt 0 By 1199060we denote the closure of supp119906 that is

1199060

= 119909 isin 119877 119906(119909) gt 0 Let 119864 be the collection of all fuzzysets of 119877 We call 119906 isin 119864 a fuzzy number if it satisfies thefollowing conditions [22]

Abstract and Applied Analysis 3

(1) 119906 is normal that is there exists 1199090

isin 119877 such that119906(1199090) = 1

(2) 119906 is fuzzy convex that is 119906(120582119909 + (1 minus 120582)119910) ge

min119906(119909) 119906(119910) for any 119909 119910 isin 119877 0 le 120582 le 1(3) 119906 is upper semicontinuous that is 119906(119909

0) ge

lim119896rarrinfin

119906(119909119896) for any 119909

119896isin 119877 (119896 = 0 1 2 ) 119909

119896rarr

1199090

(4) 1199060= 119909 isin 119877 119906(119909) gt 0 is compact

We denote the collection of all fuzzy numbers by 119886 is said to be a nonnegative fuzzy number if supp 119886 =

119909 isin 119877 | 119886(119909) gt 0 sub 119877+We can define the semiorder and the

distance in space [22] Let 119886 isin Then 119886 le if 119886120582

le 119887120582

and 119886120582

le 119887120582for 120582 isin (0 1] 119886 + = 119888 if 119886

120582+ 119887120582

= 119888120582for

120582 isin (0 1] 119863(119886 ) = sup120582isin[01]

119863(119886120582 119887120582) is the distance of 119886

and Let 119886119899sub If 119863(119886

119899 119886) rarr 0 then 119886

119899rarr 119886

119877+ denotes the collection of all nonnegative fuzzy num-bers

Lemma 3 (see [22]) Let 119886 isin Then

(1) 119886120582is a nonempty bounded and closed interval for each

120582 isin [0 1](2) if 0 le 120582

1le 1205822le 1 then 119886

1205822

sub 1198861205821

(3) if 120582119899ge 0 and 120582

119899rarr 120582 (120582 isin (01]) then⋂

infin

119899=1119886120582119899

= 119886120582

Conversely there exists a 119887120582

sub 119877 for each 120582 isin (0 1]which satisfies the conditions (1)ndash(3) then there exists a uniquefuzzy number 119886 isin such that 119887

120582= 119886120582 120582 isin (0 1] and

1198860= ⋃120582isin(01]

119886120582sub 1198870

3 Choquet Integrals of Interval-ValuedFunctions and Fuzzy-Number-ValuedFunctions Based on Nonadditive Measures

In this section we will give the definitions and the characteri-zations of the Choquet integrals of interval-valued functionsand fuzzy-number-valued functions based on the nonaddi-tive Sugeno measure Furthermore the operational schemesof above several classes of integrals on a discrete sets areinvestigated which enable us to calculate Choquet integralsin applications

We first introduce the concept of the Choquet integralsfor the interval-valued functions as follows

Definition 4 (see [23]) An interval-valued function 119865 119883 rarr

119868(119877+) is said to be measurable if both 119865(119909) and 119865(119909) are

measurable functions where 119865(119909) = [119865(119909) 119865(119909)] 119865(119909) is theleft end point of interval 119865(119909) and 119865(119909) is the right end pointof interval 119865(119909)

Interval-valued function 119865 119883 rarr 119868(119877+) is 119862-integrally

bounded if there exists a Choquet integrable function ℎ

119883 rarr 119877+ such that || le ℎ(119905) for every selection isin 119865

We denote 1198750(119883) = 119864 | 119864 sub 119883 and 119864 = 0 Let (119883A 120583)

be a measure space 119860 119861 sub A and let 119865 119860 rarr 1198750(119861) be a

measurable set valued mapping 119891 is said to be a measurable

selection of 119865 if there exists a measurable mapping 119891 119860 rarr

119861 such that 119891(119909) isin 119865(119909) for every 119909 isin 119860

Definition 5 Let (119883A 120583) be a nonadditive measure spaceAssume that 119865 119883 rarr 119868(119877

+) is measurable 119862-integrally

bounded interval-valued function and 119864 isin A 119865 is said tobe 119862-integrable if

(119888) int119864

119865119889120583 = (119888) int119864

119892119889120583 | 119892 isin 119878119865(119909)

(4)

is a closed interval on 119868(119877+) where

119878119865(119909)

= 119892 | 119892 119883 997888rarr 119877+ is a measurable selection of 119865 (119909)

(5)

Theorem 6 Let (119883A 120583) be a nonadditive measure spaceSuppose that 120583 is a fuzzymeasure119864 isin A and119865 is nonnegativemeasurable 119862-integrally bounded interval-valued functionthen 119865 is 119862-integrable on 119864 and

(119888) int119864

119865119889120583 = [(119888) int119864

119865119889120583 (119888) int119864

119865119889120583] (6)

Proof Since 119865 is nonnegative measurable on 119864 119865 and 119865 aremeasurable on119864Thus 119865 and 119865 are twomeasurable selectionsof 119865 On the other hand 119865 is 119862-integrally bounded we have

119865 le ℎ 119865 le ℎ (7)

By the properties of Choquet integral of real valued functionswe know that 119865 and 119865 are 119862-integrable on 119864 Let 119872 isin

(119888) int119864119865119889120583 That is to say there exists a measurable selection

119892 of 119865 such that (119888) int119864119892119889120583 = 119872 We can prove that 119872 isin

[(119888) int119864119865119889120583 (119888) int

119864119865119889120583] Indeed notice that 119865 le 119892 le 119865 and

by the properties of Choquet integral of real-valued functionswe have

(119888) int119864

119865119889120583 le (119888) int119864

119892119889120583 le (119888) int119864

119865119889120583 (8)

This follows that

119872 isin [(119888) int119864

119865119889120583 (119888) int119864

119865119889120583] (9)

Thus

(119888) int119864

119892119889120583 | 119892 isin 119878119865(119909)

sub [(119888) int119864

119865119889120583 (119888) int119864

119865119889120583] (10)

Conversely we can show that

[(119888) int119864

119865119889120583 (119888) int119864

119865119889120583] sub (119888) int119864

119865119889120583 (11)

For measurable selections 119901(119909)119898(119909) isin 119865(119909) we get that

119905119901 (119909) + (1 minus 119905)119898 (119909) isin 119865 (119909) (12)

4 Abstract and Applied Analysis

is measurable selection of 119865 for any 0 le 119905 le 1 Therefore

119905 sdot (119888) int119864

119901 (119909) 119889120583 + (1 minus 119905) sdot (119888) int119864

119898(119909) 119889120583

= (119888) int119864

[119905119901 (119909) + (1 minus 119905)119898 (119909)] 119889120583

isin (119888) int119864

119892119889120583 | 119892 isin 119878119865(119909)

(13)

It implies that (119888) int119864119865119889120583 = (119888) int

119864119892119889120583 | 119892 isin 119878

119865(119909) is a convex

set On the other hand since 119865 and 119865 are two measurableselections of 119865 and

(119888) int119864

119865119889120583 isin (119888) int119864

119865119889120583 (119888) int119864

119865119889120583 isin (119888) int119864

119865119889120583 (14)

we have

[(119888) int119864

119865119889120583 (119888) int119864

119865119889120583] sub (119888) int119864

119892119889120583 | 119892 isin 119878119865(119909)

(15)

Hence

(119888) int119864

119865119889120583 = [(119888) int119864

119865119889119892120583 (119888) int119864

119865119889120583] (16)

That is 119865 is 119862-integrable

From the above theorem we know that interval-valuedfunction 119865 is 119862-integrable on 119864 if (119888) int

119864119865119889120583 and (119888) int

119864119865119889120583

exist and are boundedNext we will introduce the concept of the Choquet

integrals for the Fuzzy-number-valued functions as followsFuzzy-number-valued function 119865 119883 rarr

+ on119883 is saidto be measurable if 119865

120582and 119865

120582are measurable functions with

respect to 119909 isin 119883 for any 120582 isin [0 1]Fuzzy-number-valued function 119865 119883 rarr

+ is said tobe 119862-integrally bounded if there exists a Choquet integrablefunction ℎ 119883 rarr 119877

+ such that || le ℎ(119905) for every selection isin [119865(119905)]

0

Definition 7 Let (119883A 120583) be a nonadditive measure spaceAssume that 119864 isin A 119865 119883 rarr

+ is measurable and 119862-integrally bounded function 119865 is said to be 119865119862-integrable if

[(119888) int119864

119865119889120583]

120582

= (119888) int119864

119892119889120583 | 119892 isin 119878119865120582

0 le 120582 le 1 (17)

determines a unique fuzzy number 119886 isin + which is denoted

by (119888) int119864119865119889120583 = 119886 where 119878

119865120582

= 119892 119883 rarr 119877+ 119892 isin 119865

120582is a

measurable selection of 119865120582

Theorem 8 Let (119883A 120583) be a nonadditive measurable spaceAssume 120583 is a continuous fuzzy measure 119864 isin A 119865 119883 rarr

+

ismeasurable and119862-integrally bounded function then119865 is119865119862-integrable on 119864 if and only if 119865

120582and 119865

120582are 119862-integrable on 119864

and

[(119888) int119864

119865119889120583]

120582

= [(119888) int119864

119865120582119889120583 (119888) int

119864

119865120582119889120583] (18)

for any 120582 isin [0 1]

Proof For the necessity since 119865 is measurable and 119862-integrally bounded function 119865

120582and 119865

120582are measurable for

every 120582 isin [0 1] and there exists a Choquet integrable func-tion ℎ(119909) such that 0 le 119865

120582le ℎ(119909) 0 le 119865

120582le ℎ(119909) and then

(119888) int119864

119865120582119889120583 le (119888) int

119864

119865120582119889120583 le (119888) int

119864

ℎ (119909) 119889120583 lt infin (19)

That is 119865120582 119865120582are 119862-integrable and

[(119888) int119864

119865119889120583]

120582

= [(119888) int119864

119865120582119889120583 (119888) int

119864

119865120582119889120583] (20)

for any 120582 isin [0 1]For the sufficiency let 119865 be a Fuzzy-number-valued func-

tion Note that

[(119888) int119864

119865119889120583]

120582

= [(119888) int119864

119865120582119889120583 (119888) int

119864

119865120582119889120583] (21)

we need only to prove that the interval family

[(119888) int119864

119865119889120583]

120582

= [(119888) int119864

119865120582119889120583 (119888) int

119864

119865120582119889120583] 120582 isin [0 1]

(22)

determines a unique fuzzy number Indeed the interval fam-ily

[(119888) int119864

119865120582119889120583 (119888) int

119864

119865120582119889120583] (23)

satisfies the conditions of Lemma 3

(1) 119865 is a measurable fuzzy-number-valued function foreach 120582 isin [0 1] we have 119865

120582(119909) le 119865

120582(119909) and therefore

(119888) int119864

119865120582119889120583 le (119888) int

119864

119865120582119889120583 (24)

(2) Since 1198651205822

sub 1198651205821

for 0 le 1205821le 1205822le 1 that is

1198651205821

(119909) le 1198651205822

(119909) 1198651205821(119909) ge 119865

1205822(119909) (25)

we have

(119888) int119864

1198651205821

119889120583 le (119888) int119864

1198651205822

119889120583

(119888) int119864

1198651205821

119889120583 ge (119888) int119864

1198651205822

119889120583

[(119888) int119864

1198651205821

119889120583 (119888) int119864

1198651205821

119889120583]

sup [(119888) int119864

1198651205822

119889120583 (119888) int119864

1198651205822

119889120583]

(26)

(3) For each 120582119899uarr 120582 isin (0 1] ⋂infin

119899=1119865120582119899

(119909) = 119865120582(119909) that is

lim119899rarrinfin

119865120582119899

= 119865120582 lim119899rarrinfin

119865120582119899

= 119865120582 It is easy to see

Abstract and Applied Analysis 5

that 1198651205821

le 119865120582119899

le 119865120582119899

le 1198651205821

1198651205821

1198651205821

are integrableand by the continuity of 120583 then

infin

⋂

119899=1

[(119888) int119864

119865120582119899

119889120583 (119888) int119864

119865120582119899

119889120583]

= [(119888) int119864

119865120582119889120583 (119888) int

119864

119865120582119889120583]

(27)

In conclusion there exists a unique fuzzy number 119886 isin +

such that

119886120582= [(119888) int

119864

119865120582119889120583 (119888) int

119864

119865120582119889120583] (28)

Furthermore we get that 119865 is 119865119862-integrable on 119864 and

[(119888) int119864

119865119889120583]

120582

= [(119888) int119864

119865120582119889120583 (119888) int

119864

119865120582119889120583] (29)

In the last part of the section we will investigate theoperational schemes of above several classes of integrals ona discrete set

Let119883 = 1199091 1199092 119909

119899 be a discrete setThenwewill give

a new scheme to calculate the value of the Choquet integral

Theorem 9 (see [8]) Let 119891 be a real-valued function on 119883 =

1199091 1199092 119909

119899 Then Choquet integral of 119891 with respect to a

fuzzy measure 120583 on 119883 is given by

(119888) int119883

119891119889120583 =

119899

sum

119894=1

[119891 (1199091015840

119894) minus 119891 (119909

1015840

119894minus1)] 120583 (119883

1015840

119894) (30)

or equivalently by

(119888) int119883

119891119889120583 =

119899

sum

119894=1

[120583 (1198831015840

119894) minus 120583 (119883

1015840

119894+1)] 119891 (119909

1015840

119894) (31)

where 1199091015840

1 1199091015840

2 119909

1015840

119899is a permutation of 119909

1 1199092 119909

119899such

that 119891(1199091015840

0) le 119891(119909

1015840

1) le sdot sdot sdot le 119891(119909

1015840

119899) 119891(119909

1015840

0) = 0 119883

1015840

119894=

1199091015840

119894 1199091015840

119894+1 119909

1015840

119899 119894 = 1 2 119899 and 119883

1015840

119899+1= 0

Theorem 10 Let 119865 be an interval-valued function on 119883 =

1199091 1199092 119909

119899 Then Choquet integral of 119865 with respect to a

fuzzy measure 120583 on 119883 is given by

(119888) int119883

119865119889120583 =

119899

sum

119894=1

[120583 (1198831015840

119894) minus 120583 (119883

1015840

119894+1)] 119865 (119909

1015840

119894) (32)

where 1199091015840

1 1199091015840

2 119909

1015840

119899is a permutation of 119909

1 1199092 119909

119899such

that 119865(1199091015840

0) le 119865(119909

1015840

1) le sdot sdot sdot le 119865(119909

1015840

119899) 119865(119909

1015840

0) = [0 0] 1198831015840

119894=

1199091015840

119894 1199091015840

119894+1 119909

1015840

119899 119894 = 1 2 119899 and 119883

1015840

119899+1= 0

Proof Since 119865 is an interval-valued function on119883 in view ofTheorem 6 we have

(119888) int119883

119865119889120583 = [(119888) int119883

119865119889120583 (119888) int119883

119865119889120583] (33)

Note that 119865 and 119865 are real-valued function on119883 respectivelyByTheorem 9 we get

(119888) int119883

119865119889120583 = [

119899

sum

119894=1

[120583 (1198831015840

119894) minus 120583 (119883

1015840

119894+1)] 119865 (119909

1015840

119894)

119899

sum

119894=1

[120583 (1198831015840

119894) minus 120583 (119883

1015840

119894+1)] 119865 (119909

1015840

119894)]

=

119899

sum

119894=1

[120583 (1198831015840

119894) minus 120583 (119883

1015840

119894+1)] 119865 (119909

1015840

119894)

(34)

where 1199091015840

1 1199091015840

2 119909

1015840

119899is a permutation of 119909

1 1199092 119909

119899such

that 119865(1199091015840

0) le 119865(119909

1015840

1) le sdot sdot sdot le 119865(119909

1015840

119899) 119865(119909

1015840

0) = [0 0] 1198831015840

119894=

1199091015840

119894 1199091015840

119894+1 119909

1015840

119899 119894 = 1 2 119899 and 119883

1015840

119899+1= 0

Theorem 11 Let119865 be a fuzzy-number-valued function on119883 =

1199091 1199092 119909

119899 Then Choquet integral of 119865 with respect to a

fuzzy measure 120583 on 119883 is given by

(119888) int119883

119865119889120583 =

119899

sum

119894=1

[120583 (1198831015840

119894) minus 120583 (119883

1015840

119894+1)] 119865 (119909

1015840

119894) (35)

where 1199091015840

1 1199091015840

2 119909

1015840

119899is a permutation of 119909

1 1199092 119909

119899such

that 119865(1199091015840

0) le 119865(119909

1015840

1) le sdot sdot sdot le 119865(119909

1015840

119899) 119865(119909

1015840

0) = 0 119883

1015840

119894=

1199091015840

119894 1199091015840

119894+1 119909

1015840

119899 119894 = 1 2 119899 and 119883

1015840

119899+1= 0

Proof Since 119865 is a fuzzy-number-valued on 119883 in view ofTheorem 8 we have

[(119888) int119883

119865119889120583]

120582

= [(119888) int119883

119865120582119889120583 (119888) int

119883

119865120582119889120583] (36)

for any 120582 isin [0 1] By the semiorder in space (ie let 119886 isin

Then 119886 le if 119886120582le 119887120582and 119886

120582le 119887120582) andTheorem 10 there

is a sequence 1199091015840

1 1199091015840

2 119909

1015840

119899on 119883 such that 119865(119909

1015840

0) le 119865(119909

1015840

1) le

sdot sdot sdot le 119865(1199091015840

119899) 119865(119909

1015840

0) = 0 1198831015840

119894= 1199091015840

119894 1199091015840

119894+1 119909

1015840

119899 119894 = 1 2 119899

1198831015840

119899+1= 0 and 119909

1015840

1 1199091015840

2 119909

1015840

119899is a permutation of 119909

1 1199092 119909

119899

Consequently

[(119888) int119883

119865119889120583]

120582

= [

119899

sum

119894=1

[120583 (1198831015840

119894) minus 120583 (119883

1015840

119894+1)] 119865120582(1199091015840

119894)

119899

sum

119894=1

[120583 (1198831015840

119894) minus 120583 (119883

1015840

119894+1)] 119865120582(1199091015840

119894)]

(37)

for any 120582 isin [0 1] Therefore

(119888) int119883

119865119889120583 =

119899

sum

119894=1

[120583 (1198831015840

119894) minus 120583 (119883

1015840

119894+1)] 119865 (119909

1015840

119894) (38)

4 The Representation of Choquet Integral ofFuzzy-Number-Valued Functions

Sugeno has described carefully the representation of Cho-quet integral of real-valued increasing functions and some

6 Abstract and Applied Analysis

important conclusions have been obtained [17] Motivated bythis we will discuss the representation of Choquet integral offuzzy-number-valued functions in this section

Definition 12 Fuzzy-number-valued function 119865 119883 rarr +

is said to be continuous on 119883 if for every 1199090isin 119883 there are

lim119909rarr119909

0

119865120582(119909) = 119865

120582(1199090)

lim119909rarr119909

0

119865120582(119909) = 119865

120582(1199090)

(39)

where 120582 isin [0 1]

Definition 13 Fuzzy-number-valued function 119865 119883 rarr +

is said to be increasing on 119883 if for every 1199091

le 1199092there are

119865120582(1199091) le 119865120582(1199092) and 119865

120582(1199091) le 119865120582(1199092) where 119909

1 1199092isin 119883 and

120582 isin [0 1]LetF+ be a class ofmeasurable nonnegative continuous

and increasing fuzzy-number-valued functionsLet ] be a Lebesgue measure for [119886 119887] sub [0infin) ][119886 119887] =

119887 minus 119886

Definition 14 (see [17]) Let 119898 119877+

rarr 119877+ be a continuous

and increasing function and 119898(0) = 0 A fuzzy measure 120583119898

a distorted Lebesgue measure is defined by 120583119898(sdot) = 119898(](sdot))

Definition 15 (see [24]) A fuzzy-number-valued function 119865

[119886 119887] rarr is differentiable at 119909 isin [119886 119887] if there exist fuzzynumber-valued functions 1198651015840(119909) such that

[1198651015840(119909)]120582= [1198651015840

120582(119909) 119865

1015840

120582(119909)] (40)

for every 120582 isin [0 1] where 1198651015840(119909) is the fuzzy derivative of

119865(119909)Note that 120583

119898is induced from the Lebesguemeasure ] by a

monotone transformation where 120583119898([119886 119887]) = 119898(]([119886 119887])) =

119898(119887 minus 119886) Apparently it loses additivity unless 119898(119905) = 119905 butreserves monotonicity In what follows we assume that 119898(119905)

is differentiableIn this section we consider the calculation of Choquet

integrals Let 120583 be a general fuzzy measure and consider120583([120591 119905]) for a closed interval [120591 119905] then 120583([120591 119905]) is decreasingfor 120591 and increasing for 119905 Throughout the paper we assumethat the functions 119898(119905) 119865(119905) and 119866(119905) are continuouslydifferentiable We also assume that 120583([120591 119905]) is continuouslydifferentiable with respect to 120591 on [0 119905] for every 119905 gt 0 Inaddition we require the regularity condition that 120583(119905) = 0

holds for every 119905 ge 0 We write 1205831015840([120591 119905]) = (120597120597120591)120583([120591 119905])

where we note that 1205831015840([120591 119905]) le 0 for 120591 le 119905 If 120583 = 120583119898then

1205831015840([120591 119905]) = minus119898

1015840(119905 minus 120591) where 119898

1015840(119905) = 119889119898(119905)119889119905 First we

consider a case that 119865(119905) is strictly increasingFuzzy-number-valued function 119865 119883 rarr is said to

be 119871-integrally bounded if there exists a Lebesgue integrablefunction ℎ 119883 rarr 119877

+ such that || le ℎ(119905) for every section isin [119865(119905)]

0

Definition 16 (see [22]) Let fuzzy-number-valued functions119865 [119886 +infin) rarr be measurable and 119871-integrally bounded119865 is called Kaleva-integrable if

[int

+infin

119886

119865(119905)119889119905]

120582

= (119871) int

+infin

119886

119892 (119905) 119889119905 | 119892 (119905) isin 119878[119865(119905)]

120582

0 le 120582 le 1

(41)

determines a unique fuzzy number 119886 isin which is denotedby (119870) int

+infin

119886119865(119905) 119889120583 = 119886 where 119878

[119865(119905)]120582

= 119892(119905) isin [119865(119905)]120582is a

measurable selection

Lemma 17 (see [22]) Let 119865 [119886 +infin) rarr be measurableand C-integrally bounded fuzzy-number-valued function then119865 is Kaleva integrable on [119886 +infin) if and only if 119865

120582(119905) and119865

120582(119905)

are 119871-integrable on [119886 +infin) and

[(119870)int

119905

0

119865(120591)119889120591]

120582

= [(119871) int

119905

0

119865120582(120591) 119889120591 (119871) int

119905

0

119865120582(120591) 119889120591]

(42)

for any 120582 isin [0 1]

Theorem 18 Let 119865(119905) isin F+ Then minus1205831015840([120591 119905])119865(120591) is Kaleva

integrable on [0 119905] and

[(119870)int

119905

0

[minus1205831015840([120591 119905])]119865(120591)119889120591]

120582

= [(119871) int

119905

0

[minus1205831015840([120591 119905])] 119865

120582(120591) 119889120591

(119871) int

119905

0

[minus1205831015840([120591 119905])] 119865

120582(120591) 119889120591]

(43)

Proof Since119865(119905) isin F+ for every120582 isin [0 1]we get that119865120582and

119865120582are measurable nonnegative continuous and increasing

real-valued functions on [0 119905] It follows that 119865120582and 119865

120582are

119871-integrable on [0 119905] 120583([120591 119905]) is 119871-integrable on [0 119905] as itis continuously differentiable with respect to 120591 on [0 119905] forevery 119905 gt 0 In view of Lemma 17 [minus120583

1015840([120591 119905])]119865(120591) is Kaleva

integrable on [0 119905] and

[(119870)int

119905

0

[minus1205831015840([120591 119905])] 119865 (120591) 119889120591]

120582

= [(119871) int

119905

0

[minus1205831015840([120591 119905])] 119865

120582(120591) 119889120591

(119871) int

119905

0

[minus1205831015840([120591 119905])] 119865

120582(120591) 119889120591]

(44)

Lemma 19 (see [17]) Let 119891(119905) be a real strictly increasingfunction Then the Choquet integral of 119891 with respect to 120583 on[0 119905] is represented as

(119888) int[0119905]

119891 (120591) 119889120583 (120591) = minusint

119905

0

1205831015840([120591 119905]) 119891 (120591) 119889120591 (45)

Abstract and Applied Analysis 7

In particular for 120583 = 120583119898

(119888) int[0119905]

119891 (120591) 119889120583119898

(120591) = int

119905

0

1198981015840(119905 minus 120591) 119891 (120591) 119889120591 (46)

Lemma 20 (see [17]) Let 119891(119905) be a constant real-valuedfunction 119891(119905) = 119862 forall119905 isin 119877

+= (0 +infin) Then

(119888) int[0119905]

119891 (120591) 119889120583 (120591) = minusint

119905

0

1205831015840([120591 119905]) 119862119889120591 (47)

In particular for 120583 = 120583119898

(119888) int[0119905]

119891 (120591) 119889120583119898

(120591) = 119862119898 (119905) (48)

Lemma 21 (see [17]) Let 119891(119905) be a measurable nonnegativecontinuous and increasing real-valued function Then theChoquet integral of 119891 with respect to 120583 on [0 119905] is representedas

(119888) int[0119905]

119891 (120591) 119889120583 (120591) = int

infin

0

120583 (120591 | 119891 (120591) ge 119903 cap [0 119905]) 119889119903

= minusint

119905

0

1205831015840([120591 119905]) 119891 (120591) 119889120591

(49)

In particular for 120583 = 120583119898

(119888) int[0119905]

119891 (120591) 119889120583119898

(120591) = int

infin

0

120583 (120591 | 119891 (120591) ge 119903 cap [0 119905]) 119889119903

= int

119905

0

1198981015840(119905 minus 120591) 119891 (120591) 119889120591

(50)

Theorem 22 Let 119865(119905) be a strictly increasing fuzzy-number-valued functionThen the Choquet integral of 119865with respect to120583 on [0 119905] is represented as

(119888) int[0119905]

119865 (120591) 119889120583 (120591) = (119870)int

119905

0

[minus1205831015840([120591 119905])] 119865 (120591) 119889120591 (51)

In particular for 120583 = 120583119898

(119888) int[0119905]

119865 (120591) 119889120583119898

(120591) = (119870)int

119905

0

1198981015840(119905 minus 120591) 119865 (120591) 119889120591 (52)

Proof ByTheorem 8 we have

[(119888) int[0119905]

119865 (120591) 119889120583 (120591)]

120582

= [(119888) int[0119905]

119865120582(120591) 119889120583 (120591) (119888) int

[0119905]

119865120582(120591) 119889120583 (120591)]

(53)

for any 120582 isin [0 1] Since 119865(119905) is strictly increasing 119865120582(119905) and

119865120582(119905) are strictly increasing real-valued functions respec-

tively In view of Lemma 19 we have

(119888) int[0119905]

119865120582(120591) 119889120583 (120591) = minusint

119905

0

1205831015840([120591 119905]) 119865

120582(120591) 119889120591

(119888) int[0119905]

119865120582(120591) 119889120583 (120591) = minusint

119905

0

1205831015840([120591 119905]) 119865

120582(120591) 119889120591

(54)

It follows that

[(119888) int[0119905]

119865(120591)119889120583(120591)]

120582

= [minusint

119905

0

1205831015840([120591 119905]) 119865

120582(120591) 119889120591 minusint

119905

0

1205831015840([120591 119905]) 119865120582 (120591) 119889120591]

(55)

On the other hand

int

119905

0

[minus1205831015840([120591 119905])] 119865120582 (120591) 119889120591

= int

119905

0

[minus1205831015840([120591 119905])] [119865

120582(120591) 119865

120582(120591)] 119889120591

= [minusint

119905

0

1205831015840([120591 119905]) 119865

120582(120591) 119889120591 minusint

119905

0

1205831015840([120591 119905]) 119865120582 (120591) 119889120591]

(56)

Therefore

[(119888) int[0119905]

119865(120591)119889120583(120591)]

120582

= int

119905

0

[minus1205831015840([120591 119905])] 119865120582 (120591) 119889120591 (57)

By the arbitrary of 120582 we have

(119888) int[0119905]

119865 (120591) 119889120583 (120591) = (119870)int

119905

0

[minus1205831015840([120591 119905])] 119865 (120591) 119889120591 (58)

For 120583 = 120583119898 we note that minus120583

1015840([120591 119905]) = 119898

1015840(119905 minus 120591) hence

(119888) int[0119905]

119865 (120591) 119889120583119898

(120591) = (119870)int

119905

0

1198981015840(119905 minus 120591) 119865 (120591) 119889120591 (59)

Theorem 23 Let 119865(119905) be a constant fuzzy-valued functionthat is 119865(119905) = 119862 forall119905 isin 119877

+= (0 +infin) Then

(119888) int[0119905]

119865 (120591) 119889120583 (120591)

= (119870)int

119905

0

[minus1205831015840([120591 119905])] 119862119889120591 = 119862120583 ([0 119905])

(60)

In particular for 120583 = 120583119898

(119888) int[0119905]

119865 (120591) 119889120583119898

(120591) = 119862119898 (119905) (61)

Proof Note that119862 = [119862 119862] where119862 and119862 are constant real-valued functions respectively In view of Lemma 20 we have

(119888) int[0119905]

119862119889120583 (120591) = minusint

119905

0

1205831015840([120591 119905]) 119862119889120591

(119888) int[0119905]

119862119889120583 (120591) = minusint

119905

0

1205831015840([120591 119905]) 119862119889120591

(62)

8 Abstract and Applied Analysis

The rest of the proof follows in exactly the same way as thatof Theorem 22 Hence

(119888) int[0119905]

119865 (120591) 119889120583 (120591)

= (119870)int

119905

0

[minus1205831015840([120591 119905])] 119862119889120591 = 119862120583 ([0 119905])

(63)

For 120583 = 120583119898 we obtain 120583

1015840([120591 119905]) = minus119898

1015840(119905 minus 120591) It follows that

(119888) int[0119905]

119862119889120583 (120591) = (119870)int

119905

0

[minus1205831015840([120591 119905])] 119862119889120591

= int

119905

0

1198981015840(119905 minus 120591) 119862119889120591

= (119870)119862int

119905

0

1198981015840(119905 minus 120591) 119889120591 = 119862119898 (119905)

(64)

where 119898(0) = 0

Theorem 24 Let 119865(119905) isin F+ Then the Choquet integral of 119865with respect to 120583 on [0 119905] is represented as

(119888) int[0119905]

119865 (120591) 119889120583 (120591) = (119870)int

119905

0

[minus1205831015840([120591 119905])] 119865 (120591) 119889120591 (65)

In particular for 120583 = 120583119898

(119888) int[0119905]

119865 (120591) 119889120583119898

(120591) = (119870)int

119905

0

1198981015840(119905 minus 120591) 119865 (120591) 119889120591 (66)

Proof The theorem has been proved when 119865(119905) is a strictlyincreasing or constant fuzzy-number-valued function byTheorem 22 or Theorem 23 Without loss of generality weconsider a continuous and increasing function such that

119865 (119905) =

1198651(119905) 0 le 119905 lt 119905

1

1198652(119905) 119905

1le 119905 lt 119905

2

1198653(119905) 119905

2le 119905 lt infin

(67)

where 1198651(119905) and 119865

3(119905) are strictly increasing 119865

2(119905) is constant

and 1198651(1199051) = 1198652(119905) = 119865

3(1199052)

(i) For 0 le 119905 lt 1199051

Note that 1198651(119905) is strictly increasing on [0 119905

1) from

Theorem 22 we obtain

(119888) int[0119905]

119865 (120591) 119889120583 (120591) = (119870)int

119905

0

[minus1205831015840([120591 119905])] 1198651 (120591) 119889120591 (68)

(ii) For 1199051le 119905 lt 119905

2

In view of the construction of 119865(119905) on [0 119905] we have

(119888) int[0119905]

119865 (120591) 119889120583 (120591) = (119888) int[01199051]

1198651(120591) 119889120583 (120591)

+ (119888) int[1199051119905]

1198652(120591) 119889120583 (120591)

(69)

Since 1198651(119905) is strictly increasing on [0 119905

1) and 119865

2(119905) is

constant on [1199051 1199052) by Theorems 22 and 23

(119888) int[0119905]

119865 (120591) 119889120583 (120591) = (119870)int

1199051

0

[minus1205831015840([120591 119905])] 119865

1(120591) 119889120591

+ (119870)int

119905

1199051

[minus1205831015840([120591 119905])] 119865

2(120591) 119889120591

= (119870)int

119905

0

[minus1205831015840([120591 119905])] 119865 (120591) 119889120591

(70)

(iii) For 1199052le 119905 lt infin

Since1198651(119905) is strictly increasing on [0 119905

1) 1198652(119905) is constant

on [1199051 1199052) and 119865

3(119905) is strictly increasing on [119905

1infin) by

Theorems 22 and 23 we have

(119888) int[0119905]

119865 (120591) 119889120583 (120591)

= (119888) int[01199051]

1198651(120591) 119889120583 (120591) + (119888) int

[11990511199052]

1198652(120591) 119889120583 (120591)

+ (119888) int[1199052119905]

1198653(120591) 119889120583 (120591)

= (119870)int

1199051

0

[minus1205831015840([120591 119905])] 119865

1(120591) 119889120591

+ (119870)int

1199052

1199051

[minus1205831015840([120591 119905])] 119865

2(120591) 119889120591

+ (119870)int

119905

1199052

[minus1205831015840([120591 119905])] 119865

3(120591) 119889120591

= (119870)int

119905

0

[minus1205831015840([120591 119905])] 119865 (120591) 119889120591

(71)

For 120583 = 120583119898 we obtain that

(119888) int[0119905]

119865 (120591) 119889120583119898

(120591) = (119870)int

119905

0

1198981015840(119905 minus 120591) 119865 (120591) 119889120591 (72)

Remark 25 Note that Theorem 24 holds only for a continu-ous and increasing 119865 but in the case 120583

119898= ] it holds for any

119865

Remark 26 Let us consider the representation of theChoquetintegral for a continuous case shown in Theorem 24 inrelation with a discrete case Now [0 119905] is transformed intothe discrete set 119909

1 1199092 119909

119899 Since119865 is an increasing fuzzy-

number-valued function we have 119909119894= 1199091015840

119894 119894 = 1 2 119899 in

Theorem 11 For the sake of simplicity let 0 = 1199090lt 1199091lt sdot sdot sdot lt

Abstract and Applied Analysis 9

119909119899

= 119905 119865(1199090) = 0 and [119909

119894 119905] = 119909

119894 119909119894+1

119909119899 It follows

that

(119888) int[0119905]

119865 (120591) 119889120583 (120591)

=

119899

sum

119894=1

[120583 ([119909119894 119905]) minus 120583 ([119909

119894+1 119905])] 119865 (119909

1015840

119894)

=

119899

sum

119894=1

minus [120583 ([119909119894 119905]) minus 120583 ([119909

119894+1 119905])]

Δ119909119894

119865 (1199091015840

119894) Δ119909119894

= (119870)int

119905

0

[minus1205831015840([120591 119905])] 119865 (120591) 119889120591

(73)

5 Choquet Integral Equations ofFuzzy-Number-Valued Functions

In this section let us consider a Choquet integral equationbased onTheorem 24 as shown below Given continuous andincreasing fuzzy-number-valued functions 119866(119905) with 119866(0) =

0 let us find continuous and increasing fuzzy-number-valuedfunctions 119865(119905) such that

119866 (119905) = (119888) int[0119905]

119865 (120591) 119889120583119898

(119905) (74)

which is expressed as

119866 (119905) = (119870)int

119905

0

1198981015840(119905 minus 120591) 119865 (120591) 119889120591 (75)

Definition 27 Let 119891(119905) be a real-valued function defined on[0 +infin) The following

F (119904) = int

+infin

0

119890minus119904119905

119891 (119905) 119889119905 (76)

is said to be the Laplace transformation of 119891(119905) if the infiniteintegralint+infin

0119890minus119904119905

119891(119905)119889119905 is convergentwith respect to the valueof parameter 119904 We denote its Laplace transformation asF(119904) = 119871[119891(119905)] and the inverse Laplace transformation as119891(119905) = 119871

minus1[F(119904)]

Definition 28 Let119865(119909 119910) be a fuzzy-number-valued functiondefined on [0 +infin) times [0 +infin) (119870) int

infin

0119865(119909 119910)119889119909 is said to be

convergent with respect to the parameter 119910 if for every fixed119910 isin [0 +infin) (119870) int

+infin

0119865(119909 119910)119889119909 exists

Definition 29 Let 119865(119905) be a fuzzy-number-valued functiondefined on [0 +infin) The following

F (119904) = (119870)int[0+infin)

119890minus119904119905

119865 (119905) 119889119905 (77)

is said to be the Laplace transformation of 119865(119905) if the Kalevaintegral intinfin

0119890minus119904119905

119865(119905)119889119905 is convergent with respect to the valueof parameter 119904 We denote its Laplace transformation asF(119904) = 119871[119865(119905)] and the inverse Laplace transformation as119865(119905) = 119871

minus1[F(119904)]

Remark 30 In view of Theorem 8 we have

F120582(119904) = [int

+infin

0

119890minus119904119905

119865120582(119905) 119889119905 int

+infin

0

119890minus119904119905

119865120582(119905) 119889119905] (78)

for all 120582 isin [0 1]

Theorem 31 Let 119866(119905) = (119888) int[0119905]

119865(120591)119889120583119898(119905) Then

G (119904) = 119904M (119904) F (119904)

119866 (119904) = 119871minus1

[119904M (119904) F (119904)]

(79)

where G(119904) = 119871[119866(119905)]M(s) = 119871[119898(119905)] and F(119904) = 119871[119865(119905)]

Proof Since 119866(119905) = (119888) int[0119905]

119865(120591)119889120583119898(119905) = (119870) int

119905

01198981015840(119905 minus

120591)119865(120591)119889120591 it follows easily from the condition that

G120582(119904) = [int

+infin

0

119890minus119904119905

119866120582(119905) 119889119905 int

+infin

0

119890minus119904119905

119866120582(119905) 119889119905]

= [int

+infin

0

119890minus119904119905

int

119905

0

1198981015840(119905 minus 120591) 119865

120582(120591) 119889120591 119889119905

int

+infin

0

119890minus119904119905

int

119905

0

1198981015840(119905 minus 120591) 119865

120582(120591) 119889120591 119889119905]

(80)

for all 120582 isin [0 1]Notice that

int

+infin

0

119890minus119904119905

int

119905

0

1198981015840(119905 minus 120591) 119865

120582(120591) 119889120591 119889119905

= int

+infin

0

int

+infin

120591

119890minus119904119905

1198981015840(119905 minus 120591) 119865

120582(120591) 119889119905 119889120591

= int

+infin

0

119865120582(120591) int

+infin

120591

119890minus119904119905

1198981015840(119905 minus 120591) 119889119905 119889120591

= int

+infin

0

119865120582(120591) [119890

minus119904119905119898(119905 minus 120591)

10038161003816100381610038161003816

+infin

120591minus int

+infin

120591

119898(119905 minus 120591) 119889119890minus119904119905

] 119889120591

= int

+infin

0

119865120582(120591) [minusint

+infin

120591

119898(119905 minus 120591) (minus119904) 119890minus119904119905

119889119905] 119889120591

= 119904int

+infin

0

119865120582(120591) [int

+infin

0

119898(119905 minus 120591) (minus119904) 119890minus119904(119905minus120591)

119890minus119904120591

119889 (119905 minus 120591)] 119889120591

= 119904int

+infin

0

119865120582(120591) [119890

minus119904120591int

+infin

0

119890minus119904(119905minus120591)

119898(119905 minus 120591) 119889 (119905 minus 120591)] 119889120591

= 119904int

+infin

0

119865120582(120591) 119890minus119904120591

M (119904) 119889120591

= 119904M (119904) int

+infin

0

119890minus119904120591

119865120582(120591) 119889120591 = 119904M (119904)F

120582(119904)

(81)

Following the same argument we can prove that

int

+infin

0

119890minus119904119905

int

119905

0

1198981015840(119905 minus 120591) 119865

120582(120591) 119889120591 119889119905 = 119904M (119904)F

120582(119904) (82)

10 Abstract and Applied Analysis

Hence

G (119904) = 119904M (119904) F (119904)

119866 (119904) = 119871minus1

[119904M (119904) F (119904)]

(83)

Theorem 32 For continuous and increasing fuzzy-valuedfunctions 119866(119905) with 119866(0) = 0

F (119904) =G (119904)

119904M (119904) 119865 (119904) = 119871

minus[

G (119904)

119904M (119904)] (84)

where G(119904) = 119871[119866(119905)]M(119904) = 119871[119898(119905)] and F(119904) = 119871[119865(119905)]

Proof It follows easily fromTheorem 31 that

G (119904) = 119904M (119904) F (119904)

119866 (119904) = 119871minus1

[119904M (119904) F (119904)]

(85)

It is obvious that

F (119904) =G (119904)

119904M (119904)

119865 (119904) = 119871minus1

[G (119904)

119904M (119904)]

(86)

where G(119904) = 119871[119866(119905)]M(119904) = 119871[119898(119905)] and F(119904) = 119871[119865(119905)]The Dirac delta function is defined by the properties

120575 (119905) = 0 for 119905 = 0

undefined at 119905 = 0(87)

and int+infin

minusinfin120575(119905)119889119905 = 1 that is the function has unit area

The Laplace transform of the Dirac Delta function 120575(119905) isshown as follows

119871 [120575 (119905)] = int

+infin

0

119890minus119904119905

120575 (119905) 119889119905 = 1 (88)

So

119871minus1

[1] = 120575 (119905) (89)

Example 33 Let 119898(119905) = 119905 and let the membership functionof 119866(119905) be

119898119866(119905)

(119909) = 119909 119909 isin [0 1]

0 119909 notin [0 1] 119905 le 1 (90)

Then by Theorem 32 we can solve the Choquet integralequation

119866 (119905) = (119888) int[0119905]

119865 (120591) 119889120583119898

(119905) (91)

Indeed let 119898(119905) = 119905 thenM(119904) = 11199042 The 120582-cut of 119866(119905) is

represented by interval 119866120582(119905) = 119909 | 119898

119866(119905)(119909) ge 120582 = [120582 1]

that is 119866120582(119905) = 120582 119866

120582(119905) = 1 In view of Remark 30 we have

G120582(119904) = [int

+infin

0119890minus119904119905

119866120582(119905)119889119905 int

+infin

0119890minus119904119905

119866120582(119905)119889119905] = [120582119878 1119878]

Then

F120582(119904) =

G120582(119904)

119904M (119904)= 120582

F120582(119904) =

G120582(119904)

119904M (119904)= 1

(92)

Furthermore

119865120582(119905) = 119871

minus1(F120582(119904)) = 120582120575 (119905)

119865120582(119905) = 119871

minus1(F120582(119904)) = 120575 (119905)

(93)

Therefore for every 120582 isin [0 1]

119865120582(119905) = [119865

120582(119905) 119865120582(119905)] = [120582120575 (119905) 120575 (119905)] (94)

Then we have the membership function of 119865(119905) as

119898119865(119905)

(119909) =

119909

120575 (119905) 119909 isin [0 120575 (119905)] 119905 = 0

0 otherwise(95)

6 Conclusions and Remarks

In this paper we have considered the Choquet integrals offuzzy-number-valued functions based on the nonnegativereal lineWe have discussed the Choquet integrals of interval-valued functions and fuzzy-number-valued functions basedon nonadditive Sugeno measures and showed the repre-sentation theorem of them respectively We also gave theoperational schemes of above several classes of integralson discrete sets Then we have given a representation ofthe Choquet integral of a nonnegative continuous andincreasing fuzzy-number-valued function with respect to afuzzymeasure In addition in order to solve Choquet integralequations of fuzzy-number-valued function a concept of theLaplace transformation for calculation has been introducedFor distorted Lebesgue measures it was shown that Choquetintegral equations of fuzzy-number-valued function could besolved by the Laplace transformation Finally we have givenan example to illustrate themain result at the end of the paper

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The work is supported by the Natural Scientific Funds ofChina (61262022) the Natural Scientific Fund of GansuProvince of China (1208RJZA251) Lanzhou Jiaotong Univer-sity Young Scientific Research Fund Project (2011020) theFundamental Research Funds for Gansu Province Univer-sities (no 213060) and the Scientific Research Project ofNorthwestNormalUniversity (noNWNU-KJCXGC-03-61)

Abstract and Applied Analysis 11

References

[1] G Choquet ldquoTheory of capacitiesrdquoAnnales de lrsquoInstitut Fouriervol 5 pp 131ndash295 1954

[2] D Denneberg Non-Additive Measure and Integral KluwerAcademic Publishers Boston Mass USA 1994

[3] T Murofushi M Sugeno and M Machida ldquoNon-monotonicfuzzy measures and the Choquet integralrdquo Fuzzy Sets andSystems vol 64 no 1 pp 73ndash86 1994

[4] Z Y Wang K-S Leung M-L Wong and J Fang ldquoA new typeof nonlinear integrals and the computational algorithmrdquo FuzzySets and Systems vol 112 no 2 pp 223ndash231 2000

[5] D Schmeidler ldquoSubjective probability and expected utilitywithout additivityrdquo Econometrica vol 57 no 3 pp 571ndash5871989

[6] M Grabisch ldquo119896-order additive discrete fuzzy measures andtheir representationrdquo Fuzzy Sets and Systems vol 92 no 2 pp167ndash189 1997

[7] MGrabisch I Kojadinovic and PMeyer ldquoA review ofmethodsfor capacity identification in Choquet integral based multi-attribute utility theory applications of the Kappalab R packagerdquoEuropean Journal of Operational Research vol 186 no 2 pp766ndash785 2008

[8] K Xu Z Wang P Heng and K Leung ldquoClassification bynonlinear integral projectionsrdquo IEEE Transactions on FuzzySystems vol 11 no 2 pp 187ndash201 2003

[9] M J Bolanos L M de Campos Ibanez and A GonzalezMunoz ldquoConvergence properties of the monotone expectationand its application to the extension of fuzzy measuresrdquo FuzzySets and Systems vol 33 no 2 pp 201ndash212 1989

[10] T Murofushi and M Sugeno ldquoA theory of fuzzy measuresrepresentations the Choquet integral and null setsrdquo Journal ofMathematical Analysis and Applications vol 159 no 2 pp 532ndash549 1991

[11] Z Wang ldquoConvergence theorems for sequences of Choquetintegralsrdquo International Journal of General Systems vol 26 no1-2 pp 133ndash143 1997

[12] YOuyang andHZhang ldquoOn the space ofmeasurable functionsand its topology determined by the Choquet integralrdquo Interna-tional Journal of Approximate Reasoning vol 52 no 9 pp 1355ndash1362 2011

[13] J Kawabe ldquoThe Choquet integral representability of comono-tonically additive functionals in locally compact spacesrdquo Inter-national Journal of Approximate Reasoning vol 54 no 3 pp418ndash426 2013

[14] S Abbasbandy E Babolian and M Alavi ldquoNumerical methodfor solving linear Fredholm fuzzy integral equations of thesecond kindrdquo Chaos Solitons amp Fractals vol 31 no 1 pp 138ndash146 2007

[15] O Kaleva ldquoFuzzy differential equationsrdquo Fuzzy Sets and Sys-tems vol 24 no 3 pp 301ndash317 1987

[16] J Mordeson and W Newman ldquoFuzzy integral equationsrdquoInformation Sciences vol 87 no 4 pp 215ndash229 1995

[17] M Sugeno ldquoA note on derivatives of functions with respect tofuzzy measuresrdquo Fuzzy Sets and Systems vol 222 pp 1ndash17 2013

[18] I Kojadinovic J-L Marichal and M Roubens ldquoAn axiomaticapproach to the definition of the entropy of a discrete Choquetcapacityrdquo Information Sciences vol 172 no 1-2 pp 131ndash1532005

[19] E Pap Null-Additive Set Functions vol 337 of Mathematicsand Its Applications KluwerAcademic Publishers LondonUK1994

[20] M Sugeno Theory of fuzzy integrals and its applications [PhDdissertation] Tokyo Institute of Technology Tokyo Japan 1974

[21] C X Wu D L Zhang and C M Guo ldquoFuzzy number fuzzymeasures and fuzzy integralsrdquo Fuzzy Sets and Systems vol 98no 3 pp 355ndash360 1998

[22] C X Wu and M Ma The Foundament of Fuzzy AnalysisNational Defense Press Beijing China 1991 (Chinese)

[23] R Yang Z Y Wang P A Heng and K S Leung ldquoFuzzynumbers and fuzzification of the Choquet integralrdquo Fuzzy Setsand Systems vol 153 no 1 pp 95ndash113 2005

[24] S Seikkala ldquoOn the fuzzy initial value problemrdquo Fuzzy Sets andSystems vol 24 no 3 pp 319ndash330 1987

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 3

(1) 119906 is normal that is there exists 1199090

isin 119877 such that119906(1199090) = 1

(2) 119906 is fuzzy convex that is 119906(120582119909 + (1 minus 120582)119910) ge

min119906(119909) 119906(119910) for any 119909 119910 isin 119877 0 le 120582 le 1(3) 119906 is upper semicontinuous that is 119906(119909

0) ge

lim119896rarrinfin

119906(119909119896) for any 119909

119896isin 119877 (119896 = 0 1 2 ) 119909

119896rarr

1199090

(4) 1199060= 119909 isin 119877 119906(119909) gt 0 is compact

We denote the collection of all fuzzy numbers by 119886 is said to be a nonnegative fuzzy number if supp 119886 =

119909 isin 119877 | 119886(119909) gt 0 sub 119877+We can define the semiorder and the

distance in space [22] Let 119886 isin Then 119886 le if 119886120582

le 119887120582

and 119886120582

le 119887120582for 120582 isin (0 1] 119886 + = 119888 if 119886

120582+ 119887120582

= 119888120582for

120582 isin (0 1] 119863(119886 ) = sup120582isin[01]

119863(119886120582 119887120582) is the distance of 119886

and Let 119886119899sub If 119863(119886

119899 119886) rarr 0 then 119886

119899rarr 119886

119877+ denotes the collection of all nonnegative fuzzy num-bers

Lemma 3 (see [22]) Let 119886 isin Then

(1) 119886120582is a nonempty bounded and closed interval for each

120582 isin [0 1](2) if 0 le 120582

1le 1205822le 1 then 119886

1205822

sub 1198861205821

(3) if 120582119899ge 0 and 120582

119899rarr 120582 (120582 isin (01]) then⋂

infin

119899=1119886120582119899

= 119886120582

Conversely there exists a 119887120582

sub 119877 for each 120582 isin (0 1]which satisfies the conditions (1)ndash(3) then there exists a uniquefuzzy number 119886 isin such that 119887

120582= 119886120582 120582 isin (0 1] and

1198860= ⋃120582isin(01]

119886120582sub 1198870

3 Choquet Integrals of Interval-ValuedFunctions and Fuzzy-Number-ValuedFunctions Based on Nonadditive Measures

In this section we will give the definitions and the characteri-zations of the Choquet integrals of interval-valued functionsand fuzzy-number-valued functions based on the nonaddi-tive Sugeno measure Furthermore the operational schemesof above several classes of integrals on a discrete sets areinvestigated which enable us to calculate Choquet integralsin applications

We first introduce the concept of the Choquet integralsfor the interval-valued functions as follows

Definition 4 (see [23]) An interval-valued function 119865 119883 rarr

119868(119877+) is said to be measurable if both 119865(119909) and 119865(119909) are

measurable functions where 119865(119909) = [119865(119909) 119865(119909)] 119865(119909) is theleft end point of interval 119865(119909) and 119865(119909) is the right end pointof interval 119865(119909)

Interval-valued function 119865 119883 rarr 119868(119877+) is 119862-integrally

bounded if there exists a Choquet integrable function ℎ

119883 rarr 119877+ such that || le ℎ(119905) for every selection isin 119865

We denote 1198750(119883) = 119864 | 119864 sub 119883 and 119864 = 0 Let (119883A 120583)

be a measure space 119860 119861 sub A and let 119865 119860 rarr 1198750(119861) be a

measurable set valued mapping 119891 is said to be a measurable

selection of 119865 if there exists a measurable mapping 119891 119860 rarr

119861 such that 119891(119909) isin 119865(119909) for every 119909 isin 119860

Definition 5 Let (119883A 120583) be a nonadditive measure spaceAssume that 119865 119883 rarr 119868(119877

+) is measurable 119862-integrally

bounded interval-valued function and 119864 isin A 119865 is said tobe 119862-integrable if

(119888) int119864

119865119889120583 = (119888) int119864

119892119889120583 | 119892 isin 119878119865(119909)

(4)

is a closed interval on 119868(119877+) where

119878119865(119909)

= 119892 | 119892 119883 997888rarr 119877+ is a measurable selection of 119865 (119909)

(5)

Theorem 6 Let (119883A 120583) be a nonadditive measure spaceSuppose that 120583 is a fuzzymeasure119864 isin A and119865 is nonnegativemeasurable 119862-integrally bounded interval-valued functionthen 119865 is 119862-integrable on 119864 and

(119888) int119864

119865119889120583 = [(119888) int119864

119865119889120583 (119888) int119864

119865119889120583] (6)

Proof Since 119865 is nonnegative measurable on 119864 119865 and 119865 aremeasurable on119864Thus 119865 and 119865 are twomeasurable selectionsof 119865 On the other hand 119865 is 119862-integrally bounded we have

119865 le ℎ 119865 le ℎ (7)

By the properties of Choquet integral of real valued functionswe know that 119865 and 119865 are 119862-integrable on 119864 Let 119872 isin

(119888) int119864119865119889120583 That is to say there exists a measurable selection

119892 of 119865 such that (119888) int119864119892119889120583 = 119872 We can prove that 119872 isin

[(119888) int119864119865119889120583 (119888) int

119864119865119889120583] Indeed notice that 119865 le 119892 le 119865 and

by the properties of Choquet integral of real-valued functionswe have

(119888) int119864

119865119889120583 le (119888) int119864

119892119889120583 le (119888) int119864

119865119889120583 (8)

This follows that

119872 isin [(119888) int119864

119865119889120583 (119888) int119864

119865119889120583] (9)

Thus

(119888) int119864

119892119889120583 | 119892 isin 119878119865(119909)

sub [(119888) int119864

119865119889120583 (119888) int119864

119865119889120583] (10)

Conversely we can show that

[(119888) int119864

119865119889120583 (119888) int119864

119865119889120583] sub (119888) int119864

119865119889120583 (11)

For measurable selections 119901(119909)119898(119909) isin 119865(119909) we get that

119905119901 (119909) + (1 minus 119905)119898 (119909) isin 119865 (119909) (12)

4 Abstract and Applied Analysis

is measurable selection of 119865 for any 0 le 119905 le 1 Therefore

119905 sdot (119888) int119864

119901 (119909) 119889120583 + (1 minus 119905) sdot (119888) int119864

119898(119909) 119889120583

= (119888) int119864

[119905119901 (119909) + (1 minus 119905)119898 (119909)] 119889120583

isin (119888) int119864

119892119889120583 | 119892 isin 119878119865(119909)

(13)

It implies that (119888) int119864119865119889120583 = (119888) int

119864119892119889120583 | 119892 isin 119878

119865(119909) is a convex

set On the other hand since 119865 and 119865 are two measurableselections of 119865 and

(119888) int119864

119865119889120583 isin (119888) int119864

119865119889120583 (119888) int119864

119865119889120583 isin (119888) int119864

119865119889120583 (14)

we have

[(119888) int119864

119865119889120583 (119888) int119864

119865119889120583] sub (119888) int119864

119892119889120583 | 119892 isin 119878119865(119909)

(15)

Hence

(119888) int119864

119865119889120583 = [(119888) int119864

119865119889119892120583 (119888) int119864

119865119889120583] (16)

That is 119865 is 119862-integrable

From the above theorem we know that interval-valuedfunction 119865 is 119862-integrable on 119864 if (119888) int

119864119865119889120583 and (119888) int

119864119865119889120583

exist and are boundedNext we will introduce the concept of the Choquet

integrals for the Fuzzy-number-valued functions as followsFuzzy-number-valued function 119865 119883 rarr

+ on119883 is saidto be measurable if 119865

120582and 119865

120582are measurable functions with

respect to 119909 isin 119883 for any 120582 isin [0 1]Fuzzy-number-valued function 119865 119883 rarr

+ is said tobe 119862-integrally bounded if there exists a Choquet integrablefunction ℎ 119883 rarr 119877

+ such that || le ℎ(119905) for every selection isin [119865(119905)]

0

Definition 7 Let (119883A 120583) be a nonadditive measure spaceAssume that 119864 isin A 119865 119883 rarr

+ is measurable and 119862-integrally bounded function 119865 is said to be 119865119862-integrable if

[(119888) int119864

119865119889120583]

120582

= (119888) int119864

119892119889120583 | 119892 isin 119878119865120582

0 le 120582 le 1 (17)

determines a unique fuzzy number 119886 isin + which is denoted

by (119888) int119864119865119889120583 = 119886 where 119878

119865120582

= 119892 119883 rarr 119877+ 119892 isin 119865

120582is a

measurable selection of 119865120582

Theorem 8 Let (119883A 120583) be a nonadditive measurable spaceAssume 120583 is a continuous fuzzy measure 119864 isin A 119865 119883 rarr

+

ismeasurable and119862-integrally bounded function then119865 is119865119862-integrable on 119864 if and only if 119865

120582and 119865

120582are 119862-integrable on 119864

and

[(119888) int119864

119865119889120583]

120582

= [(119888) int119864

119865120582119889120583 (119888) int

119864

119865120582119889120583] (18)

for any 120582 isin [0 1]

Proof For the necessity since 119865 is measurable and 119862-integrally bounded function 119865

120582and 119865

120582are measurable for

every 120582 isin [0 1] and there exists a Choquet integrable func-tion ℎ(119909) such that 0 le 119865

120582le ℎ(119909) 0 le 119865

120582le ℎ(119909) and then

(119888) int119864

119865120582119889120583 le (119888) int

119864

119865120582119889120583 le (119888) int

119864

ℎ (119909) 119889120583 lt infin (19)

That is 119865120582 119865120582are 119862-integrable and

[(119888) int119864

119865119889120583]

120582

= [(119888) int119864

119865120582119889120583 (119888) int

119864

119865120582119889120583] (20)

for any 120582 isin [0 1]For the sufficiency let 119865 be a Fuzzy-number-valued func-

tion Note that

[(119888) int119864

119865119889120583]

120582

= [(119888) int119864

119865120582119889120583 (119888) int

119864

119865120582119889120583] (21)

we need only to prove that the interval family

[(119888) int119864

119865119889120583]

120582

= [(119888) int119864

119865120582119889120583 (119888) int

119864

119865120582119889120583] 120582 isin [0 1]

(22)

determines a unique fuzzy number Indeed the interval fam-ily

[(119888) int119864

119865120582119889120583 (119888) int

119864

119865120582119889120583] (23)

satisfies the conditions of Lemma 3

(1) 119865 is a measurable fuzzy-number-valued function foreach 120582 isin [0 1] we have 119865

120582(119909) le 119865

120582(119909) and therefore

(119888) int119864

119865120582119889120583 le (119888) int

119864

119865120582119889120583 (24)

(2) Since 1198651205822

sub 1198651205821

for 0 le 1205821le 1205822le 1 that is

1198651205821

(119909) le 1198651205822

(119909) 1198651205821(119909) ge 119865

1205822(119909) (25)

we have

(119888) int119864

1198651205821

119889120583 le (119888) int119864

1198651205822

119889120583

(119888) int119864

1198651205821

119889120583 ge (119888) int119864

1198651205822

119889120583

[(119888) int119864

1198651205821

119889120583 (119888) int119864

1198651205821

119889120583]

sup [(119888) int119864

1198651205822

119889120583 (119888) int119864

1198651205822

119889120583]

(26)

(3) For each 120582119899uarr 120582 isin (0 1] ⋂infin

119899=1119865120582119899

(119909) = 119865120582(119909) that is

lim119899rarrinfin

119865120582119899

= 119865120582 lim119899rarrinfin

119865120582119899

= 119865120582 It is easy to see

Abstract and Applied Analysis 5

that 1198651205821

le 119865120582119899

le 119865120582119899

le 1198651205821

1198651205821

1198651205821

are integrableand by the continuity of 120583 then

infin

⋂

119899=1

[(119888) int119864

119865120582119899

119889120583 (119888) int119864

119865120582119899

119889120583]

= [(119888) int119864

119865120582119889120583 (119888) int

119864

119865120582119889120583]

(27)

In conclusion there exists a unique fuzzy number 119886 isin +

such that

119886120582= [(119888) int

119864

119865120582119889120583 (119888) int

119864

119865120582119889120583] (28)

Furthermore we get that 119865 is 119865119862-integrable on 119864 and

[(119888) int119864

119865119889120583]

120582

= [(119888) int119864

119865120582119889120583 (119888) int

119864

119865120582119889120583] (29)

In the last part of the section we will investigate theoperational schemes of above several classes of integrals ona discrete set

Let119883 = 1199091 1199092 119909

119899 be a discrete setThenwewill give

a new scheme to calculate the value of the Choquet integral

Theorem 9 (see [8]) Let 119891 be a real-valued function on 119883 =

1199091 1199092 119909

119899 Then Choquet integral of 119891 with respect to a

fuzzy measure 120583 on 119883 is given by

(119888) int119883

119891119889120583 =

119899

sum

119894=1

[119891 (1199091015840

119894) minus 119891 (119909

1015840

119894minus1)] 120583 (119883

1015840

119894) (30)

or equivalently by

(119888) int119883

119891119889120583 =

119899

sum

119894=1

[120583 (1198831015840

119894) minus 120583 (119883

1015840

119894+1)] 119891 (119909

1015840

119894) (31)

where 1199091015840

1 1199091015840

2 119909

1015840

119899is a permutation of 119909

1 1199092 119909

119899such

that 119891(1199091015840

0) le 119891(119909

1015840

1) le sdot sdot sdot le 119891(119909

1015840

119899) 119891(119909

1015840

0) = 0 119883

1015840

119894=

1199091015840

119894 1199091015840

119894+1 119909

1015840

119899 119894 = 1 2 119899 and 119883

1015840

119899+1= 0

Theorem 10 Let 119865 be an interval-valued function on 119883 =

1199091 1199092 119909

119899 Then Choquet integral of 119865 with respect to a

fuzzy measure 120583 on 119883 is given by

(119888) int119883

119865119889120583 =

119899

sum

119894=1

[120583 (1198831015840

119894) minus 120583 (119883

1015840

119894+1)] 119865 (119909

1015840

119894) (32)

where 1199091015840

1 1199091015840

2 119909

1015840

119899is a permutation of 119909

1 1199092 119909

119899such

that 119865(1199091015840

0) le 119865(119909

1015840

1) le sdot sdot sdot le 119865(119909

1015840

119899) 119865(119909

1015840

0) = [0 0] 1198831015840

119894=

1199091015840

119894 1199091015840

119894+1 119909

1015840

119899 119894 = 1 2 119899 and 119883

1015840

119899+1= 0

Proof Since 119865 is an interval-valued function on119883 in view ofTheorem 6 we have

(119888) int119883

119865119889120583 = [(119888) int119883

119865119889120583 (119888) int119883

119865119889120583] (33)

Note that 119865 and 119865 are real-valued function on119883 respectivelyByTheorem 9 we get

(119888) int119883

119865119889120583 = [

119899

sum

119894=1

[120583 (1198831015840

119894) minus 120583 (119883

1015840

119894+1)] 119865 (119909

1015840

119894)

119899

sum

119894=1

[120583 (1198831015840

119894) minus 120583 (119883

1015840

119894+1)] 119865 (119909

1015840

119894)]

=

119899

sum

119894=1

[120583 (1198831015840

119894) minus 120583 (119883

1015840

119894+1)] 119865 (119909

1015840

119894)

(34)

where 1199091015840

1 1199091015840

2 119909

1015840

119899is a permutation of 119909

1 1199092 119909

119899such

that 119865(1199091015840

0) le 119865(119909

1015840

1) le sdot sdot sdot le 119865(119909

1015840

119899) 119865(119909

1015840

0) = [0 0] 1198831015840

119894=

1199091015840

119894 1199091015840

119894+1 119909

1015840

119899 119894 = 1 2 119899 and 119883

1015840

119899+1= 0

Theorem 11 Let119865 be a fuzzy-number-valued function on119883 =

1199091 1199092 119909

119899 Then Choquet integral of 119865 with respect to a

fuzzy measure 120583 on 119883 is given by

(119888) int119883

119865119889120583 =

119899

sum

119894=1

[120583 (1198831015840

119894) minus 120583 (119883

1015840

119894+1)] 119865 (119909

1015840

119894) (35)

where 1199091015840

1 1199091015840

2 119909

1015840

119899is a permutation of 119909

1 1199092 119909

119899such

that 119865(1199091015840

0) le 119865(119909

1015840

1) le sdot sdot sdot le 119865(119909

1015840

119899) 119865(119909

1015840

0) = 0 119883

1015840

119894=

1199091015840

119894 1199091015840

119894+1 119909

1015840

119899 119894 = 1 2 119899 and 119883

1015840

119899+1= 0

Proof Since 119865 is a fuzzy-number-valued on 119883 in view ofTheorem 8 we have

[(119888) int119883

119865119889120583]

120582

= [(119888) int119883

119865120582119889120583 (119888) int

119883

119865120582119889120583] (36)

for any 120582 isin [0 1] By the semiorder in space (ie let 119886 isin

Then 119886 le if 119886120582le 119887120582and 119886

120582le 119887120582) andTheorem 10 there

is a sequence 1199091015840

1 1199091015840

2 119909

1015840

119899on 119883 such that 119865(119909

1015840

0) le 119865(119909

1015840

1) le

sdot sdot sdot le 119865(1199091015840

119899) 119865(119909

1015840

0) = 0 1198831015840

119894= 1199091015840

119894 1199091015840

119894+1 119909

1015840

119899 119894 = 1 2 119899

1198831015840

119899+1= 0 and 119909

1015840

1 1199091015840

2 119909

1015840

119899is a permutation of 119909

1 1199092 119909

119899

Consequently

[(119888) int119883

119865119889120583]

120582

= [

119899

sum

119894=1

[120583 (1198831015840

119894) minus 120583 (119883

1015840

119894+1)] 119865120582(1199091015840

119894)

119899

sum

119894=1

[120583 (1198831015840

119894) minus 120583 (119883

1015840

119894+1)] 119865120582(1199091015840

119894)]

(37)

for any 120582 isin [0 1] Therefore

(119888) int119883

119865119889120583 =

119899

sum

119894=1

[120583 (1198831015840

119894) minus 120583 (119883

1015840

119894+1)] 119865 (119909

1015840

119894) (38)

4 The Representation of Choquet Integral ofFuzzy-Number-Valued Functions

Sugeno has described carefully the representation of Cho-quet integral of real-valued increasing functions and some

6 Abstract and Applied Analysis

important conclusions have been obtained [17] Motivated bythis we will discuss the representation of Choquet integral offuzzy-number-valued functions in this section

Definition 12 Fuzzy-number-valued function 119865 119883 rarr +

is said to be continuous on 119883 if for every 1199090isin 119883 there are

lim119909rarr119909

0

119865120582(119909) = 119865

120582(1199090)

lim119909rarr119909

0

119865120582(119909) = 119865

120582(1199090)

(39)

where 120582 isin [0 1]

Definition 13 Fuzzy-number-valued function 119865 119883 rarr +

is said to be increasing on 119883 if for every 1199091

le 1199092there are

119865120582(1199091) le 119865120582(1199092) and 119865

120582(1199091) le 119865120582(1199092) where 119909

1 1199092isin 119883 and

120582 isin [0 1]LetF+ be a class ofmeasurable nonnegative continuous

and increasing fuzzy-number-valued functionsLet ] be a Lebesgue measure for [119886 119887] sub [0infin) ][119886 119887] =

119887 minus 119886

Definition 14 (see [17]) Let 119898 119877+

rarr 119877+ be a continuous

and increasing function and 119898(0) = 0 A fuzzy measure 120583119898

a distorted Lebesgue measure is defined by 120583119898(sdot) = 119898(](sdot))

Definition 15 (see [24]) A fuzzy-number-valued function 119865

[119886 119887] rarr is differentiable at 119909 isin [119886 119887] if there exist fuzzynumber-valued functions 1198651015840(119909) such that

[1198651015840(119909)]120582= [1198651015840

120582(119909) 119865

1015840

120582(119909)] (40)

for every 120582 isin [0 1] where 1198651015840(119909) is the fuzzy derivative of

119865(119909)Note that 120583

119898is induced from the Lebesguemeasure ] by a

monotone transformation where 120583119898([119886 119887]) = 119898(]([119886 119887])) =

119898(119887 minus 119886) Apparently it loses additivity unless 119898(119905) = 119905 butreserves monotonicity In what follows we assume that 119898(119905)

is differentiableIn this section we consider the calculation of Choquet

integrals Let 120583 be a general fuzzy measure and consider120583([120591 119905]) for a closed interval [120591 119905] then 120583([120591 119905]) is decreasingfor 120591 and increasing for 119905 Throughout the paper we assumethat the functions 119898(119905) 119865(119905) and 119866(119905) are continuouslydifferentiable We also assume that 120583([120591 119905]) is continuouslydifferentiable with respect to 120591 on [0 119905] for every 119905 gt 0 Inaddition we require the regularity condition that 120583(119905) = 0

holds for every 119905 ge 0 We write 1205831015840([120591 119905]) = (120597120597120591)120583([120591 119905])

where we note that 1205831015840([120591 119905]) le 0 for 120591 le 119905 If 120583 = 120583119898then

1205831015840([120591 119905]) = minus119898

1015840(119905 minus 120591) where 119898

1015840(119905) = 119889119898(119905)119889119905 First we

consider a case that 119865(119905) is strictly increasingFuzzy-number-valued function 119865 119883 rarr is said to

be 119871-integrally bounded if there exists a Lebesgue integrablefunction ℎ 119883 rarr 119877

+ such that || le ℎ(119905) for every section isin [119865(119905)]

0

Definition 16 (see [22]) Let fuzzy-number-valued functions119865 [119886 +infin) rarr be measurable and 119871-integrally bounded119865 is called Kaleva-integrable if

[int

+infin

119886

119865(119905)119889119905]

120582

= (119871) int

+infin

119886

119892 (119905) 119889119905 | 119892 (119905) isin 119878[119865(119905)]

120582

0 le 120582 le 1

(41)

determines a unique fuzzy number 119886 isin which is denotedby (119870) int

+infin

119886119865(119905) 119889120583 = 119886 where 119878

[119865(119905)]120582

= 119892(119905) isin [119865(119905)]120582is a

measurable selection

Lemma 17 (see [22]) Let 119865 [119886 +infin) rarr be measurableand C-integrally bounded fuzzy-number-valued function then119865 is Kaleva integrable on [119886 +infin) if and only if 119865

120582(119905) and119865

120582(119905)

are 119871-integrable on [119886 +infin) and

[(119870)int

119905

0

119865(120591)119889120591]

120582

= [(119871) int

119905

0

119865120582(120591) 119889120591 (119871) int

119905

0

119865120582(120591) 119889120591]

(42)

for any 120582 isin [0 1]

Theorem 18 Let 119865(119905) isin F+ Then minus1205831015840([120591 119905])119865(120591) is Kaleva

integrable on [0 119905] and

[(119870)int

119905

0

[minus1205831015840([120591 119905])]119865(120591)119889120591]

120582

= [(119871) int

119905

0

[minus1205831015840([120591 119905])] 119865

120582(120591) 119889120591

(119871) int

119905

0

[minus1205831015840([120591 119905])] 119865

120582(120591) 119889120591]

(43)

Proof Since119865(119905) isin F+ for every120582 isin [0 1]we get that119865120582and

119865120582are measurable nonnegative continuous and increasing

real-valued functions on [0 119905] It follows that 119865120582and 119865

120582are

119871-integrable on [0 119905] 120583([120591 119905]) is 119871-integrable on [0 119905] as itis continuously differentiable with respect to 120591 on [0 119905] forevery 119905 gt 0 In view of Lemma 17 [minus120583

1015840([120591 119905])]119865(120591) is Kaleva

integrable on [0 119905] and

[(119870)int

119905

0

[minus1205831015840([120591 119905])] 119865 (120591) 119889120591]

120582

= [(119871) int

119905

0

[minus1205831015840([120591 119905])] 119865

120582(120591) 119889120591

(119871) int

119905

0

[minus1205831015840([120591 119905])] 119865

120582(120591) 119889120591]

(44)

Lemma 19 (see [17]) Let 119891(119905) be a real strictly increasingfunction Then the Choquet integral of 119891 with respect to 120583 on[0 119905] is represented as

(119888) int[0119905]

119891 (120591) 119889120583 (120591) = minusint

119905

0

1205831015840([120591 119905]) 119891 (120591) 119889120591 (45)

Abstract and Applied Analysis 7

In particular for 120583 = 120583119898

(119888) int[0119905]

119891 (120591) 119889120583119898

(120591) = int

119905

0

1198981015840(119905 minus 120591) 119891 (120591) 119889120591 (46)

Lemma 20 (see [17]) Let 119891(119905) be a constant real-valuedfunction 119891(119905) = 119862 forall119905 isin 119877

+= (0 +infin) Then

(119888) int[0119905]

119891 (120591) 119889120583 (120591) = minusint

119905

0

1205831015840([120591 119905]) 119862119889120591 (47)

In particular for 120583 = 120583119898

(119888) int[0119905]

119891 (120591) 119889120583119898

(120591) = 119862119898 (119905) (48)

Lemma 21 (see [17]) Let 119891(119905) be a measurable nonnegativecontinuous and increasing real-valued function Then theChoquet integral of 119891 with respect to 120583 on [0 119905] is representedas

(119888) int[0119905]

119891 (120591) 119889120583 (120591) = int

infin

0

120583 (120591 | 119891 (120591) ge 119903 cap [0 119905]) 119889119903

= minusint

119905

0

1205831015840([120591 119905]) 119891 (120591) 119889120591

(49)

In particular for 120583 = 120583119898

(119888) int[0119905]

119891 (120591) 119889120583119898

(120591) = int

infin

0

120583 (120591 | 119891 (120591) ge 119903 cap [0 119905]) 119889119903

= int

119905

0

1198981015840(119905 minus 120591) 119891 (120591) 119889120591

(50)

Theorem 22 Let 119865(119905) be a strictly increasing fuzzy-number-valued functionThen the Choquet integral of 119865with respect to120583 on [0 119905] is represented as

(119888) int[0119905]

119865 (120591) 119889120583 (120591) = (119870)int

119905

0

[minus1205831015840([120591 119905])] 119865 (120591) 119889120591 (51)

In particular for 120583 = 120583119898

(119888) int[0119905]

119865 (120591) 119889120583119898

(120591) = (119870)int

119905

0

1198981015840(119905 minus 120591) 119865 (120591) 119889120591 (52)

Proof ByTheorem 8 we have

[(119888) int[0119905]

119865 (120591) 119889120583 (120591)]

120582

= [(119888) int[0119905]

119865120582(120591) 119889120583 (120591) (119888) int

[0119905]

119865120582(120591) 119889120583 (120591)]

(53)

for any 120582 isin [0 1] Since 119865(119905) is strictly increasing 119865120582(119905) and

119865120582(119905) are strictly increasing real-valued functions respec-

tively In view of Lemma 19 we have

(119888) int[0119905]

119865120582(120591) 119889120583 (120591) = minusint

119905

0

1205831015840([120591 119905]) 119865

120582(120591) 119889120591

(119888) int[0119905]

119865120582(120591) 119889120583 (120591) = minusint

119905

0

1205831015840([120591 119905]) 119865

120582(120591) 119889120591

(54)

It follows that

[(119888) int[0119905]

119865(120591)119889120583(120591)]

120582

= [minusint

119905

0

1205831015840([120591 119905]) 119865

120582(120591) 119889120591 minusint

119905

0

1205831015840([120591 119905]) 119865120582 (120591) 119889120591]

(55)

On the other hand

int

119905

0

[minus1205831015840([120591 119905])] 119865120582 (120591) 119889120591

= int

119905

0

[minus1205831015840([120591 119905])] [119865

120582(120591) 119865

120582(120591)] 119889120591

= [minusint

119905

0

1205831015840([120591 119905]) 119865

120582(120591) 119889120591 minusint

119905

0

1205831015840([120591 119905]) 119865120582 (120591) 119889120591]

(56)

Therefore

[(119888) int[0119905]

119865(120591)119889120583(120591)]

120582

= int

119905

0

[minus1205831015840([120591 119905])] 119865120582 (120591) 119889120591 (57)

By the arbitrary of 120582 we have

(119888) int[0119905]

119865 (120591) 119889120583 (120591) = (119870)int

119905

0

[minus1205831015840([120591 119905])] 119865 (120591) 119889120591 (58)

For 120583 = 120583119898 we note that minus120583

1015840([120591 119905]) = 119898

1015840(119905 minus 120591) hence

(119888) int[0119905]

119865 (120591) 119889120583119898

(120591) = (119870)int

119905

0

1198981015840(119905 minus 120591) 119865 (120591) 119889120591 (59)

Theorem 23 Let 119865(119905) be a constant fuzzy-valued functionthat is 119865(119905) = 119862 forall119905 isin 119877

+= (0 +infin) Then

(119888) int[0119905]

119865 (120591) 119889120583 (120591)

= (119870)int

119905

0

[minus1205831015840([120591 119905])] 119862119889120591 = 119862120583 ([0 119905])

(60)

In particular for 120583 = 120583119898

(119888) int[0119905]

119865 (120591) 119889120583119898

(120591) = 119862119898 (119905) (61)

Proof Note that119862 = [119862 119862] where119862 and119862 are constant real-valued functions respectively In view of Lemma 20 we have

(119888) int[0119905]

119862119889120583 (120591) = minusint

119905

0

1205831015840([120591 119905]) 119862119889120591

(119888) int[0119905]

119862119889120583 (120591) = minusint

119905

0

1205831015840([120591 119905]) 119862119889120591

(62)

8 Abstract and Applied Analysis

The rest of the proof follows in exactly the same way as thatof Theorem 22 Hence

(119888) int[0119905]

119865 (120591) 119889120583 (120591)

= (119870)int

119905

0

[minus1205831015840([120591 119905])] 119862119889120591 = 119862120583 ([0 119905])

(63)

For 120583 = 120583119898 we obtain 120583

1015840([120591 119905]) = minus119898

1015840(119905 minus 120591) It follows that

(119888) int[0119905]

119862119889120583 (120591) = (119870)int

119905

0

[minus1205831015840([120591 119905])] 119862119889120591

= int

119905

0

1198981015840(119905 minus 120591) 119862119889120591

= (119870)119862int

119905

0

1198981015840(119905 minus 120591) 119889120591 = 119862119898 (119905)

(64)

where 119898(0) = 0

Theorem 24 Let 119865(119905) isin F+ Then the Choquet integral of 119865with respect to 120583 on [0 119905] is represented as

(119888) int[0119905]

119865 (120591) 119889120583 (120591) = (119870)int

119905

0

[minus1205831015840([120591 119905])] 119865 (120591) 119889120591 (65)

In particular for 120583 = 120583119898

(119888) int[0119905]

119865 (120591) 119889120583119898

(120591) = (119870)int

119905

0

1198981015840(119905 minus 120591) 119865 (120591) 119889120591 (66)

Proof The theorem has been proved when 119865(119905) is a strictlyincreasing or constant fuzzy-number-valued function byTheorem 22 or Theorem 23 Without loss of generality weconsider a continuous and increasing function such that

119865 (119905) =

1198651(119905) 0 le 119905 lt 119905

1

1198652(119905) 119905

1le 119905 lt 119905

2

1198653(119905) 119905

2le 119905 lt infin

(67)

where 1198651(119905) and 119865

3(119905) are strictly increasing 119865

2(119905) is constant

and 1198651(1199051) = 1198652(119905) = 119865

3(1199052)

(i) For 0 le 119905 lt 1199051

Note that 1198651(119905) is strictly increasing on [0 119905

1) from

Theorem 22 we obtain

(119888) int[0119905]

119865 (120591) 119889120583 (120591) = (119870)int

119905

0

[minus1205831015840([120591 119905])] 1198651 (120591) 119889120591 (68)

(ii) For 1199051le 119905 lt 119905

2

In view of the construction of 119865(119905) on [0 119905] we have

(119888) int[0119905]

119865 (120591) 119889120583 (120591) = (119888) int[01199051]

1198651(120591) 119889120583 (120591)

+ (119888) int[1199051119905]

1198652(120591) 119889120583 (120591)

(69)

Since 1198651(119905) is strictly increasing on [0 119905

1) and 119865

2(119905) is

constant on [1199051 1199052) by Theorems 22 and 23

(119888) int[0119905]

119865 (120591) 119889120583 (120591) = (119870)int

1199051

0

[minus1205831015840([120591 119905])] 119865

1(120591) 119889120591

+ (119870)int

119905

1199051

[minus1205831015840([120591 119905])] 119865

2(120591) 119889120591

= (119870)int

119905

0

[minus1205831015840([120591 119905])] 119865 (120591) 119889120591

(70)

(iii) For 1199052le 119905 lt infin

Since1198651(119905) is strictly increasing on [0 119905

1) 1198652(119905) is constant

on [1199051 1199052) and 119865

3(119905) is strictly increasing on [119905

1infin) by

Theorems 22 and 23 we have

(119888) int[0119905]

119865 (120591) 119889120583 (120591)

= (119888) int[01199051]

1198651(120591) 119889120583 (120591) + (119888) int

[11990511199052]

1198652(120591) 119889120583 (120591)

+ (119888) int[1199052119905]

1198653(120591) 119889120583 (120591)

= (119870)int

1199051

0

[minus1205831015840([120591 119905])] 119865

1(120591) 119889120591

+ (119870)int

1199052

1199051

[minus1205831015840([120591 119905])] 119865

2(120591) 119889120591

+ (119870)int

119905

1199052

[minus1205831015840([120591 119905])] 119865

3(120591) 119889120591

= (119870)int

119905

0

[minus1205831015840([120591 119905])] 119865 (120591) 119889120591

(71)

For 120583 = 120583119898 we obtain that

(119888) int[0119905]

119865 (120591) 119889120583119898

(120591) = (119870)int

119905

0

1198981015840(119905 minus 120591) 119865 (120591) 119889120591 (72)

Remark 25 Note that Theorem 24 holds only for a continu-ous and increasing 119865 but in the case 120583

119898= ] it holds for any

119865

Remark 26 Let us consider the representation of theChoquetintegral for a continuous case shown in Theorem 24 inrelation with a discrete case Now [0 119905] is transformed intothe discrete set 119909

1 1199092 119909

119899 Since119865 is an increasing fuzzy-

number-valued function we have 119909119894= 1199091015840

119894 119894 = 1 2 119899 in

Theorem 11 For the sake of simplicity let 0 = 1199090lt 1199091lt sdot sdot sdot lt

Abstract and Applied Analysis 9

119909119899

= 119905 119865(1199090) = 0 and [119909

119894 119905] = 119909

119894 119909119894+1

119909119899 It follows

that

(119888) int[0119905]

119865 (120591) 119889120583 (120591)

=

119899

sum

119894=1

[120583 ([119909119894 119905]) minus 120583 ([119909

119894+1 119905])] 119865 (119909

1015840

119894)

=

119899

sum

119894=1

minus [120583 ([119909119894 119905]) minus 120583 ([119909

119894+1 119905])]

Δ119909119894

119865 (1199091015840

119894) Δ119909119894

= (119870)int

119905

0

[minus1205831015840([120591 119905])] 119865 (120591) 119889120591

(73)

5 Choquet Integral Equations ofFuzzy-Number-Valued Functions

In this section let us consider a Choquet integral equationbased onTheorem 24 as shown below Given continuous andincreasing fuzzy-number-valued functions 119866(119905) with 119866(0) =

0 let us find continuous and increasing fuzzy-number-valuedfunctions 119865(119905) such that

119866 (119905) = (119888) int[0119905]

119865 (120591) 119889120583119898

(119905) (74)

which is expressed as

119866 (119905) = (119870)int

119905

0

1198981015840(119905 minus 120591) 119865 (120591) 119889120591 (75)

Definition 27 Let 119891(119905) be a real-valued function defined on[0 +infin) The following

F (119904) = int

+infin

0

119890minus119904119905

119891 (119905) 119889119905 (76)

is said to be the Laplace transformation of 119891(119905) if the infiniteintegralint+infin

0119890minus119904119905

119891(119905)119889119905 is convergentwith respect to the valueof parameter 119904 We denote its Laplace transformation asF(119904) = 119871[119891(119905)] and the inverse Laplace transformation as119891(119905) = 119871

minus1[F(119904)]

Definition 28 Let119865(119909 119910) be a fuzzy-number-valued functiondefined on [0 +infin) times [0 +infin) (119870) int

infin

0119865(119909 119910)119889119909 is said to be

convergent with respect to the parameter 119910 if for every fixed119910 isin [0 +infin) (119870) int

+infin

0119865(119909 119910)119889119909 exists

Definition 29 Let 119865(119905) be a fuzzy-number-valued functiondefined on [0 +infin) The following

F (119904) = (119870)int[0+infin)

119890minus119904119905

119865 (119905) 119889119905 (77)

is said to be the Laplace transformation of 119865(119905) if the Kalevaintegral intinfin

0119890minus119904119905

119865(119905)119889119905 is convergent with respect to the valueof parameter 119904 We denote its Laplace transformation asF(119904) = 119871[119865(119905)] and the inverse Laplace transformation as119865(119905) = 119871

minus1[F(119904)]

Remark 30 In view of Theorem 8 we have

F120582(119904) = [int

+infin

0

119890minus119904119905

119865120582(119905) 119889119905 int

+infin

0

119890minus119904119905

119865120582(119905) 119889119905] (78)

for all 120582 isin [0 1]

Theorem 31 Let 119866(119905) = (119888) int[0119905]

119865(120591)119889120583119898(119905) Then

G (119904) = 119904M (119904) F (119904)

119866 (119904) = 119871minus1

[119904M (119904) F (119904)]

(79)

where G(119904) = 119871[119866(119905)]M(s) = 119871[119898(119905)] and F(119904) = 119871[119865(119905)]

Proof Since 119866(119905) = (119888) int[0119905]

119865(120591)119889120583119898(119905) = (119870) int

119905

01198981015840(119905 minus

120591)119865(120591)119889120591 it follows easily from the condition that

G120582(119904) = [int

+infin

0

119890minus119904119905

119866120582(119905) 119889119905 int

+infin

0

119890minus119904119905

119866120582(119905) 119889119905]

= [int

+infin

0

119890minus119904119905

int

119905

0

1198981015840(119905 minus 120591) 119865

120582(120591) 119889120591 119889119905

int

+infin

0

119890minus119904119905

int

119905

0

1198981015840(119905 minus 120591) 119865

120582(120591) 119889120591 119889119905]

(80)

for all 120582 isin [0 1]Notice that

int

+infin

0

119890minus119904119905

int

119905

0

1198981015840(119905 minus 120591) 119865

120582(120591) 119889120591 119889119905

= int

+infin

0

int

+infin

120591

119890minus119904119905

1198981015840(119905 minus 120591) 119865

120582(120591) 119889119905 119889120591

= int

+infin

0

119865120582(120591) int

+infin

120591

119890minus119904119905

1198981015840(119905 minus 120591) 119889119905 119889120591

= int

+infin

0

119865120582(120591) [119890

minus119904119905119898(119905 minus 120591)

10038161003816100381610038161003816

+infin

120591minus int

+infin

120591

119898(119905 minus 120591) 119889119890minus119904119905

] 119889120591

= int

+infin

0

119865120582(120591) [minusint

+infin

120591

119898(119905 minus 120591) (minus119904) 119890minus119904119905

119889119905] 119889120591

= 119904int

+infin

0

119865120582(120591) [int

+infin

0

119898(119905 minus 120591) (minus119904) 119890minus119904(119905minus120591)

119890minus119904120591

119889 (119905 minus 120591)] 119889120591

= 119904int

+infin

0

119865120582(120591) [119890

minus119904120591int

+infin

0

119890minus119904(119905minus120591)

119898(119905 minus 120591) 119889 (119905 minus 120591)] 119889120591

= 119904int

+infin

0

119865120582(120591) 119890minus119904120591

M (119904) 119889120591

= 119904M (119904) int

+infin

0

119890minus119904120591

119865120582(120591) 119889120591 = 119904M (119904)F

120582(119904)

(81)

Following the same argument we can prove that

int

+infin

0

119890minus119904119905

int

119905

0

1198981015840(119905 minus 120591) 119865

120582(120591) 119889120591 119889119905 = 119904M (119904)F

120582(119904) (82)

10 Abstract and Applied Analysis

Hence

G (119904) = 119904M (119904) F (119904)

119866 (119904) = 119871minus1

[119904M (119904) F (119904)]

(83)

Theorem 32 For continuous and increasing fuzzy-valuedfunctions 119866(119905) with 119866(0) = 0

F (119904) =G (119904)

119904M (119904) 119865 (119904) = 119871

minus[

G (119904)

119904M (119904)] (84)

where G(119904) = 119871[119866(119905)]M(119904) = 119871[119898(119905)] and F(119904) = 119871[119865(119905)]

Proof It follows easily fromTheorem 31 that

G (119904) = 119904M (119904) F (119904)

119866 (119904) = 119871minus1

[119904M (119904) F (119904)]

(85)

It is obvious that

F (119904) =G (119904)

119904M (119904)

119865 (119904) = 119871minus1

[G (119904)

119904M (119904)]

(86)

where G(119904) = 119871[119866(119905)]M(119904) = 119871[119898(119905)] and F(119904) = 119871[119865(119905)]The Dirac delta function is defined by the properties

120575 (119905) = 0 for 119905 = 0

undefined at 119905 = 0(87)

and int+infin

minusinfin120575(119905)119889119905 = 1 that is the function has unit area

The Laplace transform of the Dirac Delta function 120575(119905) isshown as follows

119871 [120575 (119905)] = int

+infin

0

119890minus119904119905

120575 (119905) 119889119905 = 1 (88)

So

119871minus1

[1] = 120575 (119905) (89)

Example 33 Let 119898(119905) = 119905 and let the membership functionof 119866(119905) be

119898119866(119905)

(119909) = 119909 119909 isin [0 1]

0 119909 notin [0 1] 119905 le 1 (90)

Then by Theorem 32 we can solve the Choquet integralequation

119866 (119905) = (119888) int[0119905]

119865 (120591) 119889120583119898

(119905) (91)

Indeed let 119898(119905) = 119905 thenM(119904) = 11199042 The 120582-cut of 119866(119905) is

represented by interval 119866120582(119905) = 119909 | 119898

119866(119905)(119909) ge 120582 = [120582 1]

that is 119866120582(119905) = 120582 119866

120582(119905) = 1 In view of Remark 30 we have

G120582(119904) = [int

+infin

0119890minus119904119905

119866120582(119905)119889119905 int

+infin

0119890minus119904119905

119866120582(119905)119889119905] = [120582119878 1119878]

Then

F120582(119904) =

G120582(119904)

119904M (119904)= 120582

F120582(119904) =

G120582(119904)

119904M (119904)= 1

(92)

Furthermore

119865120582(119905) = 119871

minus1(F120582(119904)) = 120582120575 (119905)

119865120582(119905) = 119871

minus1(F120582(119904)) = 120575 (119905)

(93)

Therefore for every 120582 isin [0 1]

119865120582(119905) = [119865

120582(119905) 119865120582(119905)] = [120582120575 (119905) 120575 (119905)] (94)

Then we have the membership function of 119865(119905) as

119898119865(119905)

(119909) =

119909

120575 (119905) 119909 isin [0 120575 (119905)] 119905 = 0

0 otherwise(95)

6 Conclusions and Remarks

In this paper we have considered the Choquet integrals offuzzy-number-valued functions based on the nonnegativereal lineWe have discussed the Choquet integrals of interval-valued functions and fuzzy-number-valued functions basedon nonadditive Sugeno measures and showed the repre-sentation theorem of them respectively We also gave theoperational schemes of above several classes of integralson discrete sets Then we have given a representation ofthe Choquet integral of a nonnegative continuous andincreasing fuzzy-number-valued function with respect to afuzzymeasure In addition in order to solve Choquet integralequations of fuzzy-number-valued function a concept of theLaplace transformation for calculation has been introducedFor distorted Lebesgue measures it was shown that Choquetintegral equations of fuzzy-number-valued function could besolved by the Laplace transformation Finally we have givenan example to illustrate themain result at the end of the paper

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The work is supported by the Natural Scientific Funds ofChina (61262022) the Natural Scientific Fund of GansuProvince of China (1208RJZA251) Lanzhou Jiaotong Univer-sity Young Scientific Research Fund Project (2011020) theFundamental Research Funds for Gansu Province Univer-sities (no 213060) and the Scientific Research Project ofNorthwestNormalUniversity (noNWNU-KJCXGC-03-61)

Abstract and Applied Analysis 11

References

[1] G Choquet ldquoTheory of capacitiesrdquoAnnales de lrsquoInstitut Fouriervol 5 pp 131ndash295 1954

[2] D Denneberg Non-Additive Measure and Integral KluwerAcademic Publishers Boston Mass USA 1994

[3] T Murofushi M Sugeno and M Machida ldquoNon-monotonicfuzzy measures and the Choquet integralrdquo Fuzzy Sets andSystems vol 64 no 1 pp 73ndash86 1994

[4] Z Y Wang K-S Leung M-L Wong and J Fang ldquoA new typeof nonlinear integrals and the computational algorithmrdquo FuzzySets and Systems vol 112 no 2 pp 223ndash231 2000

[5] D Schmeidler ldquoSubjective probability and expected utilitywithout additivityrdquo Econometrica vol 57 no 3 pp 571ndash5871989

[6] M Grabisch ldquo119896-order additive discrete fuzzy measures andtheir representationrdquo Fuzzy Sets and Systems vol 92 no 2 pp167ndash189 1997

[7] MGrabisch I Kojadinovic and PMeyer ldquoA review ofmethodsfor capacity identification in Choquet integral based multi-attribute utility theory applications of the Kappalab R packagerdquoEuropean Journal of Operational Research vol 186 no 2 pp766ndash785 2008

[8] K Xu Z Wang P Heng and K Leung ldquoClassification bynonlinear integral projectionsrdquo IEEE Transactions on FuzzySystems vol 11 no 2 pp 187ndash201 2003

[9] M J Bolanos L M de Campos Ibanez and A GonzalezMunoz ldquoConvergence properties of the monotone expectationand its application to the extension of fuzzy measuresrdquo FuzzySets and Systems vol 33 no 2 pp 201ndash212 1989

[10] T Murofushi and M Sugeno ldquoA theory of fuzzy measuresrepresentations the Choquet integral and null setsrdquo Journal ofMathematical Analysis and Applications vol 159 no 2 pp 532ndash549 1991

[11] Z Wang ldquoConvergence theorems for sequences of Choquetintegralsrdquo International Journal of General Systems vol 26 no1-2 pp 133ndash143 1997

[12] YOuyang andHZhang ldquoOn the space ofmeasurable functionsand its topology determined by the Choquet integralrdquo Interna-tional Journal of Approximate Reasoning vol 52 no 9 pp 1355ndash1362 2011

[13] J Kawabe ldquoThe Choquet integral representability of comono-tonically additive functionals in locally compact spacesrdquo Inter-national Journal of Approximate Reasoning vol 54 no 3 pp418ndash426 2013

[14] S Abbasbandy E Babolian and M Alavi ldquoNumerical methodfor solving linear Fredholm fuzzy integral equations of thesecond kindrdquo Chaos Solitons amp Fractals vol 31 no 1 pp 138ndash146 2007

[15] O Kaleva ldquoFuzzy differential equationsrdquo Fuzzy Sets and Sys-tems vol 24 no 3 pp 301ndash317 1987

[16] J Mordeson and W Newman ldquoFuzzy integral equationsrdquoInformation Sciences vol 87 no 4 pp 215ndash229 1995

[17] M Sugeno ldquoA note on derivatives of functions with respect tofuzzy measuresrdquo Fuzzy Sets and Systems vol 222 pp 1ndash17 2013

[18] I Kojadinovic J-L Marichal and M Roubens ldquoAn axiomaticapproach to the definition of the entropy of a discrete Choquetcapacityrdquo Information Sciences vol 172 no 1-2 pp 131ndash1532005

[19] E Pap Null-Additive Set Functions vol 337 of Mathematicsand Its Applications KluwerAcademic Publishers LondonUK1994

[20] M Sugeno Theory of fuzzy integrals and its applications [PhDdissertation] Tokyo Institute of Technology Tokyo Japan 1974

[21] C X Wu D L Zhang and C M Guo ldquoFuzzy number fuzzymeasures and fuzzy integralsrdquo Fuzzy Sets and Systems vol 98no 3 pp 355ndash360 1998

[22] C X Wu and M Ma The Foundament of Fuzzy AnalysisNational Defense Press Beijing China 1991 (Chinese)

[23] R Yang Z Y Wang P A Heng and K S Leung ldquoFuzzynumbers and fuzzification of the Choquet integralrdquo Fuzzy Setsand Systems vol 153 no 1 pp 95ndash113 2005

[24] S Seikkala ldquoOn the fuzzy initial value problemrdquo Fuzzy Sets andSystems vol 24 no 3 pp 319ndash330 1987

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Abstract and Applied Analysis

is measurable selection of 119865 for any 0 le 119905 le 1 Therefore

119905 sdot (119888) int119864

119901 (119909) 119889120583 + (1 minus 119905) sdot (119888) int119864

119898(119909) 119889120583

= (119888) int119864

[119905119901 (119909) + (1 minus 119905)119898 (119909)] 119889120583

isin (119888) int119864

119892119889120583 | 119892 isin 119878119865(119909)

(13)

It implies that (119888) int119864119865119889120583 = (119888) int

119864119892119889120583 | 119892 isin 119878

119865(119909) is a convex

set On the other hand since 119865 and 119865 are two measurableselections of 119865 and

(119888) int119864

119865119889120583 isin (119888) int119864

119865119889120583 (119888) int119864

119865119889120583 isin (119888) int119864

119865119889120583 (14)

we have

[(119888) int119864

119865119889120583 (119888) int119864

119865119889120583] sub (119888) int119864

119892119889120583 | 119892 isin 119878119865(119909)

(15)

Hence

(119888) int119864

119865119889120583 = [(119888) int119864

119865119889119892120583 (119888) int119864

119865119889120583] (16)

That is 119865 is 119862-integrable

From the above theorem we know that interval-valuedfunction 119865 is 119862-integrable on 119864 if (119888) int

119864119865119889120583 and (119888) int

119864119865119889120583

exist and are boundedNext we will introduce the concept of the Choquet

integrals for the Fuzzy-number-valued functions as followsFuzzy-number-valued function 119865 119883 rarr

+ on119883 is saidto be measurable if 119865

120582and 119865

120582are measurable functions with

respect to 119909 isin 119883 for any 120582 isin [0 1]Fuzzy-number-valued function 119865 119883 rarr

+ is said tobe 119862-integrally bounded if there exists a Choquet integrablefunction ℎ 119883 rarr 119877

+ such that || le ℎ(119905) for every selection isin [119865(119905)]

0

Definition 7 Let (119883A 120583) be a nonadditive measure spaceAssume that 119864 isin A 119865 119883 rarr

+ is measurable and 119862-integrally bounded function 119865 is said to be 119865119862-integrable if

[(119888) int119864

119865119889120583]

120582

= (119888) int119864

119892119889120583 | 119892 isin 119878119865120582

0 le 120582 le 1 (17)

determines a unique fuzzy number 119886 isin + which is denoted

by (119888) int119864119865119889120583 = 119886 where 119878

119865120582

= 119892 119883 rarr 119877+ 119892 isin 119865

120582is a

measurable selection of 119865120582

Theorem 8 Let (119883A 120583) be a nonadditive measurable spaceAssume 120583 is a continuous fuzzy measure 119864 isin A 119865 119883 rarr

+

ismeasurable and119862-integrally bounded function then119865 is119865119862-integrable on 119864 if and only if 119865

120582and 119865

120582are 119862-integrable on 119864

and

[(119888) int119864

119865119889120583]

120582

= [(119888) int119864

119865120582119889120583 (119888) int

119864

119865120582119889120583] (18)

for any 120582 isin [0 1]

Proof For the necessity since 119865 is measurable and 119862-integrally bounded function 119865

120582and 119865

120582are measurable for

every 120582 isin [0 1] and there exists a Choquet integrable func-tion ℎ(119909) such that 0 le 119865

120582le ℎ(119909) 0 le 119865

120582le ℎ(119909) and then

(119888) int119864

119865120582119889120583 le (119888) int

119864

119865120582119889120583 le (119888) int

119864

ℎ (119909) 119889120583 lt infin (19)

That is 119865120582 119865120582are 119862-integrable and

[(119888) int119864

119865119889120583]

120582

= [(119888) int119864

119865120582119889120583 (119888) int

119864

119865120582119889120583] (20)

for any 120582 isin [0 1]For the sufficiency let 119865 be a Fuzzy-number-valued func-

tion Note that

[(119888) int119864

119865119889120583]

120582

= [(119888) int119864

119865120582119889120583 (119888) int

119864

119865120582119889120583] (21)

we need only to prove that the interval family

[(119888) int119864

119865119889120583]

120582

= [(119888) int119864

119865120582119889120583 (119888) int

119864

119865120582119889120583] 120582 isin [0 1]

(22)

determines a unique fuzzy number Indeed the interval fam-ily

[(119888) int119864

119865120582119889120583 (119888) int

119864

119865120582119889120583] (23)

satisfies the conditions of Lemma 3

(1) 119865 is a measurable fuzzy-number-valued function foreach 120582 isin [0 1] we have 119865

120582(119909) le 119865

120582(119909) and therefore

(119888) int119864

119865120582119889120583 le (119888) int

119864

119865120582119889120583 (24)

(2) Since 1198651205822

sub 1198651205821

for 0 le 1205821le 1205822le 1 that is

1198651205821

(119909) le 1198651205822

(119909) 1198651205821(119909) ge 119865

1205822(119909) (25)

we have

(119888) int119864

1198651205821

119889120583 le (119888) int119864

1198651205822

119889120583

(119888) int119864

1198651205821

119889120583 ge (119888) int119864

1198651205822

119889120583

[(119888) int119864

1198651205821

119889120583 (119888) int119864

1198651205821

119889120583]

sup [(119888) int119864

1198651205822

119889120583 (119888) int119864

1198651205822

119889120583]

(26)

(3) For each 120582119899uarr 120582 isin (0 1] ⋂infin

119899=1119865120582119899

(119909) = 119865120582(119909) that is

lim119899rarrinfin

119865120582119899

= 119865120582 lim119899rarrinfin

119865120582119899

= 119865120582 It is easy to see

Abstract and Applied Analysis 5

that 1198651205821

le 119865120582119899

le 119865120582119899

le 1198651205821

1198651205821

1198651205821

are integrableand by the continuity of 120583 then

infin

⋂

119899=1

[(119888) int119864

119865120582119899

119889120583 (119888) int119864

119865120582119899

119889120583]

= [(119888) int119864

119865120582119889120583 (119888) int

119864

119865120582119889120583]

(27)

In conclusion there exists a unique fuzzy number 119886 isin +

such that

119886120582= [(119888) int

119864

119865120582119889120583 (119888) int

119864

119865120582119889120583] (28)

Furthermore we get that 119865 is 119865119862-integrable on 119864 and

[(119888) int119864

119865119889120583]

120582

= [(119888) int119864

119865120582119889120583 (119888) int

119864

119865120582119889120583] (29)

In the last part of the section we will investigate theoperational schemes of above several classes of integrals ona discrete set

Let119883 = 1199091 1199092 119909

119899 be a discrete setThenwewill give

a new scheme to calculate the value of the Choquet integral

Theorem 9 (see [8]) Let 119891 be a real-valued function on 119883 =

1199091 1199092 119909

119899 Then Choquet integral of 119891 with respect to a

fuzzy measure 120583 on 119883 is given by

(119888) int119883

119891119889120583 =

119899

sum

119894=1

[119891 (1199091015840

119894) minus 119891 (119909

1015840

119894minus1)] 120583 (119883

1015840

119894) (30)

or equivalently by

(119888) int119883

119891119889120583 =

119899

sum

119894=1

[120583 (1198831015840

119894) minus 120583 (119883

1015840

119894+1)] 119891 (119909

1015840

119894) (31)

where 1199091015840

1 1199091015840

2 119909

1015840

119899is a permutation of 119909

1 1199092 119909

119899such

that 119891(1199091015840

0) le 119891(119909

1015840

1) le sdot sdot sdot le 119891(119909

1015840

119899) 119891(119909

1015840

0) = 0 119883

1015840

119894=

1199091015840

119894 1199091015840

119894+1 119909

1015840

119899 119894 = 1 2 119899 and 119883

1015840

119899+1= 0

Theorem 10 Let 119865 be an interval-valued function on 119883 =

1199091 1199092 119909

119899 Then Choquet integral of 119865 with respect to a

fuzzy measure 120583 on 119883 is given by

(119888) int119883

119865119889120583 =

119899

sum

119894=1

[120583 (1198831015840

119894) minus 120583 (119883

1015840

119894+1)] 119865 (119909

1015840

119894) (32)

where 1199091015840

1 1199091015840

2 119909

1015840

119899is a permutation of 119909

1 1199092 119909

119899such

that 119865(1199091015840

0) le 119865(119909

1015840

1) le sdot sdot sdot le 119865(119909

1015840

119899) 119865(119909

1015840

0) = [0 0] 1198831015840

119894=

1199091015840

119894 1199091015840

119894+1 119909

1015840

119899 119894 = 1 2 119899 and 119883

1015840

119899+1= 0

Proof Since 119865 is an interval-valued function on119883 in view ofTheorem 6 we have

(119888) int119883

119865119889120583 = [(119888) int119883

119865119889120583 (119888) int119883

119865119889120583] (33)

Note that 119865 and 119865 are real-valued function on119883 respectivelyByTheorem 9 we get

(119888) int119883

119865119889120583 = [

119899

sum

119894=1

[120583 (1198831015840

119894) minus 120583 (119883

1015840

119894+1)] 119865 (119909

1015840

119894)

119899

sum

119894=1

[120583 (1198831015840

119894) minus 120583 (119883

1015840

119894+1)] 119865 (119909

1015840

119894)]

=

119899

sum

119894=1

[120583 (1198831015840

119894) minus 120583 (119883

1015840

119894+1)] 119865 (119909

1015840

119894)

(34)

where 1199091015840

1 1199091015840

2 119909

1015840

119899is a permutation of 119909

1 1199092 119909

119899such

that 119865(1199091015840

0) le 119865(119909

1015840

1) le sdot sdot sdot le 119865(119909

1015840

119899) 119865(119909

1015840

0) = [0 0] 1198831015840

119894=

1199091015840

119894 1199091015840

119894+1 119909

1015840

119899 119894 = 1 2 119899 and 119883

1015840

119899+1= 0

Theorem 11 Let119865 be a fuzzy-number-valued function on119883 =

1199091 1199092 119909

119899 Then Choquet integral of 119865 with respect to a

fuzzy measure 120583 on 119883 is given by

(119888) int119883

119865119889120583 =

119899

sum

119894=1

[120583 (1198831015840

119894) minus 120583 (119883

1015840

119894+1)] 119865 (119909

1015840

119894) (35)

where 1199091015840

1 1199091015840

2 119909

1015840

119899is a permutation of 119909

1 1199092 119909

119899such

that 119865(1199091015840

0) le 119865(119909

1015840

1) le sdot sdot sdot le 119865(119909

1015840

119899) 119865(119909

1015840

0) = 0 119883

1015840

119894=

1199091015840

119894 1199091015840

119894+1 119909

1015840

119899 119894 = 1 2 119899 and 119883

1015840

119899+1= 0

Proof Since 119865 is a fuzzy-number-valued on 119883 in view ofTheorem 8 we have

[(119888) int119883

119865119889120583]

120582

= [(119888) int119883

119865120582119889120583 (119888) int

119883

119865120582119889120583] (36)

for any 120582 isin [0 1] By the semiorder in space (ie let 119886 isin

Then 119886 le if 119886120582le 119887120582and 119886

120582le 119887120582) andTheorem 10 there

is a sequence 1199091015840

1 1199091015840

2 119909

1015840

119899on 119883 such that 119865(119909

1015840

0) le 119865(119909

1015840

1) le

sdot sdot sdot le 119865(1199091015840

119899) 119865(119909

1015840

0) = 0 1198831015840

119894= 1199091015840

119894 1199091015840

119894+1 119909

1015840

119899 119894 = 1 2 119899

1198831015840

119899+1= 0 and 119909

1015840

1 1199091015840

2 119909

1015840

119899is a permutation of 119909

1 1199092 119909

119899

Consequently

[(119888) int119883

119865119889120583]

120582

= [

119899

sum

119894=1

[120583 (1198831015840

119894) minus 120583 (119883

1015840

119894+1)] 119865120582(1199091015840

119894)

119899

sum

119894=1

[120583 (1198831015840

119894) minus 120583 (119883

1015840

119894+1)] 119865120582(1199091015840

119894)]

(37)

for any 120582 isin [0 1] Therefore

(119888) int119883

119865119889120583 =

119899

sum

119894=1

[120583 (1198831015840

119894) minus 120583 (119883

1015840

119894+1)] 119865 (119909

1015840

119894) (38)

4 The Representation of Choquet Integral ofFuzzy-Number-Valued Functions

Sugeno has described carefully the representation of Cho-quet integral of real-valued increasing functions and some

6 Abstract and Applied Analysis

important conclusions have been obtained [17] Motivated bythis we will discuss the representation of Choquet integral offuzzy-number-valued functions in this section

Definition 12 Fuzzy-number-valued function 119865 119883 rarr +

is said to be continuous on 119883 if for every 1199090isin 119883 there are

lim119909rarr119909

0

119865120582(119909) = 119865

120582(1199090)

lim119909rarr119909

0

119865120582(119909) = 119865

120582(1199090)

(39)

where 120582 isin [0 1]

Definition 13 Fuzzy-number-valued function 119865 119883 rarr +

is said to be increasing on 119883 if for every 1199091

le 1199092there are

119865120582(1199091) le 119865120582(1199092) and 119865

120582(1199091) le 119865120582(1199092) where 119909

1 1199092isin 119883 and

120582 isin [0 1]LetF+ be a class ofmeasurable nonnegative continuous

and increasing fuzzy-number-valued functionsLet ] be a Lebesgue measure for [119886 119887] sub [0infin) ][119886 119887] =

119887 minus 119886

Definition 14 (see [17]) Let 119898 119877+

rarr 119877+ be a continuous

and increasing function and 119898(0) = 0 A fuzzy measure 120583119898

a distorted Lebesgue measure is defined by 120583119898(sdot) = 119898(](sdot))

Definition 15 (see [24]) A fuzzy-number-valued function 119865

[119886 119887] rarr is differentiable at 119909 isin [119886 119887] if there exist fuzzynumber-valued functions 1198651015840(119909) such that

[1198651015840(119909)]120582= [1198651015840

120582(119909) 119865

1015840

120582(119909)] (40)

for every 120582 isin [0 1] where 1198651015840(119909) is the fuzzy derivative of

119865(119909)Note that 120583

119898is induced from the Lebesguemeasure ] by a

monotone transformation where 120583119898([119886 119887]) = 119898(]([119886 119887])) =

119898(119887 minus 119886) Apparently it loses additivity unless 119898(119905) = 119905 butreserves monotonicity In what follows we assume that 119898(119905)

is differentiableIn this section we consider the calculation of Choquet

integrals Let 120583 be a general fuzzy measure and consider120583([120591 119905]) for a closed interval [120591 119905] then 120583([120591 119905]) is decreasingfor 120591 and increasing for 119905 Throughout the paper we assumethat the functions 119898(119905) 119865(119905) and 119866(119905) are continuouslydifferentiable We also assume that 120583([120591 119905]) is continuouslydifferentiable with respect to 120591 on [0 119905] for every 119905 gt 0 Inaddition we require the regularity condition that 120583(119905) = 0

holds for every 119905 ge 0 We write 1205831015840([120591 119905]) = (120597120597120591)120583([120591 119905])

where we note that 1205831015840([120591 119905]) le 0 for 120591 le 119905 If 120583 = 120583119898then

1205831015840([120591 119905]) = minus119898

1015840(119905 minus 120591) where 119898

1015840(119905) = 119889119898(119905)119889119905 First we

consider a case that 119865(119905) is strictly increasingFuzzy-number-valued function 119865 119883 rarr is said to

be 119871-integrally bounded if there exists a Lebesgue integrablefunction ℎ 119883 rarr 119877

+ such that || le ℎ(119905) for every section isin [119865(119905)]

0

Definition 16 (see [22]) Let fuzzy-number-valued functions119865 [119886 +infin) rarr be measurable and 119871-integrally bounded119865 is called Kaleva-integrable if

[int

+infin

119886

119865(119905)119889119905]

120582

= (119871) int

+infin

119886

119892 (119905) 119889119905 | 119892 (119905) isin 119878[119865(119905)]

120582

0 le 120582 le 1

(41)

determines a unique fuzzy number 119886 isin which is denotedby (119870) int

+infin

119886119865(119905) 119889120583 = 119886 where 119878

[119865(119905)]120582

= 119892(119905) isin [119865(119905)]120582is a

measurable selection

Lemma 17 (see [22]) Let 119865 [119886 +infin) rarr be measurableand C-integrally bounded fuzzy-number-valued function then119865 is Kaleva integrable on [119886 +infin) if and only if 119865

120582(119905) and119865

120582(119905)

are 119871-integrable on [119886 +infin) and

[(119870)int

119905

0

119865(120591)119889120591]

120582

= [(119871) int

119905

0

119865120582(120591) 119889120591 (119871) int

119905

0

119865120582(120591) 119889120591]

(42)

for any 120582 isin [0 1]

Theorem 18 Let 119865(119905) isin F+ Then minus1205831015840([120591 119905])119865(120591) is Kaleva

integrable on [0 119905] and

[(119870)int

119905

0

[minus1205831015840([120591 119905])]119865(120591)119889120591]

120582

= [(119871) int

119905

0

[minus1205831015840([120591 119905])] 119865

120582(120591) 119889120591

(119871) int

119905

0

[minus1205831015840([120591 119905])] 119865

120582(120591) 119889120591]

(43)

Proof Since119865(119905) isin F+ for every120582 isin [0 1]we get that119865120582and

119865120582are measurable nonnegative continuous and increasing

real-valued functions on [0 119905] It follows that 119865120582and 119865

120582are

119871-integrable on [0 119905] 120583([120591 119905]) is 119871-integrable on [0 119905] as itis continuously differentiable with respect to 120591 on [0 119905] forevery 119905 gt 0 In view of Lemma 17 [minus120583

1015840([120591 119905])]119865(120591) is Kaleva

integrable on [0 119905] and

[(119870)int

119905

0

[minus1205831015840([120591 119905])] 119865 (120591) 119889120591]

120582

= [(119871) int

119905

0

[minus1205831015840([120591 119905])] 119865

120582(120591) 119889120591

(119871) int

119905

0

[minus1205831015840([120591 119905])] 119865

120582(120591) 119889120591]

(44)

Lemma 19 (see [17]) Let 119891(119905) be a real strictly increasingfunction Then the Choquet integral of 119891 with respect to 120583 on[0 119905] is represented as

(119888) int[0119905]

119891 (120591) 119889120583 (120591) = minusint

119905

0

1205831015840([120591 119905]) 119891 (120591) 119889120591 (45)

Abstract and Applied Analysis 7

In particular for 120583 = 120583119898

(119888) int[0119905]

119891 (120591) 119889120583119898

(120591) = int

119905

0

1198981015840(119905 minus 120591) 119891 (120591) 119889120591 (46)

Lemma 20 (see [17]) Let 119891(119905) be a constant real-valuedfunction 119891(119905) = 119862 forall119905 isin 119877

+= (0 +infin) Then

(119888) int[0119905]

119891 (120591) 119889120583 (120591) = minusint

119905

0

1205831015840([120591 119905]) 119862119889120591 (47)

In particular for 120583 = 120583119898

(119888) int[0119905]

119891 (120591) 119889120583119898

(120591) = 119862119898 (119905) (48)

Lemma 21 (see [17]) Let 119891(119905) be a measurable nonnegativecontinuous and increasing real-valued function Then theChoquet integral of 119891 with respect to 120583 on [0 119905] is representedas

(119888) int[0119905]

119891 (120591) 119889120583 (120591) = int

infin

0

120583 (120591 | 119891 (120591) ge 119903 cap [0 119905]) 119889119903

= minusint

119905

0

1205831015840([120591 119905]) 119891 (120591) 119889120591

(49)

In particular for 120583 = 120583119898

(119888) int[0119905]

119891 (120591) 119889120583119898

(120591) = int

infin

0

120583 (120591 | 119891 (120591) ge 119903 cap [0 119905]) 119889119903

= int

119905

0

1198981015840(119905 minus 120591) 119891 (120591) 119889120591

(50)

Theorem 22 Let 119865(119905) be a strictly increasing fuzzy-number-valued functionThen the Choquet integral of 119865with respect to120583 on [0 119905] is represented as

(119888) int[0119905]

119865 (120591) 119889120583 (120591) = (119870)int

119905

0

[minus1205831015840([120591 119905])] 119865 (120591) 119889120591 (51)

In particular for 120583 = 120583119898

(119888) int[0119905]

119865 (120591) 119889120583119898

(120591) = (119870)int

119905

0

1198981015840(119905 minus 120591) 119865 (120591) 119889120591 (52)

Proof ByTheorem 8 we have

[(119888) int[0119905]

119865 (120591) 119889120583 (120591)]

120582

= [(119888) int[0119905]

119865120582(120591) 119889120583 (120591) (119888) int

[0119905]

119865120582(120591) 119889120583 (120591)]

(53)

for any 120582 isin [0 1] Since 119865(119905) is strictly increasing 119865120582(119905) and

119865120582(119905) are strictly increasing real-valued functions respec-

tively In view of Lemma 19 we have

(119888) int[0119905]

119865120582(120591) 119889120583 (120591) = minusint

119905

0

1205831015840([120591 119905]) 119865

120582(120591) 119889120591

(119888) int[0119905]

119865120582(120591) 119889120583 (120591) = minusint

119905

0

1205831015840([120591 119905]) 119865

120582(120591) 119889120591

(54)

It follows that

[(119888) int[0119905]

119865(120591)119889120583(120591)]

120582

= [minusint

119905

0

1205831015840([120591 119905]) 119865

120582(120591) 119889120591 minusint

119905

0

1205831015840([120591 119905]) 119865120582 (120591) 119889120591]

(55)

On the other hand

int

119905

0

[minus1205831015840([120591 119905])] 119865120582 (120591) 119889120591

= int

119905

0

[minus1205831015840([120591 119905])] [119865

120582(120591) 119865

120582(120591)] 119889120591

= [minusint

119905

0

1205831015840([120591 119905]) 119865

120582(120591) 119889120591 minusint

119905

0

1205831015840([120591 119905]) 119865120582 (120591) 119889120591]

(56)

Therefore

[(119888) int[0119905]

119865(120591)119889120583(120591)]

120582

= int

119905

0

[minus1205831015840([120591 119905])] 119865120582 (120591) 119889120591 (57)

By the arbitrary of 120582 we have

(119888) int[0119905]

119865 (120591) 119889120583 (120591) = (119870)int

119905

0

[minus1205831015840([120591 119905])] 119865 (120591) 119889120591 (58)

For 120583 = 120583119898 we note that minus120583

1015840([120591 119905]) = 119898

1015840(119905 minus 120591) hence

(119888) int[0119905]

119865 (120591) 119889120583119898

(120591) = (119870)int

119905

0

1198981015840(119905 minus 120591) 119865 (120591) 119889120591 (59)

Theorem 23 Let 119865(119905) be a constant fuzzy-valued functionthat is 119865(119905) = 119862 forall119905 isin 119877

+= (0 +infin) Then

(119888) int[0119905]

119865 (120591) 119889120583 (120591)

= (119870)int

119905

0

[minus1205831015840([120591 119905])] 119862119889120591 = 119862120583 ([0 119905])

(60)

In particular for 120583 = 120583119898

(119888) int[0119905]

119865 (120591) 119889120583119898

(120591) = 119862119898 (119905) (61)

Proof Note that119862 = [119862 119862] where119862 and119862 are constant real-valued functions respectively In view of Lemma 20 we have

(119888) int[0119905]

119862119889120583 (120591) = minusint

119905

0

1205831015840([120591 119905]) 119862119889120591

(119888) int[0119905]

119862119889120583 (120591) = minusint

119905

0

1205831015840([120591 119905]) 119862119889120591

(62)

8 Abstract and Applied Analysis

The rest of the proof follows in exactly the same way as thatof Theorem 22 Hence

(119888) int[0119905]

119865 (120591) 119889120583 (120591)

= (119870)int

119905

0

[minus1205831015840([120591 119905])] 119862119889120591 = 119862120583 ([0 119905])

(63)

For 120583 = 120583119898 we obtain 120583

1015840([120591 119905]) = minus119898

1015840(119905 minus 120591) It follows that

(119888) int[0119905]

119862119889120583 (120591) = (119870)int

119905

0

[minus1205831015840([120591 119905])] 119862119889120591

= int

119905

0

1198981015840(119905 minus 120591) 119862119889120591

= (119870)119862int

119905

0

1198981015840(119905 minus 120591) 119889120591 = 119862119898 (119905)

(64)

where 119898(0) = 0

Theorem 24 Let 119865(119905) isin F+ Then the Choquet integral of 119865with respect to 120583 on [0 119905] is represented as

(119888) int[0119905]

119865 (120591) 119889120583 (120591) = (119870)int

119905

0

[minus1205831015840([120591 119905])] 119865 (120591) 119889120591 (65)

In particular for 120583 = 120583119898

(119888) int[0119905]

119865 (120591) 119889120583119898

(120591) = (119870)int

119905

0

1198981015840(119905 minus 120591) 119865 (120591) 119889120591 (66)

Proof The theorem has been proved when 119865(119905) is a strictlyincreasing or constant fuzzy-number-valued function byTheorem 22 or Theorem 23 Without loss of generality weconsider a continuous and increasing function such that

119865 (119905) =

1198651(119905) 0 le 119905 lt 119905

1

1198652(119905) 119905

1le 119905 lt 119905

2

1198653(119905) 119905

2le 119905 lt infin

(67)

where 1198651(119905) and 119865

3(119905) are strictly increasing 119865

2(119905) is constant

and 1198651(1199051) = 1198652(119905) = 119865

3(1199052)

(i) For 0 le 119905 lt 1199051

Note that 1198651(119905) is strictly increasing on [0 119905

1) from

Theorem 22 we obtain

(119888) int[0119905]

119865 (120591) 119889120583 (120591) = (119870)int

119905

0

[minus1205831015840([120591 119905])] 1198651 (120591) 119889120591 (68)

(ii) For 1199051le 119905 lt 119905

2

In view of the construction of 119865(119905) on [0 119905] we have

(119888) int[0119905]

119865 (120591) 119889120583 (120591) = (119888) int[01199051]

1198651(120591) 119889120583 (120591)

+ (119888) int[1199051119905]

1198652(120591) 119889120583 (120591)

(69)

Since 1198651(119905) is strictly increasing on [0 119905

1) and 119865

2(119905) is

constant on [1199051 1199052) by Theorems 22 and 23

(119888) int[0119905]

119865 (120591) 119889120583 (120591) = (119870)int

1199051

0

[minus1205831015840([120591 119905])] 119865

1(120591) 119889120591

+ (119870)int

119905

1199051

[minus1205831015840([120591 119905])] 119865

2(120591) 119889120591

= (119870)int

119905

0

[minus1205831015840([120591 119905])] 119865 (120591) 119889120591

(70)

(iii) For 1199052le 119905 lt infin

Since1198651(119905) is strictly increasing on [0 119905

1) 1198652(119905) is constant

on [1199051 1199052) and 119865

3(119905) is strictly increasing on [119905

1infin) by

Theorems 22 and 23 we have

(119888) int[0119905]

119865 (120591) 119889120583 (120591)

= (119888) int[01199051]

1198651(120591) 119889120583 (120591) + (119888) int

[11990511199052]

1198652(120591) 119889120583 (120591)

+ (119888) int[1199052119905]

1198653(120591) 119889120583 (120591)

= (119870)int

1199051

0

[minus1205831015840([120591 119905])] 119865

1(120591) 119889120591

+ (119870)int

1199052

1199051

[minus1205831015840([120591 119905])] 119865

2(120591) 119889120591

+ (119870)int

119905

1199052

[minus1205831015840([120591 119905])] 119865

3(120591) 119889120591

= (119870)int

119905

0

[minus1205831015840([120591 119905])] 119865 (120591) 119889120591

(71)

For 120583 = 120583119898 we obtain that

(119888) int[0119905]

119865 (120591) 119889120583119898

(120591) = (119870)int

119905

0

1198981015840(119905 minus 120591) 119865 (120591) 119889120591 (72)

Remark 25 Note that Theorem 24 holds only for a continu-ous and increasing 119865 but in the case 120583

119898= ] it holds for any

119865

Remark 26 Let us consider the representation of theChoquetintegral for a continuous case shown in Theorem 24 inrelation with a discrete case Now [0 119905] is transformed intothe discrete set 119909

1 1199092 119909

119899 Since119865 is an increasing fuzzy-

number-valued function we have 119909119894= 1199091015840

119894 119894 = 1 2 119899 in

Theorem 11 For the sake of simplicity let 0 = 1199090lt 1199091lt sdot sdot sdot lt

Abstract and Applied Analysis 9

119909119899

= 119905 119865(1199090) = 0 and [119909

119894 119905] = 119909

119894 119909119894+1

119909119899 It follows

that

(119888) int[0119905]

119865 (120591) 119889120583 (120591)

=

119899

sum

119894=1

[120583 ([119909119894 119905]) minus 120583 ([119909

119894+1 119905])] 119865 (119909

1015840

119894)

=

119899

sum

119894=1

minus [120583 ([119909119894 119905]) minus 120583 ([119909

119894+1 119905])]

Δ119909119894

119865 (1199091015840

119894) Δ119909119894

= (119870)int

119905

0

[minus1205831015840([120591 119905])] 119865 (120591) 119889120591

(73)

5 Choquet Integral Equations ofFuzzy-Number-Valued Functions

In this section let us consider a Choquet integral equationbased onTheorem 24 as shown below Given continuous andincreasing fuzzy-number-valued functions 119866(119905) with 119866(0) =

0 let us find continuous and increasing fuzzy-number-valuedfunctions 119865(119905) such that

119866 (119905) = (119888) int[0119905]

119865 (120591) 119889120583119898

(119905) (74)

which is expressed as

119866 (119905) = (119870)int

119905

0

1198981015840(119905 minus 120591) 119865 (120591) 119889120591 (75)

Definition 27 Let 119891(119905) be a real-valued function defined on[0 +infin) The following

F (119904) = int

+infin

0

119890minus119904119905

119891 (119905) 119889119905 (76)

is said to be the Laplace transformation of 119891(119905) if the infiniteintegralint+infin

0119890minus119904119905

119891(119905)119889119905 is convergentwith respect to the valueof parameter 119904 We denote its Laplace transformation asF(119904) = 119871[119891(119905)] and the inverse Laplace transformation as119891(119905) = 119871

minus1[F(119904)]

Definition 28 Let119865(119909 119910) be a fuzzy-number-valued functiondefined on [0 +infin) times [0 +infin) (119870) int

infin

0119865(119909 119910)119889119909 is said to be

convergent with respect to the parameter 119910 if for every fixed119910 isin [0 +infin) (119870) int

+infin

0119865(119909 119910)119889119909 exists

Definition 29 Let 119865(119905) be a fuzzy-number-valued functiondefined on [0 +infin) The following

F (119904) = (119870)int[0+infin)

119890minus119904119905

119865 (119905) 119889119905 (77)

is said to be the Laplace transformation of 119865(119905) if the Kalevaintegral intinfin

0119890minus119904119905

119865(119905)119889119905 is convergent with respect to the valueof parameter 119904 We denote its Laplace transformation asF(119904) = 119871[119865(119905)] and the inverse Laplace transformation as119865(119905) = 119871

minus1[F(119904)]

Remark 30 In view of Theorem 8 we have

F120582(119904) = [int

+infin

0

119890minus119904119905

119865120582(119905) 119889119905 int

+infin

0

119890minus119904119905

119865120582(119905) 119889119905] (78)

for all 120582 isin [0 1]

Theorem 31 Let 119866(119905) = (119888) int[0119905]

119865(120591)119889120583119898(119905) Then

G (119904) = 119904M (119904) F (119904)

119866 (119904) = 119871minus1

[119904M (119904) F (119904)]

(79)

where G(119904) = 119871[119866(119905)]M(s) = 119871[119898(119905)] and F(119904) = 119871[119865(119905)]

Proof Since 119866(119905) = (119888) int[0119905]

119865(120591)119889120583119898(119905) = (119870) int

119905

01198981015840(119905 minus

120591)119865(120591)119889120591 it follows easily from the condition that

G120582(119904) = [int

+infin

0

119890minus119904119905

119866120582(119905) 119889119905 int

+infin

0

119890minus119904119905

119866120582(119905) 119889119905]

= [int

+infin

0

119890minus119904119905

int

119905

0

1198981015840(119905 minus 120591) 119865

120582(120591) 119889120591 119889119905

int

+infin

0

119890minus119904119905

int

119905

0

1198981015840(119905 minus 120591) 119865

120582(120591) 119889120591 119889119905]

(80)

for all 120582 isin [0 1]Notice that

int

+infin

0

119890minus119904119905

int

119905

0

1198981015840(119905 minus 120591) 119865

120582(120591) 119889120591 119889119905

= int

+infin

0

int

+infin

120591

119890minus119904119905

1198981015840(119905 minus 120591) 119865

120582(120591) 119889119905 119889120591

= int

+infin

0

119865120582(120591) int

+infin

120591

119890minus119904119905

1198981015840(119905 minus 120591) 119889119905 119889120591

= int

+infin

0

119865120582(120591) [119890

minus119904119905119898(119905 minus 120591)

10038161003816100381610038161003816

+infin

120591minus int

+infin

120591

119898(119905 minus 120591) 119889119890minus119904119905

] 119889120591

= int

+infin

0

119865120582(120591) [minusint

+infin

120591

119898(119905 minus 120591) (minus119904) 119890minus119904119905

119889119905] 119889120591

= 119904int

+infin

0

119865120582(120591) [int

+infin

0

119898(119905 minus 120591) (minus119904) 119890minus119904(119905minus120591)

119890minus119904120591

119889 (119905 minus 120591)] 119889120591

= 119904int

+infin

0

119865120582(120591) [119890

minus119904120591int

+infin

0

119890minus119904(119905minus120591)

119898(119905 minus 120591) 119889 (119905 minus 120591)] 119889120591

= 119904int

+infin

0

119865120582(120591) 119890minus119904120591

M (119904) 119889120591

= 119904M (119904) int

+infin

0

119890minus119904120591

119865120582(120591) 119889120591 = 119904M (119904)F

120582(119904)

(81)

Following the same argument we can prove that

int

+infin

0

119890minus119904119905

int

119905

0

1198981015840(119905 minus 120591) 119865

120582(120591) 119889120591 119889119905 = 119904M (119904)F

120582(119904) (82)

10 Abstract and Applied Analysis

Hence

G (119904) = 119904M (119904) F (119904)

119866 (119904) = 119871minus1

[119904M (119904) F (119904)]

(83)

Theorem 32 For continuous and increasing fuzzy-valuedfunctions 119866(119905) with 119866(0) = 0

F (119904) =G (119904)

119904M (119904) 119865 (119904) = 119871

minus[

G (119904)

119904M (119904)] (84)

where G(119904) = 119871[119866(119905)]M(119904) = 119871[119898(119905)] and F(119904) = 119871[119865(119905)]

Proof It follows easily fromTheorem 31 that

G (119904) = 119904M (119904) F (119904)

119866 (119904) = 119871minus1

[119904M (119904) F (119904)]

(85)

It is obvious that

F (119904) =G (119904)

119904M (119904)

119865 (119904) = 119871minus1

[G (119904)

119904M (119904)]

(86)

where G(119904) = 119871[119866(119905)]M(119904) = 119871[119898(119905)] and F(119904) = 119871[119865(119905)]The Dirac delta function is defined by the properties

120575 (119905) = 0 for 119905 = 0

undefined at 119905 = 0(87)

and int+infin

minusinfin120575(119905)119889119905 = 1 that is the function has unit area

The Laplace transform of the Dirac Delta function 120575(119905) isshown as follows

119871 [120575 (119905)] = int

+infin

0

119890minus119904119905

120575 (119905) 119889119905 = 1 (88)

So

119871minus1

[1] = 120575 (119905) (89)

Example 33 Let 119898(119905) = 119905 and let the membership functionof 119866(119905) be

119898119866(119905)

(119909) = 119909 119909 isin [0 1]

0 119909 notin [0 1] 119905 le 1 (90)

Then by Theorem 32 we can solve the Choquet integralequation

119866 (119905) = (119888) int[0119905]

119865 (120591) 119889120583119898

(119905) (91)

Indeed let 119898(119905) = 119905 thenM(119904) = 11199042 The 120582-cut of 119866(119905) is

represented by interval 119866120582(119905) = 119909 | 119898

119866(119905)(119909) ge 120582 = [120582 1]

that is 119866120582(119905) = 120582 119866

120582(119905) = 1 In view of Remark 30 we have

G120582(119904) = [int

+infin

0119890minus119904119905

119866120582(119905)119889119905 int

+infin

0119890minus119904119905

119866120582(119905)119889119905] = [120582119878 1119878]

Then

F120582(119904) =

G120582(119904)

119904M (119904)= 120582

F120582(119904) =

G120582(119904)

119904M (119904)= 1

(92)

Furthermore

119865120582(119905) = 119871

minus1(F120582(119904)) = 120582120575 (119905)

119865120582(119905) = 119871

minus1(F120582(119904)) = 120575 (119905)

(93)

Therefore for every 120582 isin [0 1]

119865120582(119905) = [119865

120582(119905) 119865120582(119905)] = [120582120575 (119905) 120575 (119905)] (94)

Then we have the membership function of 119865(119905) as

119898119865(119905)

(119909) =

119909

120575 (119905) 119909 isin [0 120575 (119905)] 119905 = 0

0 otherwise(95)

6 Conclusions and Remarks

In this paper we have considered the Choquet integrals offuzzy-number-valued functions based on the nonnegativereal lineWe have discussed the Choquet integrals of interval-valued functions and fuzzy-number-valued functions basedon nonadditive Sugeno measures and showed the repre-sentation theorem of them respectively We also gave theoperational schemes of above several classes of integralson discrete sets Then we have given a representation ofthe Choquet integral of a nonnegative continuous andincreasing fuzzy-number-valued function with respect to afuzzymeasure In addition in order to solve Choquet integralequations of fuzzy-number-valued function a concept of theLaplace transformation for calculation has been introducedFor distorted Lebesgue measures it was shown that Choquetintegral equations of fuzzy-number-valued function could besolved by the Laplace transformation Finally we have givenan example to illustrate themain result at the end of the paper

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The work is supported by the Natural Scientific Funds ofChina (61262022) the Natural Scientific Fund of GansuProvince of China (1208RJZA251) Lanzhou Jiaotong Univer-sity Young Scientific Research Fund Project (2011020) theFundamental Research Funds for Gansu Province Univer-sities (no 213060) and the Scientific Research Project ofNorthwestNormalUniversity (noNWNU-KJCXGC-03-61)

Abstract and Applied Analysis 11

References

[1] G Choquet ldquoTheory of capacitiesrdquoAnnales de lrsquoInstitut Fouriervol 5 pp 131ndash295 1954

[2] D Denneberg Non-Additive Measure and Integral KluwerAcademic Publishers Boston Mass USA 1994

[3] T Murofushi M Sugeno and M Machida ldquoNon-monotonicfuzzy measures and the Choquet integralrdquo Fuzzy Sets andSystems vol 64 no 1 pp 73ndash86 1994

[4] Z Y Wang K-S Leung M-L Wong and J Fang ldquoA new typeof nonlinear integrals and the computational algorithmrdquo FuzzySets and Systems vol 112 no 2 pp 223ndash231 2000

[5] D Schmeidler ldquoSubjective probability and expected utilitywithout additivityrdquo Econometrica vol 57 no 3 pp 571ndash5871989

[6] M Grabisch ldquo119896-order additive discrete fuzzy measures andtheir representationrdquo Fuzzy Sets and Systems vol 92 no 2 pp167ndash189 1997

[7] MGrabisch I Kojadinovic and PMeyer ldquoA review ofmethodsfor capacity identification in Choquet integral based multi-attribute utility theory applications of the Kappalab R packagerdquoEuropean Journal of Operational Research vol 186 no 2 pp766ndash785 2008

[8] K Xu Z Wang P Heng and K Leung ldquoClassification bynonlinear integral projectionsrdquo IEEE Transactions on FuzzySystems vol 11 no 2 pp 187ndash201 2003

[9] M J Bolanos L M de Campos Ibanez and A GonzalezMunoz ldquoConvergence properties of the monotone expectationand its application to the extension of fuzzy measuresrdquo FuzzySets and Systems vol 33 no 2 pp 201ndash212 1989

[10] T Murofushi and M Sugeno ldquoA theory of fuzzy measuresrepresentations the Choquet integral and null setsrdquo Journal ofMathematical Analysis and Applications vol 159 no 2 pp 532ndash549 1991

[11] Z Wang ldquoConvergence theorems for sequences of Choquetintegralsrdquo International Journal of General Systems vol 26 no1-2 pp 133ndash143 1997

[12] YOuyang andHZhang ldquoOn the space ofmeasurable functionsand its topology determined by the Choquet integralrdquo Interna-tional Journal of Approximate Reasoning vol 52 no 9 pp 1355ndash1362 2011

[13] J Kawabe ldquoThe Choquet integral representability of comono-tonically additive functionals in locally compact spacesrdquo Inter-national Journal of Approximate Reasoning vol 54 no 3 pp418ndash426 2013

[14] S Abbasbandy E Babolian and M Alavi ldquoNumerical methodfor solving linear Fredholm fuzzy integral equations of thesecond kindrdquo Chaos Solitons amp Fractals vol 31 no 1 pp 138ndash146 2007

[15] O Kaleva ldquoFuzzy differential equationsrdquo Fuzzy Sets and Sys-tems vol 24 no 3 pp 301ndash317 1987

[16] J Mordeson and W Newman ldquoFuzzy integral equationsrdquoInformation Sciences vol 87 no 4 pp 215ndash229 1995

[17] M Sugeno ldquoA note on derivatives of functions with respect tofuzzy measuresrdquo Fuzzy Sets and Systems vol 222 pp 1ndash17 2013

[18] I Kojadinovic J-L Marichal and M Roubens ldquoAn axiomaticapproach to the definition of the entropy of a discrete Choquetcapacityrdquo Information Sciences vol 172 no 1-2 pp 131ndash1532005

[19] E Pap Null-Additive Set Functions vol 337 of Mathematicsand Its Applications KluwerAcademic Publishers LondonUK1994

[20] M Sugeno Theory of fuzzy integrals and its applications [PhDdissertation] Tokyo Institute of Technology Tokyo Japan 1974

[21] C X Wu D L Zhang and C M Guo ldquoFuzzy number fuzzymeasures and fuzzy integralsrdquo Fuzzy Sets and Systems vol 98no 3 pp 355ndash360 1998

[22] C X Wu and M Ma The Foundament of Fuzzy AnalysisNational Defense Press Beijing China 1991 (Chinese)

[23] R Yang Z Y Wang P A Heng and K S Leung ldquoFuzzynumbers and fuzzification of the Choquet integralrdquo Fuzzy Setsand Systems vol 153 no 1 pp 95ndash113 2005

[24] S Seikkala ldquoOn the fuzzy initial value problemrdquo Fuzzy Sets andSystems vol 24 no 3 pp 319ndash330 1987

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 5

that 1198651205821

le 119865120582119899

le 119865120582119899

le 1198651205821

1198651205821

1198651205821

are integrableand by the continuity of 120583 then

infin

⋂

119899=1

[(119888) int119864

119865120582119899

119889120583 (119888) int119864

119865120582119899

119889120583]

= [(119888) int119864

119865120582119889120583 (119888) int

119864

119865120582119889120583]

(27)

In conclusion there exists a unique fuzzy number 119886 isin +

such that

119886120582= [(119888) int

119864

119865120582119889120583 (119888) int

119864

119865120582119889120583] (28)

Furthermore we get that 119865 is 119865119862-integrable on 119864 and

[(119888) int119864

119865119889120583]

120582

= [(119888) int119864

119865120582119889120583 (119888) int

119864

119865120582119889120583] (29)

In the last part of the section we will investigate theoperational schemes of above several classes of integrals ona discrete set

Let119883 = 1199091 1199092 119909

119899 be a discrete setThenwewill give

a new scheme to calculate the value of the Choquet integral

Theorem 9 (see [8]) Let 119891 be a real-valued function on 119883 =

1199091 1199092 119909

119899 Then Choquet integral of 119891 with respect to a

fuzzy measure 120583 on 119883 is given by

(119888) int119883

119891119889120583 =

119899

sum

119894=1

[119891 (1199091015840

119894) minus 119891 (119909

1015840

119894minus1)] 120583 (119883

1015840

119894) (30)

or equivalently by

(119888) int119883

119891119889120583 =

119899

sum

119894=1

[120583 (1198831015840

119894) minus 120583 (119883

1015840

119894+1)] 119891 (119909

1015840

119894) (31)

where 1199091015840

1 1199091015840

2 119909

1015840

119899is a permutation of 119909

1 1199092 119909

119899such

that 119891(1199091015840

0) le 119891(119909

1015840

1) le sdot sdot sdot le 119891(119909

1015840

119899) 119891(119909

1015840

0) = 0 119883

1015840

119894=

1199091015840

119894 1199091015840

119894+1 119909

1015840

119899 119894 = 1 2 119899 and 119883

1015840

119899+1= 0

Theorem 10 Let 119865 be an interval-valued function on 119883 =

1199091 1199092 119909

119899 Then Choquet integral of 119865 with respect to a

fuzzy measure 120583 on 119883 is given by

(119888) int119883

119865119889120583 =

119899

sum

119894=1

[120583 (1198831015840

119894) minus 120583 (119883

1015840

119894+1)] 119865 (119909

1015840

119894) (32)

where 1199091015840

1 1199091015840

2 119909

1015840

119899is a permutation of 119909

1 1199092 119909

119899such

that 119865(1199091015840

0) le 119865(119909

1015840

1) le sdot sdot sdot le 119865(119909

1015840

119899) 119865(119909

1015840

0) = [0 0] 1198831015840

119894=

1199091015840

119894 1199091015840

119894+1 119909

1015840

119899 119894 = 1 2 119899 and 119883

1015840

119899+1= 0

Proof Since 119865 is an interval-valued function on119883 in view ofTheorem 6 we have

(119888) int119883

119865119889120583 = [(119888) int119883

119865119889120583 (119888) int119883

119865119889120583] (33)

Note that 119865 and 119865 are real-valued function on119883 respectivelyByTheorem 9 we get

(119888) int119883

119865119889120583 = [

119899

sum

119894=1

[120583 (1198831015840

119894) minus 120583 (119883

1015840

119894+1)] 119865 (119909

1015840

119894)

119899

sum

119894=1

[120583 (1198831015840

119894) minus 120583 (119883

1015840

119894+1)] 119865 (119909

1015840

119894)]

=

119899

sum

119894=1

[120583 (1198831015840

119894) minus 120583 (119883

1015840

119894+1)] 119865 (119909

1015840

119894)

(34)

where 1199091015840

1 1199091015840

2 119909

1015840

119899is a permutation of 119909

1 1199092 119909

119899such

that 119865(1199091015840

0) le 119865(119909

1015840

1) le sdot sdot sdot le 119865(119909

1015840

119899) 119865(119909

1015840

0) = [0 0] 1198831015840

119894=

1199091015840

119894 1199091015840

119894+1 119909

1015840

119899 119894 = 1 2 119899 and 119883

1015840

119899+1= 0

Theorem 11 Let119865 be a fuzzy-number-valued function on119883 =

1199091 1199092 119909

119899 Then Choquet integral of 119865 with respect to a

fuzzy measure 120583 on 119883 is given by

(119888) int119883

119865119889120583 =

119899

sum

119894=1

[120583 (1198831015840

119894) minus 120583 (119883

1015840

119894+1)] 119865 (119909

1015840

119894) (35)

where 1199091015840

1 1199091015840

2 119909

1015840

119899is a permutation of 119909

1 1199092 119909

119899such

that 119865(1199091015840

0) le 119865(119909

1015840

1) le sdot sdot sdot le 119865(119909

1015840

119899) 119865(119909

1015840

0) = 0 119883

1015840

119894=

1199091015840

119894 1199091015840

119894+1 119909

1015840

119899 119894 = 1 2 119899 and 119883

1015840

119899+1= 0

Proof Since 119865 is a fuzzy-number-valued on 119883 in view ofTheorem 8 we have

[(119888) int119883

119865119889120583]

120582

= [(119888) int119883

119865120582119889120583 (119888) int

119883

119865120582119889120583] (36)

for any 120582 isin [0 1] By the semiorder in space (ie let 119886 isin

Then 119886 le if 119886120582le 119887120582and 119886

120582le 119887120582) andTheorem 10 there

is a sequence 1199091015840

1 1199091015840

2 119909

1015840

119899on 119883 such that 119865(119909

1015840

0) le 119865(119909

1015840

1) le

sdot sdot sdot le 119865(1199091015840

119899) 119865(119909

1015840

0) = 0 1198831015840

119894= 1199091015840

119894 1199091015840

119894+1 119909

1015840

119899 119894 = 1 2 119899

1198831015840

119899+1= 0 and 119909

1015840

1 1199091015840

2 119909

1015840

119899is a permutation of 119909

1 1199092 119909

119899

Consequently

[(119888) int119883

119865119889120583]

120582

= [

119899

sum

119894=1

[120583 (1198831015840

119894) minus 120583 (119883

1015840

119894+1)] 119865120582(1199091015840

119894)

119899

sum

119894=1

[120583 (1198831015840

119894) minus 120583 (119883

1015840

119894+1)] 119865120582(1199091015840

119894)]

(37)

for any 120582 isin [0 1] Therefore

(119888) int119883

119865119889120583 =

119899

sum

119894=1

[120583 (1198831015840

119894) minus 120583 (119883

1015840

119894+1)] 119865 (119909

1015840

119894) (38)

4 The Representation of Choquet Integral ofFuzzy-Number-Valued Functions

Sugeno has described carefully the representation of Cho-quet integral of real-valued increasing functions and some

6 Abstract and Applied Analysis

important conclusions have been obtained [17] Motivated bythis we will discuss the representation of Choquet integral offuzzy-number-valued functions in this section

Definition 12 Fuzzy-number-valued function 119865 119883 rarr +

is said to be continuous on 119883 if for every 1199090isin 119883 there are

lim119909rarr119909

0

119865120582(119909) = 119865

120582(1199090)

lim119909rarr119909

0

119865120582(119909) = 119865

120582(1199090)

(39)

where 120582 isin [0 1]

Definition 13 Fuzzy-number-valued function 119865 119883 rarr +

is said to be increasing on 119883 if for every 1199091

le 1199092there are

119865120582(1199091) le 119865120582(1199092) and 119865

120582(1199091) le 119865120582(1199092) where 119909

1 1199092isin 119883 and

120582 isin [0 1]LetF+ be a class ofmeasurable nonnegative continuous

and increasing fuzzy-number-valued functionsLet ] be a Lebesgue measure for [119886 119887] sub [0infin) ][119886 119887] =

119887 minus 119886

Definition 14 (see [17]) Let 119898 119877+

rarr 119877+ be a continuous

and increasing function and 119898(0) = 0 A fuzzy measure 120583119898

a distorted Lebesgue measure is defined by 120583119898(sdot) = 119898(](sdot))

Definition 15 (see [24]) A fuzzy-number-valued function 119865

[119886 119887] rarr is differentiable at 119909 isin [119886 119887] if there exist fuzzynumber-valued functions 1198651015840(119909) such that

[1198651015840(119909)]120582= [1198651015840

120582(119909) 119865

1015840

120582(119909)] (40)

for every 120582 isin [0 1] where 1198651015840(119909) is the fuzzy derivative of

119865(119909)Note that 120583

119898is induced from the Lebesguemeasure ] by a

monotone transformation where 120583119898([119886 119887]) = 119898(]([119886 119887])) =

119898(119887 minus 119886) Apparently it loses additivity unless 119898(119905) = 119905 butreserves monotonicity In what follows we assume that 119898(119905)

is differentiableIn this section we consider the calculation of Choquet

integrals Let 120583 be a general fuzzy measure and consider120583([120591 119905]) for a closed interval [120591 119905] then 120583([120591 119905]) is decreasingfor 120591 and increasing for 119905 Throughout the paper we assumethat the functions 119898(119905) 119865(119905) and 119866(119905) are continuouslydifferentiable We also assume that 120583([120591 119905]) is continuouslydifferentiable with respect to 120591 on [0 119905] for every 119905 gt 0 Inaddition we require the regularity condition that 120583(119905) = 0

holds for every 119905 ge 0 We write 1205831015840([120591 119905]) = (120597120597120591)120583([120591 119905])

where we note that 1205831015840([120591 119905]) le 0 for 120591 le 119905 If 120583 = 120583119898then

1205831015840([120591 119905]) = minus119898

1015840(119905 minus 120591) where 119898

1015840(119905) = 119889119898(119905)119889119905 First we

consider a case that 119865(119905) is strictly increasingFuzzy-number-valued function 119865 119883 rarr is said to

be 119871-integrally bounded if there exists a Lebesgue integrablefunction ℎ 119883 rarr 119877

+ such that || le ℎ(119905) for every section isin [119865(119905)]

0

Definition 16 (see [22]) Let fuzzy-number-valued functions119865 [119886 +infin) rarr be measurable and 119871-integrally bounded119865 is called Kaleva-integrable if

[int

+infin

119886

119865(119905)119889119905]

120582

= (119871) int

+infin

119886

119892 (119905) 119889119905 | 119892 (119905) isin 119878[119865(119905)]

120582

0 le 120582 le 1

(41)

determines a unique fuzzy number 119886 isin which is denotedby (119870) int

+infin

119886119865(119905) 119889120583 = 119886 where 119878

[119865(119905)]120582

= 119892(119905) isin [119865(119905)]120582is a

measurable selection

Lemma 17 (see [22]) Let 119865 [119886 +infin) rarr be measurableand C-integrally bounded fuzzy-number-valued function then119865 is Kaleva integrable on [119886 +infin) if and only if 119865

120582(119905) and119865

120582(119905)

are 119871-integrable on [119886 +infin) and

[(119870)int

119905

0

119865(120591)119889120591]

120582

= [(119871) int

119905

0

119865120582(120591) 119889120591 (119871) int

119905

0

119865120582(120591) 119889120591]

(42)

for any 120582 isin [0 1]

Theorem 18 Let 119865(119905) isin F+ Then minus1205831015840([120591 119905])119865(120591) is Kaleva

integrable on [0 119905] and

[(119870)int

119905

0

[minus1205831015840([120591 119905])]119865(120591)119889120591]

120582

= [(119871) int

119905

0

[minus1205831015840([120591 119905])] 119865

120582(120591) 119889120591

(119871) int

119905

0

[minus1205831015840([120591 119905])] 119865

120582(120591) 119889120591]

(43)

Proof Since119865(119905) isin F+ for every120582 isin [0 1]we get that119865120582and

119865120582are measurable nonnegative continuous and increasing

real-valued functions on [0 119905] It follows that 119865120582and 119865

120582are

119871-integrable on [0 119905] 120583([120591 119905]) is 119871-integrable on [0 119905] as itis continuously differentiable with respect to 120591 on [0 119905] forevery 119905 gt 0 In view of Lemma 17 [minus120583

1015840([120591 119905])]119865(120591) is Kaleva

integrable on [0 119905] and

[(119870)int

119905

0

[minus1205831015840([120591 119905])] 119865 (120591) 119889120591]

120582

= [(119871) int

119905

0

[minus1205831015840([120591 119905])] 119865

120582(120591) 119889120591

(119871) int

119905

0

[minus1205831015840([120591 119905])] 119865

120582(120591) 119889120591]

(44)

Lemma 19 (see [17]) Let 119891(119905) be a real strictly increasingfunction Then the Choquet integral of 119891 with respect to 120583 on[0 119905] is represented as

(119888) int[0119905]

119891 (120591) 119889120583 (120591) = minusint

119905

0

1205831015840([120591 119905]) 119891 (120591) 119889120591 (45)

Abstract and Applied Analysis 7

In particular for 120583 = 120583119898

(119888) int[0119905]

119891 (120591) 119889120583119898

(120591) = int

119905

0

1198981015840(119905 minus 120591) 119891 (120591) 119889120591 (46)

Lemma 20 (see [17]) Let 119891(119905) be a constant real-valuedfunction 119891(119905) = 119862 forall119905 isin 119877

+= (0 +infin) Then

(119888) int[0119905]

119891 (120591) 119889120583 (120591) = minusint

119905

0

1205831015840([120591 119905]) 119862119889120591 (47)

In particular for 120583 = 120583119898

(119888) int[0119905]

119891 (120591) 119889120583119898

(120591) = 119862119898 (119905) (48)

Lemma 21 (see [17]) Let 119891(119905) be a measurable nonnegativecontinuous and increasing real-valued function Then theChoquet integral of 119891 with respect to 120583 on [0 119905] is representedas

(119888) int[0119905]

119891 (120591) 119889120583 (120591) = int

infin

0

120583 (120591 | 119891 (120591) ge 119903 cap [0 119905]) 119889119903

= minusint

119905

0

1205831015840([120591 119905]) 119891 (120591) 119889120591

(49)

In particular for 120583 = 120583119898

(119888) int[0119905]

119891 (120591) 119889120583119898

(120591) = int

infin

0

120583 (120591 | 119891 (120591) ge 119903 cap [0 119905]) 119889119903

= int

119905

0

1198981015840(119905 minus 120591) 119891 (120591) 119889120591

(50)

Theorem 22 Let 119865(119905) be a strictly increasing fuzzy-number-valued functionThen the Choquet integral of 119865with respect to120583 on [0 119905] is represented as

(119888) int[0119905]

119865 (120591) 119889120583 (120591) = (119870)int

119905

0

[minus1205831015840([120591 119905])] 119865 (120591) 119889120591 (51)

In particular for 120583 = 120583119898

(119888) int[0119905]

119865 (120591) 119889120583119898

(120591) = (119870)int

119905

0

1198981015840(119905 minus 120591) 119865 (120591) 119889120591 (52)

Proof ByTheorem 8 we have

[(119888) int[0119905]

119865 (120591) 119889120583 (120591)]

120582

= [(119888) int[0119905]

119865120582(120591) 119889120583 (120591) (119888) int

[0119905]

119865120582(120591) 119889120583 (120591)]

(53)

for any 120582 isin [0 1] Since 119865(119905) is strictly increasing 119865120582(119905) and

119865120582(119905) are strictly increasing real-valued functions respec-

tively In view of Lemma 19 we have

(119888) int[0119905]

119865120582(120591) 119889120583 (120591) = minusint

119905

0

1205831015840([120591 119905]) 119865

120582(120591) 119889120591

(119888) int[0119905]

119865120582(120591) 119889120583 (120591) = minusint

119905

0

1205831015840([120591 119905]) 119865

120582(120591) 119889120591

(54)

It follows that

[(119888) int[0119905]

119865(120591)119889120583(120591)]

120582

= [minusint

119905

0

1205831015840([120591 119905]) 119865

120582(120591) 119889120591 minusint

119905

0

1205831015840([120591 119905]) 119865120582 (120591) 119889120591]

(55)

On the other hand

int

119905

0

[minus1205831015840([120591 119905])] 119865120582 (120591) 119889120591

= int

119905

0

[minus1205831015840([120591 119905])] [119865

120582(120591) 119865

120582(120591)] 119889120591

= [minusint

119905

0

1205831015840([120591 119905]) 119865

120582(120591) 119889120591 minusint

119905

0

1205831015840([120591 119905]) 119865120582 (120591) 119889120591]

(56)

Therefore

[(119888) int[0119905]

119865(120591)119889120583(120591)]

120582

= int

119905

0

[minus1205831015840([120591 119905])] 119865120582 (120591) 119889120591 (57)

By the arbitrary of 120582 we have

(119888) int[0119905]

119865 (120591) 119889120583 (120591) = (119870)int

119905

0

[minus1205831015840([120591 119905])] 119865 (120591) 119889120591 (58)

For 120583 = 120583119898 we note that minus120583

1015840([120591 119905]) = 119898

1015840(119905 minus 120591) hence

(119888) int[0119905]

119865 (120591) 119889120583119898

(120591) = (119870)int

119905

0

1198981015840(119905 minus 120591) 119865 (120591) 119889120591 (59)

Theorem 23 Let 119865(119905) be a constant fuzzy-valued functionthat is 119865(119905) = 119862 forall119905 isin 119877

+= (0 +infin) Then

(119888) int[0119905]

119865 (120591) 119889120583 (120591)

= (119870)int

119905

0

[minus1205831015840([120591 119905])] 119862119889120591 = 119862120583 ([0 119905])

(60)

In particular for 120583 = 120583119898

(119888) int[0119905]

119865 (120591) 119889120583119898

(120591) = 119862119898 (119905) (61)

Proof Note that119862 = [119862 119862] where119862 and119862 are constant real-valued functions respectively In view of Lemma 20 we have

(119888) int[0119905]

119862119889120583 (120591) = minusint

119905

0

1205831015840([120591 119905]) 119862119889120591

(119888) int[0119905]

119862119889120583 (120591) = minusint

119905

0

1205831015840([120591 119905]) 119862119889120591

(62)

8 Abstract and Applied Analysis

The rest of the proof follows in exactly the same way as thatof Theorem 22 Hence

(119888) int[0119905]

119865 (120591) 119889120583 (120591)

= (119870)int

119905

0

[minus1205831015840([120591 119905])] 119862119889120591 = 119862120583 ([0 119905])

(63)

For 120583 = 120583119898 we obtain 120583

1015840([120591 119905]) = minus119898

1015840(119905 minus 120591) It follows that

(119888) int[0119905]

119862119889120583 (120591) = (119870)int

119905

0

[minus1205831015840([120591 119905])] 119862119889120591

= int

119905

0

1198981015840(119905 minus 120591) 119862119889120591

= (119870)119862int

119905

0

1198981015840(119905 minus 120591) 119889120591 = 119862119898 (119905)

(64)

where 119898(0) = 0

Theorem 24 Let 119865(119905) isin F+ Then the Choquet integral of 119865with respect to 120583 on [0 119905] is represented as

(119888) int[0119905]

119865 (120591) 119889120583 (120591) = (119870)int

119905

0

[minus1205831015840([120591 119905])] 119865 (120591) 119889120591 (65)

In particular for 120583 = 120583119898

(119888) int[0119905]

119865 (120591) 119889120583119898

(120591) = (119870)int

119905

0

1198981015840(119905 minus 120591) 119865 (120591) 119889120591 (66)

Proof The theorem has been proved when 119865(119905) is a strictlyincreasing or constant fuzzy-number-valued function byTheorem 22 or Theorem 23 Without loss of generality weconsider a continuous and increasing function such that

119865 (119905) =

1198651(119905) 0 le 119905 lt 119905

1

1198652(119905) 119905

1le 119905 lt 119905

2

1198653(119905) 119905

2le 119905 lt infin

(67)

where 1198651(119905) and 119865

3(119905) are strictly increasing 119865

2(119905) is constant

and 1198651(1199051) = 1198652(119905) = 119865

3(1199052)

(i) For 0 le 119905 lt 1199051

Note that 1198651(119905) is strictly increasing on [0 119905

1) from

Theorem 22 we obtain

(119888) int[0119905]

119865 (120591) 119889120583 (120591) = (119870)int

119905

0

[minus1205831015840([120591 119905])] 1198651 (120591) 119889120591 (68)

(ii) For 1199051le 119905 lt 119905

2

In view of the construction of 119865(119905) on [0 119905] we have

(119888) int[0119905]

119865 (120591) 119889120583 (120591) = (119888) int[01199051]

1198651(120591) 119889120583 (120591)

+ (119888) int[1199051119905]

1198652(120591) 119889120583 (120591)

(69)

Since 1198651(119905) is strictly increasing on [0 119905

1) and 119865

2(119905) is

constant on [1199051 1199052) by Theorems 22 and 23

(119888) int[0119905]

119865 (120591) 119889120583 (120591) = (119870)int

1199051

0

[minus1205831015840([120591 119905])] 119865

1(120591) 119889120591

+ (119870)int

119905

1199051

[minus1205831015840([120591 119905])] 119865

2(120591) 119889120591

= (119870)int

119905

0

[minus1205831015840([120591 119905])] 119865 (120591) 119889120591

(70)

(iii) For 1199052le 119905 lt infin

Since1198651(119905) is strictly increasing on [0 119905

1) 1198652(119905) is constant

on [1199051 1199052) and 119865

3(119905) is strictly increasing on [119905

1infin) by

Theorems 22 and 23 we have

(119888) int[0119905]

119865 (120591) 119889120583 (120591)

= (119888) int[01199051]

1198651(120591) 119889120583 (120591) + (119888) int

[11990511199052]

1198652(120591) 119889120583 (120591)

+ (119888) int[1199052119905]

1198653(120591) 119889120583 (120591)

= (119870)int

1199051

0

[minus1205831015840([120591 119905])] 119865

1(120591) 119889120591

+ (119870)int

1199052

1199051

[minus1205831015840([120591 119905])] 119865

2(120591) 119889120591

+ (119870)int

119905

1199052

[minus1205831015840([120591 119905])] 119865

3(120591) 119889120591

= (119870)int

119905

0

[minus1205831015840([120591 119905])] 119865 (120591) 119889120591

(71)

For 120583 = 120583119898 we obtain that

(119888) int[0119905]

119865 (120591) 119889120583119898

(120591) = (119870)int

119905

0

1198981015840(119905 minus 120591) 119865 (120591) 119889120591 (72)

Remark 25 Note that Theorem 24 holds only for a continu-ous and increasing 119865 but in the case 120583

119898= ] it holds for any

119865

Remark 26 Let us consider the representation of theChoquetintegral for a continuous case shown in Theorem 24 inrelation with a discrete case Now [0 119905] is transformed intothe discrete set 119909

1 1199092 119909

119899 Since119865 is an increasing fuzzy-

number-valued function we have 119909119894= 1199091015840

119894 119894 = 1 2 119899 in

Theorem 11 For the sake of simplicity let 0 = 1199090lt 1199091lt sdot sdot sdot lt

Abstract and Applied Analysis 9

119909119899

= 119905 119865(1199090) = 0 and [119909

119894 119905] = 119909

119894 119909119894+1

119909119899 It follows

that

(119888) int[0119905]

119865 (120591) 119889120583 (120591)

=

119899

sum

119894=1

[120583 ([119909119894 119905]) minus 120583 ([119909

119894+1 119905])] 119865 (119909

1015840

119894)

=

119899

sum

119894=1

minus [120583 ([119909119894 119905]) minus 120583 ([119909

119894+1 119905])]

Δ119909119894

119865 (1199091015840

119894) Δ119909119894

= (119870)int

119905

0

[minus1205831015840([120591 119905])] 119865 (120591) 119889120591

(73)

5 Choquet Integral Equations ofFuzzy-Number-Valued Functions

In this section let us consider a Choquet integral equationbased onTheorem 24 as shown below Given continuous andincreasing fuzzy-number-valued functions 119866(119905) with 119866(0) =

0 let us find continuous and increasing fuzzy-number-valuedfunctions 119865(119905) such that

119866 (119905) = (119888) int[0119905]

119865 (120591) 119889120583119898

(119905) (74)

which is expressed as

119866 (119905) = (119870)int

119905

0

1198981015840(119905 minus 120591) 119865 (120591) 119889120591 (75)

Definition 27 Let 119891(119905) be a real-valued function defined on[0 +infin) The following

F (119904) = int

+infin

0

119890minus119904119905

119891 (119905) 119889119905 (76)

is said to be the Laplace transformation of 119891(119905) if the infiniteintegralint+infin

0119890minus119904119905

119891(119905)119889119905 is convergentwith respect to the valueof parameter 119904 We denote its Laplace transformation asF(119904) = 119871[119891(119905)] and the inverse Laplace transformation as119891(119905) = 119871

minus1[F(119904)]

Definition 28 Let119865(119909 119910) be a fuzzy-number-valued functiondefined on [0 +infin) times [0 +infin) (119870) int

infin

0119865(119909 119910)119889119909 is said to be

convergent with respect to the parameter 119910 if for every fixed119910 isin [0 +infin) (119870) int

+infin

0119865(119909 119910)119889119909 exists

Definition 29 Let 119865(119905) be a fuzzy-number-valued functiondefined on [0 +infin) The following

F (119904) = (119870)int[0+infin)

119890minus119904119905

119865 (119905) 119889119905 (77)

is said to be the Laplace transformation of 119865(119905) if the Kalevaintegral intinfin

0119890minus119904119905

119865(119905)119889119905 is convergent with respect to the valueof parameter 119904 We denote its Laplace transformation asF(119904) = 119871[119865(119905)] and the inverse Laplace transformation as119865(119905) = 119871

minus1[F(119904)]

Remark 30 In view of Theorem 8 we have

F120582(119904) = [int

+infin

0

119890minus119904119905

119865120582(119905) 119889119905 int

+infin

0

119890minus119904119905

119865120582(119905) 119889119905] (78)

for all 120582 isin [0 1]

Theorem 31 Let 119866(119905) = (119888) int[0119905]

119865(120591)119889120583119898(119905) Then

G (119904) = 119904M (119904) F (119904)

119866 (119904) = 119871minus1

[119904M (119904) F (119904)]

(79)

where G(119904) = 119871[119866(119905)]M(s) = 119871[119898(119905)] and F(119904) = 119871[119865(119905)]

Proof Since 119866(119905) = (119888) int[0119905]

119865(120591)119889120583119898(119905) = (119870) int

119905

01198981015840(119905 minus

120591)119865(120591)119889120591 it follows easily from the condition that

G120582(119904) = [int

+infin

0

119890minus119904119905

119866120582(119905) 119889119905 int

+infin

0

119890minus119904119905

119866120582(119905) 119889119905]

= [int

+infin

0

119890minus119904119905

int

119905

0

1198981015840(119905 minus 120591) 119865

120582(120591) 119889120591 119889119905

int

+infin

0

119890minus119904119905

int

119905

0

1198981015840(119905 minus 120591) 119865

120582(120591) 119889120591 119889119905]

(80)

for all 120582 isin [0 1]Notice that

int

+infin

0

119890minus119904119905

int

119905

0

1198981015840(119905 minus 120591) 119865

120582(120591) 119889120591 119889119905

= int

+infin

0

int

+infin

120591

119890minus119904119905

1198981015840(119905 minus 120591) 119865

120582(120591) 119889119905 119889120591

= int

+infin

0

119865120582(120591) int

+infin

120591

119890minus119904119905

1198981015840(119905 minus 120591) 119889119905 119889120591

= int

+infin

0

119865120582(120591) [119890

minus119904119905119898(119905 minus 120591)

10038161003816100381610038161003816

+infin

120591minus int

+infin

120591

119898(119905 minus 120591) 119889119890minus119904119905

] 119889120591

= int

+infin

0

119865120582(120591) [minusint

+infin

120591

119898(119905 minus 120591) (minus119904) 119890minus119904119905

119889119905] 119889120591

= 119904int

+infin

0

119865120582(120591) [int

+infin

0

119898(119905 minus 120591) (minus119904) 119890minus119904(119905minus120591)

119890minus119904120591

119889 (119905 minus 120591)] 119889120591

= 119904int

+infin

0

119865120582(120591) [119890

minus119904120591int

+infin

0

119890minus119904(119905minus120591)

119898(119905 minus 120591) 119889 (119905 minus 120591)] 119889120591

= 119904int

+infin

0

119865120582(120591) 119890minus119904120591

M (119904) 119889120591

= 119904M (119904) int

+infin

0

119890minus119904120591

119865120582(120591) 119889120591 = 119904M (119904)F

120582(119904)

(81)

Following the same argument we can prove that

int

+infin

0

119890minus119904119905

int

119905

0

1198981015840(119905 minus 120591) 119865

120582(120591) 119889120591 119889119905 = 119904M (119904)F

120582(119904) (82)

10 Abstract and Applied Analysis

Hence

G (119904) = 119904M (119904) F (119904)

119866 (119904) = 119871minus1

[119904M (119904) F (119904)]

(83)

Theorem 32 For continuous and increasing fuzzy-valuedfunctions 119866(119905) with 119866(0) = 0

F (119904) =G (119904)

119904M (119904) 119865 (119904) = 119871

minus[

G (119904)

119904M (119904)] (84)

where G(119904) = 119871[119866(119905)]M(119904) = 119871[119898(119905)] and F(119904) = 119871[119865(119905)]

Proof It follows easily fromTheorem 31 that

G (119904) = 119904M (119904) F (119904)

119866 (119904) = 119871minus1

[119904M (119904) F (119904)]

(85)

It is obvious that

F (119904) =G (119904)

119904M (119904)

119865 (119904) = 119871minus1

[G (119904)

119904M (119904)]

(86)

where G(119904) = 119871[119866(119905)]M(119904) = 119871[119898(119905)] and F(119904) = 119871[119865(119905)]The Dirac delta function is defined by the properties

120575 (119905) = 0 for 119905 = 0

undefined at 119905 = 0(87)

and int+infin

minusinfin120575(119905)119889119905 = 1 that is the function has unit area

The Laplace transform of the Dirac Delta function 120575(119905) isshown as follows

119871 [120575 (119905)] = int

+infin

0

119890minus119904119905

120575 (119905) 119889119905 = 1 (88)

So

119871minus1

[1] = 120575 (119905) (89)

Example 33 Let 119898(119905) = 119905 and let the membership functionof 119866(119905) be

119898119866(119905)

(119909) = 119909 119909 isin [0 1]

0 119909 notin [0 1] 119905 le 1 (90)

Then by Theorem 32 we can solve the Choquet integralequation

119866 (119905) = (119888) int[0119905]

119865 (120591) 119889120583119898

(119905) (91)

Indeed let 119898(119905) = 119905 thenM(119904) = 11199042 The 120582-cut of 119866(119905) is

represented by interval 119866120582(119905) = 119909 | 119898

119866(119905)(119909) ge 120582 = [120582 1]

that is 119866120582(119905) = 120582 119866

120582(119905) = 1 In view of Remark 30 we have

G120582(119904) = [int

+infin

0119890minus119904119905

119866120582(119905)119889119905 int

+infin

0119890minus119904119905

119866120582(119905)119889119905] = [120582119878 1119878]

Then

F120582(119904) =

G120582(119904)

119904M (119904)= 120582

F120582(119904) =

G120582(119904)

119904M (119904)= 1

(92)

Furthermore

119865120582(119905) = 119871

minus1(F120582(119904)) = 120582120575 (119905)

119865120582(119905) = 119871

minus1(F120582(119904)) = 120575 (119905)

(93)

Therefore for every 120582 isin [0 1]

119865120582(119905) = [119865

120582(119905) 119865120582(119905)] = [120582120575 (119905) 120575 (119905)] (94)

Then we have the membership function of 119865(119905) as

119898119865(119905)

(119909) =

119909

120575 (119905) 119909 isin [0 120575 (119905)] 119905 = 0

0 otherwise(95)

6 Conclusions and Remarks

In this paper we have considered the Choquet integrals offuzzy-number-valued functions based on the nonnegativereal lineWe have discussed the Choquet integrals of interval-valued functions and fuzzy-number-valued functions basedon nonadditive Sugeno measures and showed the repre-sentation theorem of them respectively We also gave theoperational schemes of above several classes of integralson discrete sets Then we have given a representation ofthe Choquet integral of a nonnegative continuous andincreasing fuzzy-number-valued function with respect to afuzzymeasure In addition in order to solve Choquet integralequations of fuzzy-number-valued function a concept of theLaplace transformation for calculation has been introducedFor distorted Lebesgue measures it was shown that Choquetintegral equations of fuzzy-number-valued function could besolved by the Laplace transformation Finally we have givenan example to illustrate themain result at the end of the paper

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The work is supported by the Natural Scientific Funds ofChina (61262022) the Natural Scientific Fund of GansuProvince of China (1208RJZA251) Lanzhou Jiaotong Univer-sity Young Scientific Research Fund Project (2011020) theFundamental Research Funds for Gansu Province Univer-sities (no 213060) and the Scientific Research Project ofNorthwestNormalUniversity (noNWNU-KJCXGC-03-61)

Abstract and Applied Analysis 11

References

[1] G Choquet ldquoTheory of capacitiesrdquoAnnales de lrsquoInstitut Fouriervol 5 pp 131ndash295 1954

[2] D Denneberg Non-Additive Measure and Integral KluwerAcademic Publishers Boston Mass USA 1994

[3] T Murofushi M Sugeno and M Machida ldquoNon-monotonicfuzzy measures and the Choquet integralrdquo Fuzzy Sets andSystems vol 64 no 1 pp 73ndash86 1994

[4] Z Y Wang K-S Leung M-L Wong and J Fang ldquoA new typeof nonlinear integrals and the computational algorithmrdquo FuzzySets and Systems vol 112 no 2 pp 223ndash231 2000

[5] D Schmeidler ldquoSubjective probability and expected utilitywithout additivityrdquo Econometrica vol 57 no 3 pp 571ndash5871989

[6] M Grabisch ldquo119896-order additive discrete fuzzy measures andtheir representationrdquo Fuzzy Sets and Systems vol 92 no 2 pp167ndash189 1997

[7] MGrabisch I Kojadinovic and PMeyer ldquoA review ofmethodsfor capacity identification in Choquet integral based multi-attribute utility theory applications of the Kappalab R packagerdquoEuropean Journal of Operational Research vol 186 no 2 pp766ndash785 2008

[8] K Xu Z Wang P Heng and K Leung ldquoClassification bynonlinear integral projectionsrdquo IEEE Transactions on FuzzySystems vol 11 no 2 pp 187ndash201 2003

[9] M J Bolanos L M de Campos Ibanez and A GonzalezMunoz ldquoConvergence properties of the monotone expectationand its application to the extension of fuzzy measuresrdquo FuzzySets and Systems vol 33 no 2 pp 201ndash212 1989

[10] T Murofushi and M Sugeno ldquoA theory of fuzzy measuresrepresentations the Choquet integral and null setsrdquo Journal ofMathematical Analysis and Applications vol 159 no 2 pp 532ndash549 1991

[11] Z Wang ldquoConvergence theorems for sequences of Choquetintegralsrdquo International Journal of General Systems vol 26 no1-2 pp 133ndash143 1997

[12] YOuyang andHZhang ldquoOn the space ofmeasurable functionsand its topology determined by the Choquet integralrdquo Interna-tional Journal of Approximate Reasoning vol 52 no 9 pp 1355ndash1362 2011

[13] J Kawabe ldquoThe Choquet integral representability of comono-tonically additive functionals in locally compact spacesrdquo Inter-national Journal of Approximate Reasoning vol 54 no 3 pp418ndash426 2013

[14] S Abbasbandy E Babolian and M Alavi ldquoNumerical methodfor solving linear Fredholm fuzzy integral equations of thesecond kindrdquo Chaos Solitons amp Fractals vol 31 no 1 pp 138ndash146 2007

[15] O Kaleva ldquoFuzzy differential equationsrdquo Fuzzy Sets and Sys-tems vol 24 no 3 pp 301ndash317 1987

[16] J Mordeson and W Newman ldquoFuzzy integral equationsrdquoInformation Sciences vol 87 no 4 pp 215ndash229 1995

[17] M Sugeno ldquoA note on derivatives of functions with respect tofuzzy measuresrdquo Fuzzy Sets and Systems vol 222 pp 1ndash17 2013

[18] I Kojadinovic J-L Marichal and M Roubens ldquoAn axiomaticapproach to the definition of the entropy of a discrete Choquetcapacityrdquo Information Sciences vol 172 no 1-2 pp 131ndash1532005

[19] E Pap Null-Additive Set Functions vol 337 of Mathematicsand Its Applications KluwerAcademic Publishers LondonUK1994

[20] M Sugeno Theory of fuzzy integrals and its applications [PhDdissertation] Tokyo Institute of Technology Tokyo Japan 1974

[21] C X Wu D L Zhang and C M Guo ldquoFuzzy number fuzzymeasures and fuzzy integralsrdquo Fuzzy Sets and Systems vol 98no 3 pp 355ndash360 1998

[22] C X Wu and M Ma The Foundament of Fuzzy AnalysisNational Defense Press Beijing China 1991 (Chinese)

[23] R Yang Z Y Wang P A Heng and K S Leung ldquoFuzzynumbers and fuzzification of the Choquet integralrdquo Fuzzy Setsand Systems vol 153 no 1 pp 95ndash113 2005

[24] S Seikkala ldquoOn the fuzzy initial value problemrdquo Fuzzy Sets andSystems vol 24 no 3 pp 319ndash330 1987

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Abstract and Applied Analysis

important conclusions have been obtained [17] Motivated bythis we will discuss the representation of Choquet integral offuzzy-number-valued functions in this section

Definition 12 Fuzzy-number-valued function 119865 119883 rarr +

is said to be continuous on 119883 if for every 1199090isin 119883 there are

lim119909rarr119909

0

119865120582(119909) = 119865

120582(1199090)

lim119909rarr119909

0

119865120582(119909) = 119865

120582(1199090)

(39)

where 120582 isin [0 1]

Definition 13 Fuzzy-number-valued function 119865 119883 rarr +

is said to be increasing on 119883 if for every 1199091

le 1199092there are

119865120582(1199091) le 119865120582(1199092) and 119865

120582(1199091) le 119865120582(1199092) where 119909

1 1199092isin 119883 and

120582 isin [0 1]LetF+ be a class ofmeasurable nonnegative continuous

and increasing fuzzy-number-valued functionsLet ] be a Lebesgue measure for [119886 119887] sub [0infin) ][119886 119887] =

119887 minus 119886

Definition 14 (see [17]) Let 119898 119877+

rarr 119877+ be a continuous

and increasing function and 119898(0) = 0 A fuzzy measure 120583119898

a distorted Lebesgue measure is defined by 120583119898(sdot) = 119898(](sdot))

Definition 15 (see [24]) A fuzzy-number-valued function 119865

[119886 119887] rarr is differentiable at 119909 isin [119886 119887] if there exist fuzzynumber-valued functions 1198651015840(119909) such that

[1198651015840(119909)]120582= [1198651015840

120582(119909) 119865

1015840

120582(119909)] (40)

for every 120582 isin [0 1] where 1198651015840(119909) is the fuzzy derivative of

119865(119909)Note that 120583

119898is induced from the Lebesguemeasure ] by a

monotone transformation where 120583119898([119886 119887]) = 119898(]([119886 119887])) =

119898(119887 minus 119886) Apparently it loses additivity unless 119898(119905) = 119905 butreserves monotonicity In what follows we assume that 119898(119905)

is differentiableIn this section we consider the calculation of Choquet

integrals Let 120583 be a general fuzzy measure and consider120583([120591 119905]) for a closed interval [120591 119905] then 120583([120591 119905]) is decreasingfor 120591 and increasing for 119905 Throughout the paper we assumethat the functions 119898(119905) 119865(119905) and 119866(119905) are continuouslydifferentiable We also assume that 120583([120591 119905]) is continuouslydifferentiable with respect to 120591 on [0 119905] for every 119905 gt 0 Inaddition we require the regularity condition that 120583(119905) = 0

holds for every 119905 ge 0 We write 1205831015840([120591 119905]) = (120597120597120591)120583([120591 119905])

where we note that 1205831015840([120591 119905]) le 0 for 120591 le 119905 If 120583 = 120583119898then

1205831015840([120591 119905]) = minus119898

1015840(119905 minus 120591) where 119898

1015840(119905) = 119889119898(119905)119889119905 First we

consider a case that 119865(119905) is strictly increasingFuzzy-number-valued function 119865 119883 rarr is said to

be 119871-integrally bounded if there exists a Lebesgue integrablefunction ℎ 119883 rarr 119877

+ such that || le ℎ(119905) for every section isin [119865(119905)]

0

Definition 16 (see [22]) Let fuzzy-number-valued functions119865 [119886 +infin) rarr be measurable and 119871-integrally bounded119865 is called Kaleva-integrable if

[int

+infin

119886

119865(119905)119889119905]

120582

= (119871) int

+infin

119886

119892 (119905) 119889119905 | 119892 (119905) isin 119878[119865(119905)]

120582

0 le 120582 le 1

(41)

determines a unique fuzzy number 119886 isin which is denotedby (119870) int

+infin

119886119865(119905) 119889120583 = 119886 where 119878

[119865(119905)]120582

= 119892(119905) isin [119865(119905)]120582is a

measurable selection

Lemma 17 (see [22]) Let 119865 [119886 +infin) rarr be measurableand C-integrally bounded fuzzy-number-valued function then119865 is Kaleva integrable on [119886 +infin) if and only if 119865

120582(119905) and119865

120582(119905)

are 119871-integrable on [119886 +infin) and

[(119870)int

119905

0

119865(120591)119889120591]

120582

= [(119871) int

119905

0

119865120582(120591) 119889120591 (119871) int

119905

0

119865120582(120591) 119889120591]

(42)

for any 120582 isin [0 1]

Theorem 18 Let 119865(119905) isin F+ Then minus1205831015840([120591 119905])119865(120591) is Kaleva

integrable on [0 119905] and

[(119870)int

119905

0

[minus1205831015840([120591 119905])]119865(120591)119889120591]

120582

= [(119871) int

119905

0

[minus1205831015840([120591 119905])] 119865

120582(120591) 119889120591

(119871) int

119905

0

[minus1205831015840([120591 119905])] 119865

120582(120591) 119889120591]

(43)

Proof Since119865(119905) isin F+ for every120582 isin [0 1]we get that119865120582and

119865120582are measurable nonnegative continuous and increasing

real-valued functions on [0 119905] It follows that 119865120582and 119865

120582are

119871-integrable on [0 119905] 120583([120591 119905]) is 119871-integrable on [0 119905] as itis continuously differentiable with respect to 120591 on [0 119905] forevery 119905 gt 0 In view of Lemma 17 [minus120583

1015840([120591 119905])]119865(120591) is Kaleva

integrable on [0 119905] and

[(119870)int

119905

0

[minus1205831015840([120591 119905])] 119865 (120591) 119889120591]

120582

= [(119871) int

119905

0

[minus1205831015840([120591 119905])] 119865

120582(120591) 119889120591

(119871) int

119905

0

[minus1205831015840([120591 119905])] 119865

120582(120591) 119889120591]

(44)

Lemma 19 (see [17]) Let 119891(119905) be a real strictly increasingfunction Then the Choquet integral of 119891 with respect to 120583 on[0 119905] is represented as

(119888) int[0119905]

119891 (120591) 119889120583 (120591) = minusint

119905

0

1205831015840([120591 119905]) 119891 (120591) 119889120591 (45)

Abstract and Applied Analysis 7

In particular for 120583 = 120583119898

(119888) int[0119905]

119891 (120591) 119889120583119898

(120591) = int

119905

0

1198981015840(119905 minus 120591) 119891 (120591) 119889120591 (46)

Lemma 20 (see [17]) Let 119891(119905) be a constant real-valuedfunction 119891(119905) = 119862 forall119905 isin 119877

+= (0 +infin) Then

(119888) int[0119905]

119891 (120591) 119889120583 (120591) = minusint

119905

0

1205831015840([120591 119905]) 119862119889120591 (47)

In particular for 120583 = 120583119898

(119888) int[0119905]

119891 (120591) 119889120583119898

(120591) = 119862119898 (119905) (48)

Lemma 21 (see [17]) Let 119891(119905) be a measurable nonnegativecontinuous and increasing real-valued function Then theChoquet integral of 119891 with respect to 120583 on [0 119905] is representedas

(119888) int[0119905]

119891 (120591) 119889120583 (120591) = int

infin

0

120583 (120591 | 119891 (120591) ge 119903 cap [0 119905]) 119889119903

= minusint

119905

0

1205831015840([120591 119905]) 119891 (120591) 119889120591

(49)

In particular for 120583 = 120583119898

(119888) int[0119905]

119891 (120591) 119889120583119898

(120591) = int

infin

0

120583 (120591 | 119891 (120591) ge 119903 cap [0 119905]) 119889119903

= int

119905

0

1198981015840(119905 minus 120591) 119891 (120591) 119889120591

(50)

Theorem 22 Let 119865(119905) be a strictly increasing fuzzy-number-valued functionThen the Choquet integral of 119865with respect to120583 on [0 119905] is represented as

(119888) int[0119905]

119865 (120591) 119889120583 (120591) = (119870)int

119905

0

[minus1205831015840([120591 119905])] 119865 (120591) 119889120591 (51)

In particular for 120583 = 120583119898

(119888) int[0119905]

119865 (120591) 119889120583119898

(120591) = (119870)int

119905

0

1198981015840(119905 minus 120591) 119865 (120591) 119889120591 (52)

Proof ByTheorem 8 we have

[(119888) int[0119905]

119865 (120591) 119889120583 (120591)]

120582

= [(119888) int[0119905]

119865120582(120591) 119889120583 (120591) (119888) int

[0119905]

119865120582(120591) 119889120583 (120591)]

(53)

for any 120582 isin [0 1] Since 119865(119905) is strictly increasing 119865120582(119905) and

119865120582(119905) are strictly increasing real-valued functions respec-

tively In view of Lemma 19 we have

(119888) int[0119905]

119865120582(120591) 119889120583 (120591) = minusint

119905

0

1205831015840([120591 119905]) 119865

120582(120591) 119889120591

(119888) int[0119905]

119865120582(120591) 119889120583 (120591) = minusint

119905

0

1205831015840([120591 119905]) 119865

120582(120591) 119889120591

(54)

It follows that

[(119888) int[0119905]

119865(120591)119889120583(120591)]

120582

= [minusint

119905

0

1205831015840([120591 119905]) 119865

120582(120591) 119889120591 minusint

119905

0

1205831015840([120591 119905]) 119865120582 (120591) 119889120591]

(55)

On the other hand

int

119905

0

[minus1205831015840([120591 119905])] 119865120582 (120591) 119889120591

= int

119905

0

[minus1205831015840([120591 119905])] [119865

120582(120591) 119865

120582(120591)] 119889120591

= [minusint

119905

0

1205831015840([120591 119905]) 119865

120582(120591) 119889120591 minusint

119905

0

1205831015840([120591 119905]) 119865120582 (120591) 119889120591]

(56)

Therefore

[(119888) int[0119905]

119865(120591)119889120583(120591)]

120582

= int

119905

0

[minus1205831015840([120591 119905])] 119865120582 (120591) 119889120591 (57)

By the arbitrary of 120582 we have

(119888) int[0119905]

119865 (120591) 119889120583 (120591) = (119870)int

119905

0

[minus1205831015840([120591 119905])] 119865 (120591) 119889120591 (58)

For 120583 = 120583119898 we note that minus120583

1015840([120591 119905]) = 119898

1015840(119905 minus 120591) hence

(119888) int[0119905]

119865 (120591) 119889120583119898

(120591) = (119870)int

119905

0

1198981015840(119905 minus 120591) 119865 (120591) 119889120591 (59)

Theorem 23 Let 119865(119905) be a constant fuzzy-valued functionthat is 119865(119905) = 119862 forall119905 isin 119877

+= (0 +infin) Then

(119888) int[0119905]

119865 (120591) 119889120583 (120591)

= (119870)int

119905

0

[minus1205831015840([120591 119905])] 119862119889120591 = 119862120583 ([0 119905])

(60)

In particular for 120583 = 120583119898

(119888) int[0119905]

119865 (120591) 119889120583119898

(120591) = 119862119898 (119905) (61)

Proof Note that119862 = [119862 119862] where119862 and119862 are constant real-valued functions respectively In view of Lemma 20 we have

(119888) int[0119905]

119862119889120583 (120591) = minusint

119905

0

1205831015840([120591 119905]) 119862119889120591

(119888) int[0119905]

119862119889120583 (120591) = minusint

119905

0

1205831015840([120591 119905]) 119862119889120591

(62)

8 Abstract and Applied Analysis

The rest of the proof follows in exactly the same way as thatof Theorem 22 Hence

(119888) int[0119905]

119865 (120591) 119889120583 (120591)

= (119870)int

119905

0

[minus1205831015840([120591 119905])] 119862119889120591 = 119862120583 ([0 119905])

(63)

For 120583 = 120583119898 we obtain 120583

1015840([120591 119905]) = minus119898

1015840(119905 minus 120591) It follows that

(119888) int[0119905]

119862119889120583 (120591) = (119870)int

119905

0

[minus1205831015840([120591 119905])] 119862119889120591

= int

119905

0

1198981015840(119905 minus 120591) 119862119889120591

= (119870)119862int

119905

0

1198981015840(119905 minus 120591) 119889120591 = 119862119898 (119905)

(64)

where 119898(0) = 0

Theorem 24 Let 119865(119905) isin F+ Then the Choquet integral of 119865with respect to 120583 on [0 119905] is represented as

(119888) int[0119905]

119865 (120591) 119889120583 (120591) = (119870)int

119905

0

[minus1205831015840([120591 119905])] 119865 (120591) 119889120591 (65)

In particular for 120583 = 120583119898

(119888) int[0119905]

119865 (120591) 119889120583119898

(120591) = (119870)int

119905

0

1198981015840(119905 minus 120591) 119865 (120591) 119889120591 (66)

Proof The theorem has been proved when 119865(119905) is a strictlyincreasing or constant fuzzy-number-valued function byTheorem 22 or Theorem 23 Without loss of generality weconsider a continuous and increasing function such that

119865 (119905) =

1198651(119905) 0 le 119905 lt 119905

1

1198652(119905) 119905

1le 119905 lt 119905

2

1198653(119905) 119905

2le 119905 lt infin

(67)

where 1198651(119905) and 119865

3(119905) are strictly increasing 119865

2(119905) is constant

and 1198651(1199051) = 1198652(119905) = 119865

3(1199052)

(i) For 0 le 119905 lt 1199051

Note that 1198651(119905) is strictly increasing on [0 119905

1) from

Theorem 22 we obtain

(119888) int[0119905]

119865 (120591) 119889120583 (120591) = (119870)int

119905

0

[minus1205831015840([120591 119905])] 1198651 (120591) 119889120591 (68)

(ii) For 1199051le 119905 lt 119905

2

In view of the construction of 119865(119905) on [0 119905] we have

(119888) int[0119905]

119865 (120591) 119889120583 (120591) = (119888) int[01199051]

1198651(120591) 119889120583 (120591)

+ (119888) int[1199051119905]

1198652(120591) 119889120583 (120591)

(69)

Since 1198651(119905) is strictly increasing on [0 119905

1) and 119865

2(119905) is

constant on [1199051 1199052) by Theorems 22 and 23

(119888) int[0119905]

119865 (120591) 119889120583 (120591) = (119870)int

1199051

0

[minus1205831015840([120591 119905])] 119865

1(120591) 119889120591

+ (119870)int

119905

1199051

[minus1205831015840([120591 119905])] 119865

2(120591) 119889120591

= (119870)int

119905

0

[minus1205831015840([120591 119905])] 119865 (120591) 119889120591

(70)

(iii) For 1199052le 119905 lt infin

Since1198651(119905) is strictly increasing on [0 119905

1) 1198652(119905) is constant

on [1199051 1199052) and 119865

3(119905) is strictly increasing on [119905

1infin) by

Theorems 22 and 23 we have

(119888) int[0119905]

119865 (120591) 119889120583 (120591)

= (119888) int[01199051]

1198651(120591) 119889120583 (120591) + (119888) int

[11990511199052]

1198652(120591) 119889120583 (120591)

+ (119888) int[1199052119905]

1198653(120591) 119889120583 (120591)

= (119870)int

1199051

0

[minus1205831015840([120591 119905])] 119865

1(120591) 119889120591

+ (119870)int

1199052

1199051

[minus1205831015840([120591 119905])] 119865

2(120591) 119889120591

+ (119870)int

119905

1199052

[minus1205831015840([120591 119905])] 119865

3(120591) 119889120591

= (119870)int

119905

0

[minus1205831015840([120591 119905])] 119865 (120591) 119889120591

(71)

For 120583 = 120583119898 we obtain that

(119888) int[0119905]

119865 (120591) 119889120583119898

(120591) = (119870)int

119905

0

1198981015840(119905 minus 120591) 119865 (120591) 119889120591 (72)

Remark 25 Note that Theorem 24 holds only for a continu-ous and increasing 119865 but in the case 120583

119898= ] it holds for any

119865

Remark 26 Let us consider the representation of theChoquetintegral for a continuous case shown in Theorem 24 inrelation with a discrete case Now [0 119905] is transformed intothe discrete set 119909

1 1199092 119909

119899 Since119865 is an increasing fuzzy-

number-valued function we have 119909119894= 1199091015840

119894 119894 = 1 2 119899 in

Theorem 11 For the sake of simplicity let 0 = 1199090lt 1199091lt sdot sdot sdot lt

Abstract and Applied Analysis 9

119909119899

= 119905 119865(1199090) = 0 and [119909

119894 119905] = 119909

119894 119909119894+1

119909119899 It follows

that

(119888) int[0119905]

119865 (120591) 119889120583 (120591)

=

119899

sum

119894=1

[120583 ([119909119894 119905]) minus 120583 ([119909

119894+1 119905])] 119865 (119909

1015840

119894)

=

119899

sum

119894=1

minus [120583 ([119909119894 119905]) minus 120583 ([119909

119894+1 119905])]

Δ119909119894

119865 (1199091015840

119894) Δ119909119894

= (119870)int

119905

0

[minus1205831015840([120591 119905])] 119865 (120591) 119889120591

(73)

5 Choquet Integral Equations ofFuzzy-Number-Valued Functions

In this section let us consider a Choquet integral equationbased onTheorem 24 as shown below Given continuous andincreasing fuzzy-number-valued functions 119866(119905) with 119866(0) =

0 let us find continuous and increasing fuzzy-number-valuedfunctions 119865(119905) such that

119866 (119905) = (119888) int[0119905]

119865 (120591) 119889120583119898

(119905) (74)

which is expressed as

119866 (119905) = (119870)int

119905

0

1198981015840(119905 minus 120591) 119865 (120591) 119889120591 (75)

Definition 27 Let 119891(119905) be a real-valued function defined on[0 +infin) The following

F (119904) = int

+infin

0

119890minus119904119905

119891 (119905) 119889119905 (76)

is said to be the Laplace transformation of 119891(119905) if the infiniteintegralint+infin

0119890minus119904119905

119891(119905)119889119905 is convergentwith respect to the valueof parameter 119904 We denote its Laplace transformation asF(119904) = 119871[119891(119905)] and the inverse Laplace transformation as119891(119905) = 119871

minus1[F(119904)]

Definition 28 Let119865(119909 119910) be a fuzzy-number-valued functiondefined on [0 +infin) times [0 +infin) (119870) int

infin

0119865(119909 119910)119889119909 is said to be

convergent with respect to the parameter 119910 if for every fixed119910 isin [0 +infin) (119870) int

+infin

0119865(119909 119910)119889119909 exists

Definition 29 Let 119865(119905) be a fuzzy-number-valued functiondefined on [0 +infin) The following

F (119904) = (119870)int[0+infin)

119890minus119904119905

119865 (119905) 119889119905 (77)

is said to be the Laplace transformation of 119865(119905) if the Kalevaintegral intinfin

0119890minus119904119905

119865(119905)119889119905 is convergent with respect to the valueof parameter 119904 We denote its Laplace transformation asF(119904) = 119871[119865(119905)] and the inverse Laplace transformation as119865(119905) = 119871

minus1[F(119904)]

Remark 30 In view of Theorem 8 we have

F120582(119904) = [int

+infin

0

119890minus119904119905

119865120582(119905) 119889119905 int

+infin

0

119890minus119904119905

119865120582(119905) 119889119905] (78)

for all 120582 isin [0 1]

Theorem 31 Let 119866(119905) = (119888) int[0119905]

119865(120591)119889120583119898(119905) Then

G (119904) = 119904M (119904) F (119904)

119866 (119904) = 119871minus1

[119904M (119904) F (119904)]

(79)

where G(119904) = 119871[119866(119905)]M(s) = 119871[119898(119905)] and F(119904) = 119871[119865(119905)]

Proof Since 119866(119905) = (119888) int[0119905]

119865(120591)119889120583119898(119905) = (119870) int

119905

01198981015840(119905 minus

120591)119865(120591)119889120591 it follows easily from the condition that

G120582(119904) = [int

+infin

0

119890minus119904119905

119866120582(119905) 119889119905 int

+infin

0

119890minus119904119905

119866120582(119905) 119889119905]

= [int

+infin

0

119890minus119904119905

int

119905

0

1198981015840(119905 minus 120591) 119865

120582(120591) 119889120591 119889119905

int

+infin

0

119890minus119904119905

int

119905

0

1198981015840(119905 minus 120591) 119865

120582(120591) 119889120591 119889119905]

(80)

for all 120582 isin [0 1]Notice that

int

+infin

0

119890minus119904119905

int

119905

0

1198981015840(119905 minus 120591) 119865

120582(120591) 119889120591 119889119905

= int

+infin

0

int

+infin

120591

119890minus119904119905

1198981015840(119905 minus 120591) 119865

120582(120591) 119889119905 119889120591

= int

+infin

0

119865120582(120591) int

+infin

120591

119890minus119904119905

1198981015840(119905 minus 120591) 119889119905 119889120591

= int

+infin

0

119865120582(120591) [119890

minus119904119905119898(119905 minus 120591)

10038161003816100381610038161003816

+infin

120591minus int

+infin

120591

119898(119905 minus 120591) 119889119890minus119904119905

] 119889120591

= int

+infin

0

119865120582(120591) [minusint

+infin

120591

119898(119905 minus 120591) (minus119904) 119890minus119904119905

119889119905] 119889120591

= 119904int

+infin

0

119865120582(120591) [int

+infin

0

119898(119905 minus 120591) (minus119904) 119890minus119904(119905minus120591)

119890minus119904120591

119889 (119905 minus 120591)] 119889120591

= 119904int

+infin

0

119865120582(120591) [119890

minus119904120591int

+infin

0

119890minus119904(119905minus120591)

119898(119905 minus 120591) 119889 (119905 minus 120591)] 119889120591

= 119904int

+infin

0

119865120582(120591) 119890minus119904120591

M (119904) 119889120591

= 119904M (119904) int

+infin

0

119890minus119904120591

119865120582(120591) 119889120591 = 119904M (119904)F

120582(119904)

(81)

Following the same argument we can prove that

int

+infin

0

119890minus119904119905

int

119905

0

1198981015840(119905 minus 120591) 119865

120582(120591) 119889120591 119889119905 = 119904M (119904)F

120582(119904) (82)

10 Abstract and Applied Analysis

Hence

G (119904) = 119904M (119904) F (119904)

119866 (119904) = 119871minus1

[119904M (119904) F (119904)]

(83)

Theorem 32 For continuous and increasing fuzzy-valuedfunctions 119866(119905) with 119866(0) = 0

F (119904) =G (119904)

119904M (119904) 119865 (119904) = 119871

minus[

G (119904)

119904M (119904)] (84)

where G(119904) = 119871[119866(119905)]M(119904) = 119871[119898(119905)] and F(119904) = 119871[119865(119905)]

Proof It follows easily fromTheorem 31 that

G (119904) = 119904M (119904) F (119904)

119866 (119904) = 119871minus1

[119904M (119904) F (119904)]

(85)

It is obvious that

F (119904) =G (119904)

119904M (119904)

119865 (119904) = 119871minus1

[G (119904)

119904M (119904)]

(86)

where G(119904) = 119871[119866(119905)]M(119904) = 119871[119898(119905)] and F(119904) = 119871[119865(119905)]The Dirac delta function is defined by the properties

120575 (119905) = 0 for 119905 = 0

undefined at 119905 = 0(87)

and int+infin

minusinfin120575(119905)119889119905 = 1 that is the function has unit area

The Laplace transform of the Dirac Delta function 120575(119905) isshown as follows

119871 [120575 (119905)] = int

+infin

0

119890minus119904119905

120575 (119905) 119889119905 = 1 (88)

So

119871minus1

[1] = 120575 (119905) (89)

Example 33 Let 119898(119905) = 119905 and let the membership functionof 119866(119905) be

119898119866(119905)

(119909) = 119909 119909 isin [0 1]

0 119909 notin [0 1] 119905 le 1 (90)

Then by Theorem 32 we can solve the Choquet integralequation

119866 (119905) = (119888) int[0119905]

119865 (120591) 119889120583119898

(119905) (91)

Indeed let 119898(119905) = 119905 thenM(119904) = 11199042 The 120582-cut of 119866(119905) is

represented by interval 119866120582(119905) = 119909 | 119898

119866(119905)(119909) ge 120582 = [120582 1]

that is 119866120582(119905) = 120582 119866

120582(119905) = 1 In view of Remark 30 we have

G120582(119904) = [int

+infin

0119890minus119904119905

119866120582(119905)119889119905 int

+infin

0119890minus119904119905

119866120582(119905)119889119905] = [120582119878 1119878]

Then

F120582(119904) =

G120582(119904)

119904M (119904)= 120582

F120582(119904) =

G120582(119904)

119904M (119904)= 1

(92)

Furthermore

119865120582(119905) = 119871

minus1(F120582(119904)) = 120582120575 (119905)

119865120582(119905) = 119871

minus1(F120582(119904)) = 120575 (119905)

(93)

Therefore for every 120582 isin [0 1]

119865120582(119905) = [119865

120582(119905) 119865120582(119905)] = [120582120575 (119905) 120575 (119905)] (94)

Then we have the membership function of 119865(119905) as

119898119865(119905)

(119909) =

119909

120575 (119905) 119909 isin [0 120575 (119905)] 119905 = 0

0 otherwise(95)

6 Conclusions and Remarks

In this paper we have considered the Choquet integrals offuzzy-number-valued functions based on the nonnegativereal lineWe have discussed the Choquet integrals of interval-valued functions and fuzzy-number-valued functions basedon nonadditive Sugeno measures and showed the repre-sentation theorem of them respectively We also gave theoperational schemes of above several classes of integralson discrete sets Then we have given a representation ofthe Choquet integral of a nonnegative continuous andincreasing fuzzy-number-valued function with respect to afuzzymeasure In addition in order to solve Choquet integralequations of fuzzy-number-valued function a concept of theLaplace transformation for calculation has been introducedFor distorted Lebesgue measures it was shown that Choquetintegral equations of fuzzy-number-valued function could besolved by the Laplace transformation Finally we have givenan example to illustrate themain result at the end of the paper

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The work is supported by the Natural Scientific Funds ofChina (61262022) the Natural Scientific Fund of GansuProvince of China (1208RJZA251) Lanzhou Jiaotong Univer-sity Young Scientific Research Fund Project (2011020) theFundamental Research Funds for Gansu Province Univer-sities (no 213060) and the Scientific Research Project ofNorthwestNormalUniversity (noNWNU-KJCXGC-03-61)

Abstract and Applied Analysis 11

References

[1] G Choquet ldquoTheory of capacitiesrdquoAnnales de lrsquoInstitut Fouriervol 5 pp 131ndash295 1954

[2] D Denneberg Non-Additive Measure and Integral KluwerAcademic Publishers Boston Mass USA 1994

[3] T Murofushi M Sugeno and M Machida ldquoNon-monotonicfuzzy measures and the Choquet integralrdquo Fuzzy Sets andSystems vol 64 no 1 pp 73ndash86 1994

[4] Z Y Wang K-S Leung M-L Wong and J Fang ldquoA new typeof nonlinear integrals and the computational algorithmrdquo FuzzySets and Systems vol 112 no 2 pp 223ndash231 2000

[5] D Schmeidler ldquoSubjective probability and expected utilitywithout additivityrdquo Econometrica vol 57 no 3 pp 571ndash5871989

[6] M Grabisch ldquo119896-order additive discrete fuzzy measures andtheir representationrdquo Fuzzy Sets and Systems vol 92 no 2 pp167ndash189 1997

[7] MGrabisch I Kojadinovic and PMeyer ldquoA review ofmethodsfor capacity identification in Choquet integral based multi-attribute utility theory applications of the Kappalab R packagerdquoEuropean Journal of Operational Research vol 186 no 2 pp766ndash785 2008

[8] K Xu Z Wang P Heng and K Leung ldquoClassification bynonlinear integral projectionsrdquo IEEE Transactions on FuzzySystems vol 11 no 2 pp 187ndash201 2003

[9] M J Bolanos L M de Campos Ibanez and A GonzalezMunoz ldquoConvergence properties of the monotone expectationand its application to the extension of fuzzy measuresrdquo FuzzySets and Systems vol 33 no 2 pp 201ndash212 1989

[10] T Murofushi and M Sugeno ldquoA theory of fuzzy measuresrepresentations the Choquet integral and null setsrdquo Journal ofMathematical Analysis and Applications vol 159 no 2 pp 532ndash549 1991

[11] Z Wang ldquoConvergence theorems for sequences of Choquetintegralsrdquo International Journal of General Systems vol 26 no1-2 pp 133ndash143 1997

[12] YOuyang andHZhang ldquoOn the space ofmeasurable functionsand its topology determined by the Choquet integralrdquo Interna-tional Journal of Approximate Reasoning vol 52 no 9 pp 1355ndash1362 2011

[13] J Kawabe ldquoThe Choquet integral representability of comono-tonically additive functionals in locally compact spacesrdquo Inter-national Journal of Approximate Reasoning vol 54 no 3 pp418ndash426 2013

[14] S Abbasbandy E Babolian and M Alavi ldquoNumerical methodfor solving linear Fredholm fuzzy integral equations of thesecond kindrdquo Chaos Solitons amp Fractals vol 31 no 1 pp 138ndash146 2007

[15] O Kaleva ldquoFuzzy differential equationsrdquo Fuzzy Sets and Sys-tems vol 24 no 3 pp 301ndash317 1987

[16] J Mordeson and W Newman ldquoFuzzy integral equationsrdquoInformation Sciences vol 87 no 4 pp 215ndash229 1995

[17] M Sugeno ldquoA note on derivatives of functions with respect tofuzzy measuresrdquo Fuzzy Sets and Systems vol 222 pp 1ndash17 2013

[18] I Kojadinovic J-L Marichal and M Roubens ldquoAn axiomaticapproach to the definition of the entropy of a discrete Choquetcapacityrdquo Information Sciences vol 172 no 1-2 pp 131ndash1532005

[19] E Pap Null-Additive Set Functions vol 337 of Mathematicsand Its Applications KluwerAcademic Publishers LondonUK1994

[20] M Sugeno Theory of fuzzy integrals and its applications [PhDdissertation] Tokyo Institute of Technology Tokyo Japan 1974

[21] C X Wu D L Zhang and C M Guo ldquoFuzzy number fuzzymeasures and fuzzy integralsrdquo Fuzzy Sets and Systems vol 98no 3 pp 355ndash360 1998

[22] C X Wu and M Ma The Foundament of Fuzzy AnalysisNational Defense Press Beijing China 1991 (Chinese)

[23] R Yang Z Y Wang P A Heng and K S Leung ldquoFuzzynumbers and fuzzification of the Choquet integralrdquo Fuzzy Setsand Systems vol 153 no 1 pp 95ndash113 2005

[24] S Seikkala ldquoOn the fuzzy initial value problemrdquo Fuzzy Sets andSystems vol 24 no 3 pp 319ndash330 1987

Submit your manuscripts athttpwwwhindawicom

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Differential EquationsInternational Journal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 7

In particular for 120583 = 120583119898

(119888) int[0119905]

119891 (120591) 119889120583119898

(120591) = int

119905

0

1198981015840(119905 minus 120591) 119891 (120591) 119889120591 (46)

Lemma 20 (see [17]) Let 119891(119905) be a constant real-valuedfunction 119891(119905) = 119862 forall119905 isin 119877

+= (0 +infin) Then

(119888) int[0119905]

119891 (120591) 119889120583 (120591) = minusint

119905

0

1205831015840([120591 119905]) 119862119889120591 (47)

In particular for 120583 = 120583119898

(119888) int[0119905]

119891 (120591) 119889120583119898

(120591) = 119862119898 (119905) (48)

Lemma 21 (see [17]) Let 119891(119905) be a measurable nonnegativecontinuous and increasing real-valued function Then theChoquet integral of 119891 with respect to 120583 on [0 119905] is representedas

(119888) int[0119905]

119891 (120591) 119889120583 (120591) = int

infin

0

120583 (120591 | 119891 (120591) ge 119903 cap [0 119905]) 119889119903

= minusint

119905

0

1205831015840([120591 119905]) 119891 (120591) 119889120591

(49)

In particular for 120583 = 120583119898

(119888) int[0119905]

119891 (120591) 119889120583119898

(120591) = int

infin

0

120583 (120591 | 119891 (120591) ge 119903 cap [0 119905]) 119889119903

= int

119905

0

1198981015840(119905 minus 120591) 119891 (120591) 119889120591

(50)

Theorem 22 Let 119865(119905) be a strictly increasing fuzzy-number-valued functionThen the Choquet integral of 119865with respect to120583 on [0 119905] is represented as

(119888) int[0119905]

119865 (120591) 119889120583 (120591) = (119870)int

119905

0

[minus1205831015840([120591 119905])] 119865 (120591) 119889120591 (51)

In particular for 120583 = 120583119898

(119888) int[0119905]

119865 (120591) 119889120583119898

(120591) = (119870)int

119905

0

1198981015840(119905 minus 120591) 119865 (120591) 119889120591 (52)

Proof ByTheorem 8 we have

[(119888) int[0119905]

119865 (120591) 119889120583 (120591)]

120582

= [(119888) int[0119905]

119865120582(120591) 119889120583 (120591) (119888) int

[0119905]

119865120582(120591) 119889120583 (120591)]

(53)

for any 120582 isin [0 1] Since 119865(119905) is strictly increasing 119865120582(119905) and

119865120582(119905) are strictly increasing real-valued functions respec-

tively In view of Lemma 19 we have

(119888) int[0119905]

119865120582(120591) 119889120583 (120591) = minusint

119905

0

1205831015840([120591 119905]) 119865

120582(120591) 119889120591

(119888) int[0119905]

119865120582(120591) 119889120583 (120591) = minusint

119905

0

1205831015840([120591 119905]) 119865

120582(120591) 119889120591

(54)

It follows that

[(119888) int[0119905]

119865(120591)119889120583(120591)]

120582

= [minusint

119905

0

1205831015840([120591 119905]) 119865

120582(120591) 119889120591 minusint

119905

0

1205831015840([120591 119905]) 119865120582 (120591) 119889120591]

(55)

On the other hand

int

119905

0

[minus1205831015840([120591 119905])] 119865120582 (120591) 119889120591

= int

119905

0

[minus1205831015840([120591 119905])] [119865

120582(120591) 119865

120582(120591)] 119889120591

= [minusint

119905

0

1205831015840([120591 119905]) 119865

120582(120591) 119889120591 minusint

119905

0

1205831015840([120591 119905]) 119865120582 (120591) 119889120591]

(56)

Therefore

[(119888) int[0119905]

119865(120591)119889120583(120591)]

120582

= int

119905

0

[minus1205831015840([120591 119905])] 119865120582 (120591) 119889120591 (57)

By the arbitrary of 120582 we have

(119888) int[0119905]

119865 (120591) 119889120583 (120591) = (119870)int

119905

0

[minus1205831015840([120591 119905])] 119865 (120591) 119889120591 (58)

For 120583 = 120583119898 we note that minus120583

1015840([120591 119905]) = 119898

1015840(119905 minus 120591) hence

(119888) int[0119905]

119865 (120591) 119889120583119898

(120591) = (119870)int

119905

0

1198981015840(119905 minus 120591) 119865 (120591) 119889120591 (59)

Theorem 23 Let 119865(119905) be a constant fuzzy-valued functionthat is 119865(119905) = 119862 forall119905 isin 119877

+= (0 +infin) Then

(119888) int[0119905]

119865 (120591) 119889120583 (120591)

= (119870)int

119905

0

[minus1205831015840([120591 119905])] 119862119889120591 = 119862120583 ([0 119905])

(60)

In particular for 120583 = 120583119898

(119888) int[0119905]

119865 (120591) 119889120583119898

(120591) = 119862119898 (119905) (61)

Proof Note that119862 = [119862 119862] where119862 and119862 are constant real-valued functions respectively In view of Lemma 20 we have

(119888) int[0119905]

119862119889120583 (120591) = minusint

119905

0

1205831015840([120591 119905]) 119862119889120591

(119888) int[0119905]

119862119889120583 (120591) = minusint

119905

0

1205831015840([120591 119905]) 119862119889120591

(62)

8 Abstract and Applied Analysis

The rest of the proof follows in exactly the same way as thatof Theorem 22 Hence

(119888) int[0119905]

119865 (120591) 119889120583 (120591)

= (119870)int

119905

0

[minus1205831015840([120591 119905])] 119862119889120591 = 119862120583 ([0 119905])

(63)

For 120583 = 120583119898 we obtain 120583

1015840([120591 119905]) = minus119898

1015840(119905 minus 120591) It follows that

(119888) int[0119905]

119862119889120583 (120591) = (119870)int

119905

0

[minus1205831015840([120591 119905])] 119862119889120591

= int

119905

0

1198981015840(119905 minus 120591) 119862119889120591

= (119870)119862int

119905

0

1198981015840(119905 minus 120591) 119889120591 = 119862119898 (119905)

(64)

where 119898(0) = 0

Theorem 24 Let 119865(119905) isin F+ Then the Choquet integral of 119865with respect to 120583 on [0 119905] is represented as

(119888) int[0119905]

119865 (120591) 119889120583 (120591) = (119870)int

119905

0

[minus1205831015840([120591 119905])] 119865 (120591) 119889120591 (65)

In particular for 120583 = 120583119898

(119888) int[0119905]

119865 (120591) 119889120583119898

(120591) = (119870)int

119905

0

1198981015840(119905 minus 120591) 119865 (120591) 119889120591 (66)

Proof The theorem has been proved when 119865(119905) is a strictlyincreasing or constant fuzzy-number-valued function byTheorem 22 or Theorem 23 Without loss of generality weconsider a continuous and increasing function such that

119865 (119905) =

1198651(119905) 0 le 119905 lt 119905

1

1198652(119905) 119905

1le 119905 lt 119905

2

1198653(119905) 119905

2le 119905 lt infin

(67)

where 1198651(119905) and 119865

3(119905) are strictly increasing 119865

2(119905) is constant

and 1198651(1199051) = 1198652(119905) = 119865

3(1199052)

(i) For 0 le 119905 lt 1199051

Note that 1198651(119905) is strictly increasing on [0 119905

1) from

Theorem 22 we obtain

(119888) int[0119905]

119865 (120591) 119889120583 (120591) = (119870)int

119905

0

[minus1205831015840([120591 119905])] 1198651 (120591) 119889120591 (68)

(ii) For 1199051le 119905 lt 119905

2

In view of the construction of 119865(119905) on [0 119905] we have

(119888) int[0119905]

119865 (120591) 119889120583 (120591) = (119888) int[01199051]

1198651(120591) 119889120583 (120591)

+ (119888) int[1199051119905]

1198652(120591) 119889120583 (120591)

(69)

Since 1198651(119905) is strictly increasing on [0 119905

1) and 119865

2(119905) is

constant on [1199051 1199052) by Theorems 22 and 23

(119888) int[0119905]

119865 (120591) 119889120583 (120591) = (119870)int

1199051

0

[minus1205831015840([120591 119905])] 119865

1(120591) 119889120591

+ (119870)int

119905

1199051

[minus1205831015840([120591 119905])] 119865

2(120591) 119889120591

= (119870)int

119905

0

[minus1205831015840([120591 119905])] 119865 (120591) 119889120591

(70)

(iii) For 1199052le 119905 lt infin

Since1198651(119905) is strictly increasing on [0 119905

1) 1198652(119905) is constant

on [1199051 1199052) and 119865

3(119905) is strictly increasing on [119905

1infin) by

Theorems 22 and 23 we have

(119888) int[0119905]

119865 (120591) 119889120583 (120591)

= (119888) int[01199051]

1198651(120591) 119889120583 (120591) + (119888) int

[11990511199052]

1198652(120591) 119889120583 (120591)

+ (119888) int[1199052119905]

1198653(120591) 119889120583 (120591)

= (119870)int

1199051

0

[minus1205831015840([120591 119905])] 119865

1(120591) 119889120591

+ (119870)int

1199052

1199051

[minus1205831015840([120591 119905])] 119865

2(120591) 119889120591

+ (119870)int

119905

1199052

[minus1205831015840([120591 119905])] 119865

3(120591) 119889120591

= (119870)int

119905

0

[minus1205831015840([120591 119905])] 119865 (120591) 119889120591

(71)

For 120583 = 120583119898 we obtain that

(119888) int[0119905]

119865 (120591) 119889120583119898

(120591) = (119870)int

119905

0

1198981015840(119905 minus 120591) 119865 (120591) 119889120591 (72)

Remark 25 Note that Theorem 24 holds only for a continu-ous and increasing 119865 but in the case 120583

119898= ] it holds for any

119865

Remark 26 Let us consider the representation of theChoquetintegral for a continuous case shown in Theorem 24 inrelation with a discrete case Now [0 119905] is transformed intothe discrete set 119909

1 1199092 119909

119899 Since119865 is an increasing fuzzy-

number-valued function we have 119909119894= 1199091015840

119894 119894 = 1 2 119899 in

Theorem 11 For the sake of simplicity let 0 = 1199090lt 1199091lt sdot sdot sdot lt

Abstract and Applied Analysis 9

119909119899

= 119905 119865(1199090) = 0 and [119909

119894 119905] = 119909

119894 119909119894+1

119909119899 It follows

that

(119888) int[0119905]

119865 (120591) 119889120583 (120591)

=

119899

sum

119894=1

[120583 ([119909119894 119905]) minus 120583 ([119909

119894+1 119905])] 119865 (119909

1015840

119894)

=

119899

sum

119894=1

minus [120583 ([119909119894 119905]) minus 120583 ([119909

119894+1 119905])]

Δ119909119894

119865 (1199091015840

119894) Δ119909119894

= (119870)int

119905

0

[minus1205831015840([120591 119905])] 119865 (120591) 119889120591

(73)

5 Choquet Integral Equations ofFuzzy-Number-Valued Functions

In this section let us consider a Choquet integral equationbased onTheorem 24 as shown below Given continuous andincreasing fuzzy-number-valued functions 119866(119905) with 119866(0) =

0 let us find continuous and increasing fuzzy-number-valuedfunctions 119865(119905) such that

119866 (119905) = (119888) int[0119905]

119865 (120591) 119889120583119898

(119905) (74)

which is expressed as

119866 (119905) = (119870)int

119905

0

1198981015840(119905 minus 120591) 119865 (120591) 119889120591 (75)

Definition 27 Let 119891(119905) be a real-valued function defined on[0 +infin) The following

F (119904) = int

+infin

0

119890minus119904119905

119891 (119905) 119889119905 (76)

is said to be the Laplace transformation of 119891(119905) if the infiniteintegralint+infin

0119890minus119904119905

119891(119905)119889119905 is convergentwith respect to the valueof parameter 119904 We denote its Laplace transformation asF(119904) = 119871[119891(119905)] and the inverse Laplace transformation as119891(119905) = 119871

minus1[F(119904)]

Definition 28 Let119865(119909 119910) be a fuzzy-number-valued functiondefined on [0 +infin) times [0 +infin) (119870) int

infin

0119865(119909 119910)119889119909 is said to be

convergent with respect to the parameter 119910 if for every fixed119910 isin [0 +infin) (119870) int

+infin

0119865(119909 119910)119889119909 exists

Definition 29 Let 119865(119905) be a fuzzy-number-valued functiondefined on [0 +infin) The following

F (119904) = (119870)int[0+infin)

119890minus119904119905

119865 (119905) 119889119905 (77)

is said to be the Laplace transformation of 119865(119905) if the Kalevaintegral intinfin

0119890minus119904119905

119865(119905)119889119905 is convergent with respect to the valueof parameter 119904 We denote its Laplace transformation asF(119904) = 119871[119865(119905)] and the inverse Laplace transformation as119865(119905) = 119871

minus1[F(119904)]

Remark 30 In view of Theorem 8 we have

F120582(119904) = [int

+infin

0

119890minus119904119905

119865120582(119905) 119889119905 int

+infin

0

119890minus119904119905

119865120582(119905) 119889119905] (78)

for all 120582 isin [0 1]

Theorem 31 Let 119866(119905) = (119888) int[0119905]

119865(120591)119889120583119898(119905) Then

G (119904) = 119904M (119904) F (119904)

119866 (119904) = 119871minus1

[119904M (119904) F (119904)]

(79)

where G(119904) = 119871[119866(119905)]M(s) = 119871[119898(119905)] and F(119904) = 119871[119865(119905)]

Proof Since 119866(119905) = (119888) int[0119905]

119865(120591)119889120583119898(119905) = (119870) int

119905

01198981015840(119905 minus

120591)119865(120591)119889120591 it follows easily from the condition that

G120582(119904) = [int

+infin

0

119890minus119904119905

119866120582(119905) 119889119905 int

+infin

0

119890minus119904119905

119866120582(119905) 119889119905]

= [int

+infin

0

119890minus119904119905

int

119905

0

1198981015840(119905 minus 120591) 119865

120582(120591) 119889120591 119889119905

int

+infin

0

119890minus119904119905

int

119905

0

1198981015840(119905 minus 120591) 119865

120582(120591) 119889120591 119889119905]

(80)

for all 120582 isin [0 1]Notice that

int

+infin

0

119890minus119904119905

int

119905

0

1198981015840(119905 minus 120591) 119865

120582(120591) 119889120591 119889119905

= int

+infin

0

int

+infin

120591

119890minus119904119905

1198981015840(119905 minus 120591) 119865

120582(120591) 119889119905 119889120591

= int

+infin

0

119865120582(120591) int

+infin

120591

119890minus119904119905

1198981015840(119905 minus 120591) 119889119905 119889120591

= int

+infin

0

119865120582(120591) [119890

minus119904119905119898(119905 minus 120591)

10038161003816100381610038161003816

+infin

120591minus int

+infin

120591

119898(119905 minus 120591) 119889119890minus119904119905

] 119889120591

= int

+infin

0

119865120582(120591) [minusint

+infin

120591

119898(119905 minus 120591) (minus119904) 119890minus119904119905

119889119905] 119889120591

= 119904int

+infin

0

119865120582(120591) [int

+infin

0

119898(119905 minus 120591) (minus119904) 119890minus119904(119905minus120591)

119890minus119904120591

119889 (119905 minus 120591)] 119889120591

= 119904int

+infin

0

119865120582(120591) [119890

minus119904120591int

+infin

0

119890minus119904(119905minus120591)

119898(119905 minus 120591) 119889 (119905 minus 120591)] 119889120591

= 119904int

+infin

0

119865120582(120591) 119890minus119904120591

M (119904) 119889120591

= 119904M (119904) int

+infin

0

119890minus119904120591

119865120582(120591) 119889120591 = 119904M (119904)F

120582(119904)

(81)

Following the same argument we can prove that

int

+infin

0

119890minus119904119905

int

119905

0

1198981015840(119905 minus 120591) 119865

120582(120591) 119889120591 119889119905 = 119904M (119904)F

120582(119904) (82)

10 Abstract and Applied Analysis

Hence

G (119904) = 119904M (119904) F (119904)

119866 (119904) = 119871minus1

[119904M (119904) F (119904)]

(83)

Theorem 32 For continuous and increasing fuzzy-valuedfunctions 119866(119905) with 119866(0) = 0

F (119904) =G (119904)

119904M (119904) 119865 (119904) = 119871

minus[

G (119904)

119904M (119904)] (84)

where G(119904) = 119871[119866(119905)]M(119904) = 119871[119898(119905)] and F(119904) = 119871[119865(119905)]

Proof It follows easily fromTheorem 31 that

G (119904) = 119904M (119904) F (119904)

119866 (119904) = 119871minus1

[119904M (119904) F (119904)]

(85)

It is obvious that

F (119904) =G (119904)

119904M (119904)

119865 (119904) = 119871minus1

[G (119904)

119904M (119904)]

(86)

where G(119904) = 119871[119866(119905)]M(119904) = 119871[119898(119905)] and F(119904) = 119871[119865(119905)]The Dirac delta function is defined by the properties

120575 (119905) = 0 for 119905 = 0

undefined at 119905 = 0(87)

and int+infin

minusinfin120575(119905)119889119905 = 1 that is the function has unit area

The Laplace transform of the Dirac Delta function 120575(119905) isshown as follows

119871 [120575 (119905)] = int

+infin

0

119890minus119904119905

120575 (119905) 119889119905 = 1 (88)

So

119871minus1

[1] = 120575 (119905) (89)

Example 33 Let 119898(119905) = 119905 and let the membership functionof 119866(119905) be

119898119866(119905)

(119909) = 119909 119909 isin [0 1]

0 119909 notin [0 1] 119905 le 1 (90)

Then by Theorem 32 we can solve the Choquet integralequation

119866 (119905) = (119888) int[0119905]

119865 (120591) 119889120583119898

(119905) (91)

Indeed let 119898(119905) = 119905 thenM(119904) = 11199042 The 120582-cut of 119866(119905) is

represented by interval 119866120582(119905) = 119909 | 119898

119866(119905)(119909) ge 120582 = [120582 1]

that is 119866120582(119905) = 120582 119866

120582(119905) = 1 In view of Remark 30 we have

G120582(119904) = [int

+infin

0119890minus119904119905

119866120582(119905)119889119905 int

+infin

0119890minus119904119905

119866120582(119905)119889119905] = [120582119878 1119878]

Then

F120582(119904) =

G120582(119904)

119904M (119904)= 120582

F120582(119904) =

G120582(119904)

119904M (119904)= 1

(92)

Furthermore

119865120582(119905) = 119871

minus1(F120582(119904)) = 120582120575 (119905)

119865120582(119905) = 119871

minus1(F120582(119904)) = 120575 (119905)

(93)

Therefore for every 120582 isin [0 1]

119865120582(119905) = [119865

120582(119905) 119865120582(119905)] = [120582120575 (119905) 120575 (119905)] (94)

Then we have the membership function of 119865(119905) as

119898119865(119905)

(119909) =

119909

120575 (119905) 119909 isin [0 120575 (119905)] 119905 = 0

0 otherwise(95)

6 Conclusions and Remarks

In this paper we have considered the Choquet integrals offuzzy-number-valued functions based on the nonnegativereal lineWe have discussed the Choquet integrals of interval-valued functions and fuzzy-number-valued functions basedon nonadditive Sugeno measures and showed the repre-sentation theorem of them respectively We also gave theoperational schemes of above several classes of integralson discrete sets Then we have given a representation ofthe Choquet integral of a nonnegative continuous andincreasing fuzzy-number-valued function with respect to afuzzymeasure In addition in order to solve Choquet integralequations of fuzzy-number-valued function a concept of theLaplace transformation for calculation has been introducedFor distorted Lebesgue measures it was shown that Choquetintegral equations of fuzzy-number-valued function could besolved by the Laplace transformation Finally we have givenan example to illustrate themain result at the end of the paper

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The work is supported by the Natural Scientific Funds ofChina (61262022) the Natural Scientific Fund of GansuProvince of China (1208RJZA251) Lanzhou Jiaotong Univer-sity Young Scientific Research Fund Project (2011020) theFundamental Research Funds for Gansu Province Univer-sities (no 213060) and the Scientific Research Project ofNorthwestNormalUniversity (noNWNU-KJCXGC-03-61)

Abstract and Applied Analysis 11

References

[1] G Choquet ldquoTheory of capacitiesrdquoAnnales de lrsquoInstitut Fouriervol 5 pp 131ndash295 1954

[2] D Denneberg Non-Additive Measure and Integral KluwerAcademic Publishers Boston Mass USA 1994

[3] T Murofushi M Sugeno and M Machida ldquoNon-monotonicfuzzy measures and the Choquet integralrdquo Fuzzy Sets andSystems vol 64 no 1 pp 73ndash86 1994

[4] Z Y Wang K-S Leung M-L Wong and J Fang ldquoA new typeof nonlinear integrals and the computational algorithmrdquo FuzzySets and Systems vol 112 no 2 pp 223ndash231 2000

[5] D Schmeidler ldquoSubjective probability and expected utilitywithout additivityrdquo Econometrica vol 57 no 3 pp 571ndash5871989

[6] M Grabisch ldquo119896-order additive discrete fuzzy measures andtheir representationrdquo Fuzzy Sets and Systems vol 92 no 2 pp167ndash189 1997

[7] MGrabisch I Kojadinovic and PMeyer ldquoA review ofmethodsfor capacity identification in Choquet integral based multi-attribute utility theory applications of the Kappalab R packagerdquoEuropean Journal of Operational Research vol 186 no 2 pp766ndash785 2008

[8] K Xu Z Wang P Heng and K Leung ldquoClassification bynonlinear integral projectionsrdquo IEEE Transactions on FuzzySystems vol 11 no 2 pp 187ndash201 2003

[9] M J Bolanos L M de Campos Ibanez and A GonzalezMunoz ldquoConvergence properties of the monotone expectationand its application to the extension of fuzzy measuresrdquo FuzzySets and Systems vol 33 no 2 pp 201ndash212 1989

[10] T Murofushi and M Sugeno ldquoA theory of fuzzy measuresrepresentations the Choquet integral and null setsrdquo Journal ofMathematical Analysis and Applications vol 159 no 2 pp 532ndash549 1991

[11] Z Wang ldquoConvergence theorems for sequences of Choquetintegralsrdquo International Journal of General Systems vol 26 no1-2 pp 133ndash143 1997

[12] YOuyang andHZhang ldquoOn the space ofmeasurable functionsand its topology determined by the Choquet integralrdquo Interna-tional Journal of Approximate Reasoning vol 52 no 9 pp 1355ndash1362 2011

[13] J Kawabe ldquoThe Choquet integral representability of comono-tonically additive functionals in locally compact spacesrdquo Inter-national Journal of Approximate Reasoning vol 54 no 3 pp418ndash426 2013

[14] S Abbasbandy E Babolian and M Alavi ldquoNumerical methodfor solving linear Fredholm fuzzy integral equations of thesecond kindrdquo Chaos Solitons amp Fractals vol 31 no 1 pp 138ndash146 2007

[15] O Kaleva ldquoFuzzy differential equationsrdquo Fuzzy Sets and Sys-tems vol 24 no 3 pp 301ndash317 1987

[16] J Mordeson and W Newman ldquoFuzzy integral equationsrdquoInformation Sciences vol 87 no 4 pp 215ndash229 1995

[17] M Sugeno ldquoA note on derivatives of functions with respect tofuzzy measuresrdquo Fuzzy Sets and Systems vol 222 pp 1ndash17 2013

[18] I Kojadinovic J-L Marichal and M Roubens ldquoAn axiomaticapproach to the definition of the entropy of a discrete Choquetcapacityrdquo Information Sciences vol 172 no 1-2 pp 131ndash1532005

[19] E Pap Null-Additive Set Functions vol 337 of Mathematicsand Its Applications KluwerAcademic Publishers LondonUK1994

[20] M Sugeno Theory of fuzzy integrals and its applications [PhDdissertation] Tokyo Institute of Technology Tokyo Japan 1974

[21] C X Wu D L Zhang and C M Guo ldquoFuzzy number fuzzymeasures and fuzzy integralsrdquo Fuzzy Sets and Systems vol 98no 3 pp 355ndash360 1998

[22] C X Wu and M Ma The Foundament of Fuzzy AnalysisNational Defense Press Beijing China 1991 (Chinese)

[23] R Yang Z Y Wang P A Heng and K S Leung ldquoFuzzynumbers and fuzzification of the Choquet integralrdquo Fuzzy Setsand Systems vol 153 no 1 pp 95ndash113 2005

[24] S Seikkala ldquoOn the fuzzy initial value problemrdquo Fuzzy Sets andSystems vol 24 no 3 pp 319ndash330 1987

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Abstract and Applied Analysis

The rest of the proof follows in exactly the same way as thatof Theorem 22 Hence

(119888) int[0119905]

119865 (120591) 119889120583 (120591)

= (119870)int

119905

0

[minus1205831015840([120591 119905])] 119862119889120591 = 119862120583 ([0 119905])

(63)

For 120583 = 120583119898 we obtain 120583

1015840([120591 119905]) = minus119898

1015840(119905 minus 120591) It follows that

(119888) int[0119905]

119862119889120583 (120591) = (119870)int

119905

0

[minus1205831015840([120591 119905])] 119862119889120591

= int

119905

0

1198981015840(119905 minus 120591) 119862119889120591

= (119870)119862int

119905

0

1198981015840(119905 minus 120591) 119889120591 = 119862119898 (119905)

(64)

where 119898(0) = 0

Theorem 24 Let 119865(119905) isin F+ Then the Choquet integral of 119865with respect to 120583 on [0 119905] is represented as

(119888) int[0119905]

119865 (120591) 119889120583 (120591) = (119870)int

119905

0

[minus1205831015840([120591 119905])] 119865 (120591) 119889120591 (65)

In particular for 120583 = 120583119898

(119888) int[0119905]

119865 (120591) 119889120583119898

(120591) = (119870)int

119905

0

1198981015840(119905 minus 120591) 119865 (120591) 119889120591 (66)

Proof The theorem has been proved when 119865(119905) is a strictlyincreasing or constant fuzzy-number-valued function byTheorem 22 or Theorem 23 Without loss of generality weconsider a continuous and increasing function such that

119865 (119905) =

1198651(119905) 0 le 119905 lt 119905

1

1198652(119905) 119905

1le 119905 lt 119905

2

1198653(119905) 119905

2le 119905 lt infin

(67)

where 1198651(119905) and 119865

3(119905) are strictly increasing 119865

2(119905) is constant

and 1198651(1199051) = 1198652(119905) = 119865

3(1199052)

(i) For 0 le 119905 lt 1199051

Note that 1198651(119905) is strictly increasing on [0 119905

1) from

Theorem 22 we obtain

(119888) int[0119905]

119865 (120591) 119889120583 (120591) = (119870)int

119905

0

[minus1205831015840([120591 119905])] 1198651 (120591) 119889120591 (68)

(ii) For 1199051le 119905 lt 119905

2

In view of the construction of 119865(119905) on [0 119905] we have

(119888) int[0119905]

119865 (120591) 119889120583 (120591) = (119888) int[01199051]

1198651(120591) 119889120583 (120591)

+ (119888) int[1199051119905]

1198652(120591) 119889120583 (120591)

(69)

Since 1198651(119905) is strictly increasing on [0 119905

1) and 119865

2(119905) is

constant on [1199051 1199052) by Theorems 22 and 23

(119888) int[0119905]

119865 (120591) 119889120583 (120591) = (119870)int

1199051

0

[minus1205831015840([120591 119905])] 119865

1(120591) 119889120591

+ (119870)int

119905

1199051

[minus1205831015840([120591 119905])] 119865

2(120591) 119889120591

= (119870)int

119905

0

[minus1205831015840([120591 119905])] 119865 (120591) 119889120591

(70)

(iii) For 1199052le 119905 lt infin

Since1198651(119905) is strictly increasing on [0 119905

1) 1198652(119905) is constant

on [1199051 1199052) and 119865

3(119905) is strictly increasing on [119905

1infin) by

Theorems 22 and 23 we have

(119888) int[0119905]

119865 (120591) 119889120583 (120591)

= (119888) int[01199051]

1198651(120591) 119889120583 (120591) + (119888) int

[11990511199052]

1198652(120591) 119889120583 (120591)

+ (119888) int[1199052119905]

1198653(120591) 119889120583 (120591)

= (119870)int

1199051

0

[minus1205831015840([120591 119905])] 119865

1(120591) 119889120591

+ (119870)int

1199052

1199051

[minus1205831015840([120591 119905])] 119865

2(120591) 119889120591

+ (119870)int

119905

1199052

[minus1205831015840([120591 119905])] 119865

3(120591) 119889120591

= (119870)int

119905

0

[minus1205831015840([120591 119905])] 119865 (120591) 119889120591

(71)

For 120583 = 120583119898 we obtain that

(119888) int[0119905]

119865 (120591) 119889120583119898

(120591) = (119870)int

119905

0

1198981015840(119905 minus 120591) 119865 (120591) 119889120591 (72)

Remark 25 Note that Theorem 24 holds only for a continu-ous and increasing 119865 but in the case 120583

119898= ] it holds for any

119865

Remark 26 Let us consider the representation of theChoquetintegral for a continuous case shown in Theorem 24 inrelation with a discrete case Now [0 119905] is transformed intothe discrete set 119909

1 1199092 119909

119899 Since119865 is an increasing fuzzy-

number-valued function we have 119909119894= 1199091015840

119894 119894 = 1 2 119899 in

Theorem 11 For the sake of simplicity let 0 = 1199090lt 1199091lt sdot sdot sdot lt

Abstract and Applied Analysis 9

119909119899

= 119905 119865(1199090) = 0 and [119909

119894 119905] = 119909

119894 119909119894+1

119909119899 It follows

that

(119888) int[0119905]

119865 (120591) 119889120583 (120591)

=

119899

sum

119894=1

[120583 ([119909119894 119905]) minus 120583 ([119909

119894+1 119905])] 119865 (119909

1015840

119894)

=

119899

sum

119894=1

minus [120583 ([119909119894 119905]) minus 120583 ([119909

119894+1 119905])]

Δ119909119894

119865 (1199091015840

119894) Δ119909119894

= (119870)int

119905

0

[minus1205831015840([120591 119905])] 119865 (120591) 119889120591

(73)

5 Choquet Integral Equations ofFuzzy-Number-Valued Functions

In this section let us consider a Choquet integral equationbased onTheorem 24 as shown below Given continuous andincreasing fuzzy-number-valued functions 119866(119905) with 119866(0) =

0 let us find continuous and increasing fuzzy-number-valuedfunctions 119865(119905) such that

119866 (119905) = (119888) int[0119905]

119865 (120591) 119889120583119898

(119905) (74)

which is expressed as

119866 (119905) = (119870)int

119905

0

1198981015840(119905 minus 120591) 119865 (120591) 119889120591 (75)

Definition 27 Let 119891(119905) be a real-valued function defined on[0 +infin) The following

F (119904) = int

+infin

0

119890minus119904119905

119891 (119905) 119889119905 (76)

is said to be the Laplace transformation of 119891(119905) if the infiniteintegralint+infin

0119890minus119904119905

119891(119905)119889119905 is convergentwith respect to the valueof parameter 119904 We denote its Laplace transformation asF(119904) = 119871[119891(119905)] and the inverse Laplace transformation as119891(119905) = 119871

minus1[F(119904)]

Definition 28 Let119865(119909 119910) be a fuzzy-number-valued functiondefined on [0 +infin) times [0 +infin) (119870) int

infin

0119865(119909 119910)119889119909 is said to be

convergent with respect to the parameter 119910 if for every fixed119910 isin [0 +infin) (119870) int

+infin

0119865(119909 119910)119889119909 exists

Definition 29 Let 119865(119905) be a fuzzy-number-valued functiondefined on [0 +infin) The following

F (119904) = (119870)int[0+infin)

119890minus119904119905

119865 (119905) 119889119905 (77)

is said to be the Laplace transformation of 119865(119905) if the Kalevaintegral intinfin

0119890minus119904119905

119865(119905)119889119905 is convergent with respect to the valueof parameter 119904 We denote its Laplace transformation asF(119904) = 119871[119865(119905)] and the inverse Laplace transformation as119865(119905) = 119871

minus1[F(119904)]

Remark 30 In view of Theorem 8 we have

F120582(119904) = [int

+infin

0

119890minus119904119905

119865120582(119905) 119889119905 int

+infin

0

119890minus119904119905

119865120582(119905) 119889119905] (78)

for all 120582 isin [0 1]

Theorem 31 Let 119866(119905) = (119888) int[0119905]

119865(120591)119889120583119898(119905) Then

G (119904) = 119904M (119904) F (119904)

119866 (119904) = 119871minus1

[119904M (119904) F (119904)]

(79)

where G(119904) = 119871[119866(119905)]M(s) = 119871[119898(119905)] and F(119904) = 119871[119865(119905)]

Proof Since 119866(119905) = (119888) int[0119905]

119865(120591)119889120583119898(119905) = (119870) int

119905

01198981015840(119905 minus

120591)119865(120591)119889120591 it follows easily from the condition that

G120582(119904) = [int

+infin

0

119890minus119904119905

119866120582(119905) 119889119905 int

+infin

0

119890minus119904119905

119866120582(119905) 119889119905]

= [int

+infin

0

119890minus119904119905

int

119905

0

1198981015840(119905 minus 120591) 119865

120582(120591) 119889120591 119889119905

int

+infin

0

119890minus119904119905

int

119905

0

1198981015840(119905 minus 120591) 119865

120582(120591) 119889120591 119889119905]

(80)

for all 120582 isin [0 1]Notice that

int

+infin

0

119890minus119904119905

int

119905

0

1198981015840(119905 minus 120591) 119865

120582(120591) 119889120591 119889119905

= int

+infin

0

int

+infin

120591

119890minus119904119905

1198981015840(119905 minus 120591) 119865

120582(120591) 119889119905 119889120591

= int

+infin

0

119865120582(120591) int

+infin

120591

119890minus119904119905

1198981015840(119905 minus 120591) 119889119905 119889120591

= int

+infin

0

119865120582(120591) [119890

minus119904119905119898(119905 minus 120591)

10038161003816100381610038161003816

+infin

120591minus int

+infin

120591

119898(119905 minus 120591) 119889119890minus119904119905

] 119889120591

= int

+infin

0

119865120582(120591) [minusint

+infin

120591

119898(119905 minus 120591) (minus119904) 119890minus119904119905

119889119905] 119889120591

= 119904int

+infin

0

119865120582(120591) [int

+infin

0

119898(119905 minus 120591) (minus119904) 119890minus119904(119905minus120591)

119890minus119904120591

119889 (119905 minus 120591)] 119889120591

= 119904int

+infin

0

119865120582(120591) [119890

minus119904120591int

+infin

0

119890minus119904(119905minus120591)

119898(119905 minus 120591) 119889 (119905 minus 120591)] 119889120591

= 119904int

+infin

0

119865120582(120591) 119890minus119904120591

M (119904) 119889120591

= 119904M (119904) int

+infin

0

119890minus119904120591

119865120582(120591) 119889120591 = 119904M (119904)F

120582(119904)

(81)

Following the same argument we can prove that

int

+infin

0

119890minus119904119905

int

119905

0

1198981015840(119905 minus 120591) 119865

120582(120591) 119889120591 119889119905 = 119904M (119904)F

120582(119904) (82)

10 Abstract and Applied Analysis

Hence

G (119904) = 119904M (119904) F (119904)

119866 (119904) = 119871minus1

[119904M (119904) F (119904)]

(83)

Theorem 32 For continuous and increasing fuzzy-valuedfunctions 119866(119905) with 119866(0) = 0

F (119904) =G (119904)

119904M (119904) 119865 (119904) = 119871

minus[

G (119904)

119904M (119904)] (84)

where G(119904) = 119871[119866(119905)]M(119904) = 119871[119898(119905)] and F(119904) = 119871[119865(119905)]

Proof It follows easily fromTheorem 31 that

G (119904) = 119904M (119904) F (119904)

119866 (119904) = 119871minus1

[119904M (119904) F (119904)]

(85)

It is obvious that

F (119904) =G (119904)

119904M (119904)

119865 (119904) = 119871minus1

[G (119904)

119904M (119904)]

(86)

where G(119904) = 119871[119866(119905)]M(119904) = 119871[119898(119905)] and F(119904) = 119871[119865(119905)]The Dirac delta function is defined by the properties

120575 (119905) = 0 for 119905 = 0

undefined at 119905 = 0(87)

and int+infin

minusinfin120575(119905)119889119905 = 1 that is the function has unit area

The Laplace transform of the Dirac Delta function 120575(119905) isshown as follows

119871 [120575 (119905)] = int

+infin

0

119890minus119904119905

120575 (119905) 119889119905 = 1 (88)

So

119871minus1

[1] = 120575 (119905) (89)

Example 33 Let 119898(119905) = 119905 and let the membership functionof 119866(119905) be

119898119866(119905)

(119909) = 119909 119909 isin [0 1]

0 119909 notin [0 1] 119905 le 1 (90)

Then by Theorem 32 we can solve the Choquet integralequation

119866 (119905) = (119888) int[0119905]

119865 (120591) 119889120583119898

(119905) (91)

Indeed let 119898(119905) = 119905 thenM(119904) = 11199042 The 120582-cut of 119866(119905) is

represented by interval 119866120582(119905) = 119909 | 119898

119866(119905)(119909) ge 120582 = [120582 1]

that is 119866120582(119905) = 120582 119866

120582(119905) = 1 In view of Remark 30 we have

G120582(119904) = [int

+infin

0119890minus119904119905

119866120582(119905)119889119905 int

+infin

0119890minus119904119905

119866120582(119905)119889119905] = [120582119878 1119878]

Then

F120582(119904) =

G120582(119904)

119904M (119904)= 120582

F120582(119904) =

G120582(119904)

119904M (119904)= 1

(92)

Furthermore

119865120582(119905) = 119871

minus1(F120582(119904)) = 120582120575 (119905)

119865120582(119905) = 119871

minus1(F120582(119904)) = 120575 (119905)

(93)

Therefore for every 120582 isin [0 1]

119865120582(119905) = [119865

120582(119905) 119865120582(119905)] = [120582120575 (119905) 120575 (119905)] (94)

Then we have the membership function of 119865(119905) as

119898119865(119905)

(119909) =

119909

120575 (119905) 119909 isin [0 120575 (119905)] 119905 = 0

0 otherwise(95)

6 Conclusions and Remarks

In this paper we have considered the Choquet integrals offuzzy-number-valued functions based on the nonnegativereal lineWe have discussed the Choquet integrals of interval-valued functions and fuzzy-number-valued functions basedon nonadditive Sugeno measures and showed the repre-sentation theorem of them respectively We also gave theoperational schemes of above several classes of integralson discrete sets Then we have given a representation ofthe Choquet integral of a nonnegative continuous andincreasing fuzzy-number-valued function with respect to afuzzymeasure In addition in order to solve Choquet integralequations of fuzzy-number-valued function a concept of theLaplace transformation for calculation has been introducedFor distorted Lebesgue measures it was shown that Choquetintegral equations of fuzzy-number-valued function could besolved by the Laplace transformation Finally we have givenan example to illustrate themain result at the end of the paper

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The work is supported by the Natural Scientific Funds ofChina (61262022) the Natural Scientific Fund of GansuProvince of China (1208RJZA251) Lanzhou Jiaotong Univer-sity Young Scientific Research Fund Project (2011020) theFundamental Research Funds for Gansu Province Univer-sities (no 213060) and the Scientific Research Project ofNorthwestNormalUniversity (noNWNU-KJCXGC-03-61)

Abstract and Applied Analysis 11

References

[1] G Choquet ldquoTheory of capacitiesrdquoAnnales de lrsquoInstitut Fouriervol 5 pp 131ndash295 1954

[2] D Denneberg Non-Additive Measure and Integral KluwerAcademic Publishers Boston Mass USA 1994

[3] T Murofushi M Sugeno and M Machida ldquoNon-monotonicfuzzy measures and the Choquet integralrdquo Fuzzy Sets andSystems vol 64 no 1 pp 73ndash86 1994

[4] Z Y Wang K-S Leung M-L Wong and J Fang ldquoA new typeof nonlinear integrals and the computational algorithmrdquo FuzzySets and Systems vol 112 no 2 pp 223ndash231 2000

[5] D Schmeidler ldquoSubjective probability and expected utilitywithout additivityrdquo Econometrica vol 57 no 3 pp 571ndash5871989

[6] M Grabisch ldquo119896-order additive discrete fuzzy measures andtheir representationrdquo Fuzzy Sets and Systems vol 92 no 2 pp167ndash189 1997

[7] MGrabisch I Kojadinovic and PMeyer ldquoA review ofmethodsfor capacity identification in Choquet integral based multi-attribute utility theory applications of the Kappalab R packagerdquoEuropean Journal of Operational Research vol 186 no 2 pp766ndash785 2008

[8] K Xu Z Wang P Heng and K Leung ldquoClassification bynonlinear integral projectionsrdquo IEEE Transactions on FuzzySystems vol 11 no 2 pp 187ndash201 2003

[9] M J Bolanos L M de Campos Ibanez and A GonzalezMunoz ldquoConvergence properties of the monotone expectationand its application to the extension of fuzzy measuresrdquo FuzzySets and Systems vol 33 no 2 pp 201ndash212 1989

[10] T Murofushi and M Sugeno ldquoA theory of fuzzy measuresrepresentations the Choquet integral and null setsrdquo Journal ofMathematical Analysis and Applications vol 159 no 2 pp 532ndash549 1991

[11] Z Wang ldquoConvergence theorems for sequences of Choquetintegralsrdquo International Journal of General Systems vol 26 no1-2 pp 133ndash143 1997

[12] YOuyang andHZhang ldquoOn the space ofmeasurable functionsand its topology determined by the Choquet integralrdquo Interna-tional Journal of Approximate Reasoning vol 52 no 9 pp 1355ndash1362 2011

[13] J Kawabe ldquoThe Choquet integral representability of comono-tonically additive functionals in locally compact spacesrdquo Inter-national Journal of Approximate Reasoning vol 54 no 3 pp418ndash426 2013

[14] S Abbasbandy E Babolian and M Alavi ldquoNumerical methodfor solving linear Fredholm fuzzy integral equations of thesecond kindrdquo Chaos Solitons amp Fractals vol 31 no 1 pp 138ndash146 2007

[15] O Kaleva ldquoFuzzy differential equationsrdquo Fuzzy Sets and Sys-tems vol 24 no 3 pp 301ndash317 1987

[16] J Mordeson and W Newman ldquoFuzzy integral equationsrdquoInformation Sciences vol 87 no 4 pp 215ndash229 1995

[17] M Sugeno ldquoA note on derivatives of functions with respect tofuzzy measuresrdquo Fuzzy Sets and Systems vol 222 pp 1ndash17 2013

[18] I Kojadinovic J-L Marichal and M Roubens ldquoAn axiomaticapproach to the definition of the entropy of a discrete Choquetcapacityrdquo Information Sciences vol 172 no 1-2 pp 131ndash1532005

[19] E Pap Null-Additive Set Functions vol 337 of Mathematicsand Its Applications KluwerAcademic Publishers LondonUK1994

[20] M Sugeno Theory of fuzzy integrals and its applications [PhDdissertation] Tokyo Institute of Technology Tokyo Japan 1974

[21] C X Wu D L Zhang and C M Guo ldquoFuzzy number fuzzymeasures and fuzzy integralsrdquo Fuzzy Sets and Systems vol 98no 3 pp 355ndash360 1998

[22] C X Wu and M Ma The Foundament of Fuzzy AnalysisNational Defense Press Beijing China 1991 (Chinese)

[23] R Yang Z Y Wang P A Heng and K S Leung ldquoFuzzynumbers and fuzzification of the Choquet integralrdquo Fuzzy Setsand Systems vol 153 no 1 pp 95ndash113 2005

[24] S Seikkala ldquoOn the fuzzy initial value problemrdquo Fuzzy Sets andSystems vol 24 no 3 pp 319ndash330 1987

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 9

119909119899

= 119905 119865(1199090) = 0 and [119909

119894 119905] = 119909

119894 119909119894+1

119909119899 It follows

that

(119888) int[0119905]

119865 (120591) 119889120583 (120591)

=

119899

sum

119894=1

[120583 ([119909119894 119905]) minus 120583 ([119909

119894+1 119905])] 119865 (119909

1015840

119894)

=

119899

sum

119894=1

minus [120583 ([119909119894 119905]) minus 120583 ([119909

119894+1 119905])]

Δ119909119894

119865 (1199091015840

119894) Δ119909119894

= (119870)int

119905

0

[minus1205831015840([120591 119905])] 119865 (120591) 119889120591

(73)

5 Choquet Integral Equations ofFuzzy-Number-Valued Functions

In this section let us consider a Choquet integral equationbased onTheorem 24 as shown below Given continuous andincreasing fuzzy-number-valued functions 119866(119905) with 119866(0) =

0 let us find continuous and increasing fuzzy-number-valuedfunctions 119865(119905) such that

119866 (119905) = (119888) int[0119905]

119865 (120591) 119889120583119898

(119905) (74)

which is expressed as

119866 (119905) = (119870)int

119905

0

1198981015840(119905 minus 120591) 119865 (120591) 119889120591 (75)

Definition 27 Let 119891(119905) be a real-valued function defined on[0 +infin) The following

F (119904) = int

+infin

0

119890minus119904119905

119891 (119905) 119889119905 (76)

is said to be the Laplace transformation of 119891(119905) if the infiniteintegralint+infin

0119890minus119904119905

119891(119905)119889119905 is convergentwith respect to the valueof parameter 119904 We denote its Laplace transformation asF(119904) = 119871[119891(119905)] and the inverse Laplace transformation as119891(119905) = 119871

minus1[F(119904)]

Definition 28 Let119865(119909 119910) be a fuzzy-number-valued functiondefined on [0 +infin) times [0 +infin) (119870) int

infin

0119865(119909 119910)119889119909 is said to be

convergent with respect to the parameter 119910 if for every fixed119910 isin [0 +infin) (119870) int

+infin

0119865(119909 119910)119889119909 exists

Definition 29 Let 119865(119905) be a fuzzy-number-valued functiondefined on [0 +infin) The following

F (119904) = (119870)int[0+infin)

119890minus119904119905

119865 (119905) 119889119905 (77)

is said to be the Laplace transformation of 119865(119905) if the Kalevaintegral intinfin

0119890minus119904119905

119865(119905)119889119905 is convergent with respect to the valueof parameter 119904 We denote its Laplace transformation asF(119904) = 119871[119865(119905)] and the inverse Laplace transformation as119865(119905) = 119871

minus1[F(119904)]

Remark 30 In view of Theorem 8 we have

F120582(119904) = [int

+infin

0

119890minus119904119905

119865120582(119905) 119889119905 int

+infin

0

119890minus119904119905

119865120582(119905) 119889119905] (78)

for all 120582 isin [0 1]

Theorem 31 Let 119866(119905) = (119888) int[0119905]

119865(120591)119889120583119898(119905) Then

G (119904) = 119904M (119904) F (119904)

119866 (119904) = 119871minus1

[119904M (119904) F (119904)]

(79)

where G(119904) = 119871[119866(119905)]M(s) = 119871[119898(119905)] and F(119904) = 119871[119865(119905)]

Proof Since 119866(119905) = (119888) int[0119905]

119865(120591)119889120583119898(119905) = (119870) int

119905

01198981015840(119905 minus

120591)119865(120591)119889120591 it follows easily from the condition that

G120582(119904) = [int

+infin

0

119890minus119904119905

119866120582(119905) 119889119905 int

+infin

0

119890minus119904119905

119866120582(119905) 119889119905]

= [int

+infin

0

119890minus119904119905

int

119905

0

1198981015840(119905 minus 120591) 119865

120582(120591) 119889120591 119889119905

int

+infin

0

119890minus119904119905

int

119905

0

1198981015840(119905 minus 120591) 119865

120582(120591) 119889120591 119889119905]

(80)

for all 120582 isin [0 1]Notice that

int

+infin

0

119890minus119904119905

int

119905

0

1198981015840(119905 minus 120591) 119865

120582(120591) 119889120591 119889119905

= int

+infin

0

int

+infin

120591

119890minus119904119905

1198981015840(119905 minus 120591) 119865

120582(120591) 119889119905 119889120591

= int

+infin

0

119865120582(120591) int

+infin

120591

119890minus119904119905

1198981015840(119905 minus 120591) 119889119905 119889120591

= int

+infin

0

119865120582(120591) [119890

minus119904119905119898(119905 minus 120591)

10038161003816100381610038161003816

+infin

120591minus int

+infin

120591

119898(119905 minus 120591) 119889119890minus119904119905

] 119889120591

= int

+infin

0

119865120582(120591) [minusint

+infin

120591

119898(119905 minus 120591) (minus119904) 119890minus119904119905

119889119905] 119889120591

= 119904int

+infin

0

119865120582(120591) [int

+infin

0

119898(119905 minus 120591) (minus119904) 119890minus119904(119905minus120591)

119890minus119904120591

119889 (119905 minus 120591)] 119889120591

= 119904int

+infin

0

119865120582(120591) [119890

minus119904120591int

+infin

0

119890minus119904(119905minus120591)

119898(119905 minus 120591) 119889 (119905 minus 120591)] 119889120591

= 119904int

+infin

0

119865120582(120591) 119890minus119904120591

M (119904) 119889120591

= 119904M (119904) int

+infin

0

119890minus119904120591

119865120582(120591) 119889120591 = 119904M (119904)F

120582(119904)

(81)

Following the same argument we can prove that

int

+infin

0

119890minus119904119905

int

119905

0

1198981015840(119905 minus 120591) 119865

120582(120591) 119889120591 119889119905 = 119904M (119904)F

120582(119904) (82)

10 Abstract and Applied Analysis

Hence

G (119904) = 119904M (119904) F (119904)

119866 (119904) = 119871minus1

[119904M (119904) F (119904)]

(83)

Theorem 32 For continuous and increasing fuzzy-valuedfunctions 119866(119905) with 119866(0) = 0

F (119904) =G (119904)

119904M (119904) 119865 (119904) = 119871

minus[

G (119904)

119904M (119904)] (84)

where G(119904) = 119871[119866(119905)]M(119904) = 119871[119898(119905)] and F(119904) = 119871[119865(119905)]

Proof It follows easily fromTheorem 31 that

G (119904) = 119904M (119904) F (119904)

119866 (119904) = 119871minus1

[119904M (119904) F (119904)]

(85)

It is obvious that

F (119904) =G (119904)

119904M (119904)

119865 (119904) = 119871minus1

[G (119904)

119904M (119904)]

(86)

where G(119904) = 119871[119866(119905)]M(119904) = 119871[119898(119905)] and F(119904) = 119871[119865(119905)]The Dirac delta function is defined by the properties

120575 (119905) = 0 for 119905 = 0

undefined at 119905 = 0(87)

and int+infin

minusinfin120575(119905)119889119905 = 1 that is the function has unit area

The Laplace transform of the Dirac Delta function 120575(119905) isshown as follows

119871 [120575 (119905)] = int

+infin

0

119890minus119904119905

120575 (119905) 119889119905 = 1 (88)

So

119871minus1

[1] = 120575 (119905) (89)

Example 33 Let 119898(119905) = 119905 and let the membership functionof 119866(119905) be

119898119866(119905)

(119909) = 119909 119909 isin [0 1]

0 119909 notin [0 1] 119905 le 1 (90)

Then by Theorem 32 we can solve the Choquet integralequation

119866 (119905) = (119888) int[0119905]

119865 (120591) 119889120583119898

(119905) (91)

Indeed let 119898(119905) = 119905 thenM(119904) = 11199042 The 120582-cut of 119866(119905) is

represented by interval 119866120582(119905) = 119909 | 119898

119866(119905)(119909) ge 120582 = [120582 1]

that is 119866120582(119905) = 120582 119866

120582(119905) = 1 In view of Remark 30 we have

G120582(119904) = [int

+infin

0119890minus119904119905

119866120582(119905)119889119905 int

+infin

0119890minus119904119905

119866120582(119905)119889119905] = [120582119878 1119878]

Then

F120582(119904) =

G120582(119904)

119904M (119904)= 120582

F120582(119904) =

G120582(119904)

119904M (119904)= 1

(92)

Furthermore

119865120582(119905) = 119871

minus1(F120582(119904)) = 120582120575 (119905)

119865120582(119905) = 119871

minus1(F120582(119904)) = 120575 (119905)

(93)

Therefore for every 120582 isin [0 1]

119865120582(119905) = [119865

120582(119905) 119865120582(119905)] = [120582120575 (119905) 120575 (119905)] (94)

Then we have the membership function of 119865(119905) as

119898119865(119905)

(119909) =

119909

120575 (119905) 119909 isin [0 120575 (119905)] 119905 = 0

0 otherwise(95)

6 Conclusions and Remarks

In this paper we have considered the Choquet integrals offuzzy-number-valued functions based on the nonnegativereal lineWe have discussed the Choquet integrals of interval-valued functions and fuzzy-number-valued functions basedon nonadditive Sugeno measures and showed the repre-sentation theorem of them respectively We also gave theoperational schemes of above several classes of integralson discrete sets Then we have given a representation ofthe Choquet integral of a nonnegative continuous andincreasing fuzzy-number-valued function with respect to afuzzymeasure In addition in order to solve Choquet integralequations of fuzzy-number-valued function a concept of theLaplace transformation for calculation has been introducedFor distorted Lebesgue measures it was shown that Choquetintegral equations of fuzzy-number-valued function could besolved by the Laplace transformation Finally we have givenan example to illustrate themain result at the end of the paper

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The work is supported by the Natural Scientific Funds ofChina (61262022) the Natural Scientific Fund of GansuProvince of China (1208RJZA251) Lanzhou Jiaotong Univer-sity Young Scientific Research Fund Project (2011020) theFundamental Research Funds for Gansu Province Univer-sities (no 213060) and the Scientific Research Project ofNorthwestNormalUniversity (noNWNU-KJCXGC-03-61)

Abstract and Applied Analysis 11

References

[1] G Choquet ldquoTheory of capacitiesrdquoAnnales de lrsquoInstitut Fouriervol 5 pp 131ndash295 1954

[2] D Denneberg Non-Additive Measure and Integral KluwerAcademic Publishers Boston Mass USA 1994

[3] T Murofushi M Sugeno and M Machida ldquoNon-monotonicfuzzy measures and the Choquet integralrdquo Fuzzy Sets andSystems vol 64 no 1 pp 73ndash86 1994

[4] Z Y Wang K-S Leung M-L Wong and J Fang ldquoA new typeof nonlinear integrals and the computational algorithmrdquo FuzzySets and Systems vol 112 no 2 pp 223ndash231 2000

[5] D Schmeidler ldquoSubjective probability and expected utilitywithout additivityrdquo Econometrica vol 57 no 3 pp 571ndash5871989

[6] M Grabisch ldquo119896-order additive discrete fuzzy measures andtheir representationrdquo Fuzzy Sets and Systems vol 92 no 2 pp167ndash189 1997

[7] MGrabisch I Kojadinovic and PMeyer ldquoA review ofmethodsfor capacity identification in Choquet integral based multi-attribute utility theory applications of the Kappalab R packagerdquoEuropean Journal of Operational Research vol 186 no 2 pp766ndash785 2008

[8] K Xu Z Wang P Heng and K Leung ldquoClassification bynonlinear integral projectionsrdquo IEEE Transactions on FuzzySystems vol 11 no 2 pp 187ndash201 2003

[9] M J Bolanos L M de Campos Ibanez and A GonzalezMunoz ldquoConvergence properties of the monotone expectationand its application to the extension of fuzzy measuresrdquo FuzzySets and Systems vol 33 no 2 pp 201ndash212 1989

[10] T Murofushi and M Sugeno ldquoA theory of fuzzy measuresrepresentations the Choquet integral and null setsrdquo Journal ofMathematical Analysis and Applications vol 159 no 2 pp 532ndash549 1991

[11] Z Wang ldquoConvergence theorems for sequences of Choquetintegralsrdquo International Journal of General Systems vol 26 no1-2 pp 133ndash143 1997

[12] YOuyang andHZhang ldquoOn the space ofmeasurable functionsand its topology determined by the Choquet integralrdquo Interna-tional Journal of Approximate Reasoning vol 52 no 9 pp 1355ndash1362 2011

[13] J Kawabe ldquoThe Choquet integral representability of comono-tonically additive functionals in locally compact spacesrdquo Inter-national Journal of Approximate Reasoning vol 54 no 3 pp418ndash426 2013

[14] S Abbasbandy E Babolian and M Alavi ldquoNumerical methodfor solving linear Fredholm fuzzy integral equations of thesecond kindrdquo Chaos Solitons amp Fractals vol 31 no 1 pp 138ndash146 2007

[15] O Kaleva ldquoFuzzy differential equationsrdquo Fuzzy Sets and Sys-tems vol 24 no 3 pp 301ndash317 1987

[16] J Mordeson and W Newman ldquoFuzzy integral equationsrdquoInformation Sciences vol 87 no 4 pp 215ndash229 1995

[17] M Sugeno ldquoA note on derivatives of functions with respect tofuzzy measuresrdquo Fuzzy Sets and Systems vol 222 pp 1ndash17 2013

[18] I Kojadinovic J-L Marichal and M Roubens ldquoAn axiomaticapproach to the definition of the entropy of a discrete Choquetcapacityrdquo Information Sciences vol 172 no 1-2 pp 131ndash1532005

[19] E Pap Null-Additive Set Functions vol 337 of Mathematicsand Its Applications KluwerAcademic Publishers LondonUK1994

[20] M Sugeno Theory of fuzzy integrals and its applications [PhDdissertation] Tokyo Institute of Technology Tokyo Japan 1974

[21] C X Wu D L Zhang and C M Guo ldquoFuzzy number fuzzymeasures and fuzzy integralsrdquo Fuzzy Sets and Systems vol 98no 3 pp 355ndash360 1998

[22] C X Wu and M Ma The Foundament of Fuzzy AnalysisNational Defense Press Beijing China 1991 (Chinese)

[23] R Yang Z Y Wang P A Heng and K S Leung ldquoFuzzynumbers and fuzzification of the Choquet integralrdquo Fuzzy Setsand Systems vol 153 no 1 pp 95ndash113 2005

[24] S Seikkala ldquoOn the fuzzy initial value problemrdquo Fuzzy Sets andSystems vol 24 no 3 pp 319ndash330 1987

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Abstract and Applied Analysis

Hence

G (119904) = 119904M (119904) F (119904)

119866 (119904) = 119871minus1

[119904M (119904) F (119904)]

(83)

Theorem 32 For continuous and increasing fuzzy-valuedfunctions 119866(119905) with 119866(0) = 0

F (119904) =G (119904)

119904M (119904) 119865 (119904) = 119871

minus[

G (119904)

119904M (119904)] (84)

where G(119904) = 119871[119866(119905)]M(119904) = 119871[119898(119905)] and F(119904) = 119871[119865(119905)]

Proof It follows easily fromTheorem 31 that

G (119904) = 119904M (119904) F (119904)

119866 (119904) = 119871minus1

[119904M (119904) F (119904)]

(85)

It is obvious that

F (119904) =G (119904)

119904M (119904)

119865 (119904) = 119871minus1

[G (119904)

119904M (119904)]

(86)

where G(119904) = 119871[119866(119905)]M(119904) = 119871[119898(119905)] and F(119904) = 119871[119865(119905)]The Dirac delta function is defined by the properties

120575 (119905) = 0 for 119905 = 0

undefined at 119905 = 0(87)

and int+infin

minusinfin120575(119905)119889119905 = 1 that is the function has unit area

The Laplace transform of the Dirac Delta function 120575(119905) isshown as follows

119871 [120575 (119905)] = int

+infin

0

119890minus119904119905

120575 (119905) 119889119905 = 1 (88)

So

119871minus1

[1] = 120575 (119905) (89)

Example 33 Let 119898(119905) = 119905 and let the membership functionof 119866(119905) be

119898119866(119905)

(119909) = 119909 119909 isin [0 1]

0 119909 notin [0 1] 119905 le 1 (90)

Then by Theorem 32 we can solve the Choquet integralequation

119866 (119905) = (119888) int[0119905]

119865 (120591) 119889120583119898

(119905) (91)

Indeed let 119898(119905) = 119905 thenM(119904) = 11199042 The 120582-cut of 119866(119905) is

represented by interval 119866120582(119905) = 119909 | 119898

119866(119905)(119909) ge 120582 = [120582 1]

that is 119866120582(119905) = 120582 119866

120582(119905) = 1 In view of Remark 30 we have

G120582(119904) = [int

+infin

0119890minus119904119905

119866120582(119905)119889119905 int

+infin

0119890minus119904119905

119866120582(119905)119889119905] = [120582119878 1119878]

Then

F120582(119904) =

G120582(119904)

119904M (119904)= 120582

F120582(119904) =

G120582(119904)

119904M (119904)= 1

(92)

Furthermore

119865120582(119905) = 119871

minus1(F120582(119904)) = 120582120575 (119905)

119865120582(119905) = 119871

minus1(F120582(119904)) = 120575 (119905)

(93)

Therefore for every 120582 isin [0 1]

119865120582(119905) = [119865

120582(119905) 119865120582(119905)] = [120582120575 (119905) 120575 (119905)] (94)

Then we have the membership function of 119865(119905) as

119898119865(119905)

(119909) =

119909

120575 (119905) 119909 isin [0 120575 (119905)] 119905 = 0

0 otherwise(95)

6 Conclusions and Remarks

In this paper we have considered the Choquet integrals offuzzy-number-valued functions based on the nonnegativereal lineWe have discussed the Choquet integrals of interval-valued functions and fuzzy-number-valued functions basedon nonadditive Sugeno measures and showed the repre-sentation theorem of them respectively We also gave theoperational schemes of above several classes of integralson discrete sets Then we have given a representation ofthe Choquet integral of a nonnegative continuous andincreasing fuzzy-number-valued function with respect to afuzzymeasure In addition in order to solve Choquet integralequations of fuzzy-number-valued function a concept of theLaplace transformation for calculation has been introducedFor distorted Lebesgue measures it was shown that Choquetintegral equations of fuzzy-number-valued function could besolved by the Laplace transformation Finally we have givenan example to illustrate themain result at the end of the paper

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The work is supported by the Natural Scientific Funds ofChina (61262022) the Natural Scientific Fund of GansuProvince of China (1208RJZA251) Lanzhou Jiaotong Univer-sity Young Scientific Research Fund Project (2011020) theFundamental Research Funds for Gansu Province Univer-sities (no 213060) and the Scientific Research Project ofNorthwestNormalUniversity (noNWNU-KJCXGC-03-61)

Abstract and Applied Analysis 11

References

[1] G Choquet ldquoTheory of capacitiesrdquoAnnales de lrsquoInstitut Fouriervol 5 pp 131ndash295 1954

[2] D Denneberg Non-Additive Measure and Integral KluwerAcademic Publishers Boston Mass USA 1994

[3] T Murofushi M Sugeno and M Machida ldquoNon-monotonicfuzzy measures and the Choquet integralrdquo Fuzzy Sets andSystems vol 64 no 1 pp 73ndash86 1994

[4] Z Y Wang K-S Leung M-L Wong and J Fang ldquoA new typeof nonlinear integrals and the computational algorithmrdquo FuzzySets and Systems vol 112 no 2 pp 223ndash231 2000

[5] D Schmeidler ldquoSubjective probability and expected utilitywithout additivityrdquo Econometrica vol 57 no 3 pp 571ndash5871989

[6] M Grabisch ldquo119896-order additive discrete fuzzy measures andtheir representationrdquo Fuzzy Sets and Systems vol 92 no 2 pp167ndash189 1997

[7] MGrabisch I Kojadinovic and PMeyer ldquoA review ofmethodsfor capacity identification in Choquet integral based multi-attribute utility theory applications of the Kappalab R packagerdquoEuropean Journal of Operational Research vol 186 no 2 pp766ndash785 2008

[8] K Xu Z Wang P Heng and K Leung ldquoClassification bynonlinear integral projectionsrdquo IEEE Transactions on FuzzySystems vol 11 no 2 pp 187ndash201 2003

[9] M J Bolanos L M de Campos Ibanez and A GonzalezMunoz ldquoConvergence properties of the monotone expectationand its application to the extension of fuzzy measuresrdquo FuzzySets and Systems vol 33 no 2 pp 201ndash212 1989

[10] T Murofushi and M Sugeno ldquoA theory of fuzzy measuresrepresentations the Choquet integral and null setsrdquo Journal ofMathematical Analysis and Applications vol 159 no 2 pp 532ndash549 1991

[11] Z Wang ldquoConvergence theorems for sequences of Choquetintegralsrdquo International Journal of General Systems vol 26 no1-2 pp 133ndash143 1997

[12] YOuyang andHZhang ldquoOn the space ofmeasurable functionsand its topology determined by the Choquet integralrdquo Interna-tional Journal of Approximate Reasoning vol 52 no 9 pp 1355ndash1362 2011

[13] J Kawabe ldquoThe Choquet integral representability of comono-tonically additive functionals in locally compact spacesrdquo Inter-national Journal of Approximate Reasoning vol 54 no 3 pp418ndash426 2013

[14] S Abbasbandy E Babolian and M Alavi ldquoNumerical methodfor solving linear Fredholm fuzzy integral equations of thesecond kindrdquo Chaos Solitons amp Fractals vol 31 no 1 pp 138ndash146 2007

[15] O Kaleva ldquoFuzzy differential equationsrdquo Fuzzy Sets and Sys-tems vol 24 no 3 pp 301ndash317 1987

[16] J Mordeson and W Newman ldquoFuzzy integral equationsrdquoInformation Sciences vol 87 no 4 pp 215ndash229 1995

[17] M Sugeno ldquoA note on derivatives of functions with respect tofuzzy measuresrdquo Fuzzy Sets and Systems vol 222 pp 1ndash17 2013

[18] I Kojadinovic J-L Marichal and M Roubens ldquoAn axiomaticapproach to the definition of the entropy of a discrete Choquetcapacityrdquo Information Sciences vol 172 no 1-2 pp 131ndash1532005

[19] E Pap Null-Additive Set Functions vol 337 of Mathematicsand Its Applications KluwerAcademic Publishers LondonUK1994

[20] M Sugeno Theory of fuzzy integrals and its applications [PhDdissertation] Tokyo Institute of Technology Tokyo Japan 1974

[21] C X Wu D L Zhang and C M Guo ldquoFuzzy number fuzzymeasures and fuzzy integralsrdquo Fuzzy Sets and Systems vol 98no 3 pp 355ndash360 1998

[22] C X Wu and M Ma The Foundament of Fuzzy AnalysisNational Defense Press Beijing China 1991 (Chinese)

[23] R Yang Z Y Wang P A Heng and K S Leung ldquoFuzzynumbers and fuzzification of the Choquet integralrdquo Fuzzy Setsand Systems vol 153 no 1 pp 95ndash113 2005

[24] S Seikkala ldquoOn the fuzzy initial value problemrdquo Fuzzy Sets andSystems vol 24 no 3 pp 319ndash330 1987

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 11

References

[1] G Choquet ldquoTheory of capacitiesrdquoAnnales de lrsquoInstitut Fouriervol 5 pp 131ndash295 1954

[2] D Denneberg Non-Additive Measure and Integral KluwerAcademic Publishers Boston Mass USA 1994

[3] T Murofushi M Sugeno and M Machida ldquoNon-monotonicfuzzy measures and the Choquet integralrdquo Fuzzy Sets andSystems vol 64 no 1 pp 73ndash86 1994

[4] Z Y Wang K-S Leung M-L Wong and J Fang ldquoA new typeof nonlinear integrals and the computational algorithmrdquo FuzzySets and Systems vol 112 no 2 pp 223ndash231 2000

[5] D Schmeidler ldquoSubjective probability and expected utilitywithout additivityrdquo Econometrica vol 57 no 3 pp 571ndash5871989

[6] M Grabisch ldquo119896-order additive discrete fuzzy measures andtheir representationrdquo Fuzzy Sets and Systems vol 92 no 2 pp167ndash189 1997

[7] MGrabisch I Kojadinovic and PMeyer ldquoA review ofmethodsfor capacity identification in Choquet integral based multi-attribute utility theory applications of the Kappalab R packagerdquoEuropean Journal of Operational Research vol 186 no 2 pp766ndash785 2008

[8] K Xu Z Wang P Heng and K Leung ldquoClassification bynonlinear integral projectionsrdquo IEEE Transactions on FuzzySystems vol 11 no 2 pp 187ndash201 2003

[9] M J Bolanos L M de Campos Ibanez and A GonzalezMunoz ldquoConvergence properties of the monotone expectationand its application to the extension of fuzzy measuresrdquo FuzzySets and Systems vol 33 no 2 pp 201ndash212 1989

[10] T Murofushi and M Sugeno ldquoA theory of fuzzy measuresrepresentations the Choquet integral and null setsrdquo Journal ofMathematical Analysis and Applications vol 159 no 2 pp 532ndash549 1991

[11] Z Wang ldquoConvergence theorems for sequences of Choquetintegralsrdquo International Journal of General Systems vol 26 no1-2 pp 133ndash143 1997

[12] YOuyang andHZhang ldquoOn the space ofmeasurable functionsand its topology determined by the Choquet integralrdquo Interna-tional Journal of Approximate Reasoning vol 52 no 9 pp 1355ndash1362 2011

[13] J Kawabe ldquoThe Choquet integral representability of comono-tonically additive functionals in locally compact spacesrdquo Inter-national Journal of Approximate Reasoning vol 54 no 3 pp418ndash426 2013

[14] S Abbasbandy E Babolian and M Alavi ldquoNumerical methodfor solving linear Fredholm fuzzy integral equations of thesecond kindrdquo Chaos Solitons amp Fractals vol 31 no 1 pp 138ndash146 2007

[15] O Kaleva ldquoFuzzy differential equationsrdquo Fuzzy Sets and Sys-tems vol 24 no 3 pp 301ndash317 1987

[16] J Mordeson and W Newman ldquoFuzzy integral equationsrdquoInformation Sciences vol 87 no 4 pp 215ndash229 1995

[17] M Sugeno ldquoA note on derivatives of functions with respect tofuzzy measuresrdquo Fuzzy Sets and Systems vol 222 pp 1ndash17 2013

[18] I Kojadinovic J-L Marichal and M Roubens ldquoAn axiomaticapproach to the definition of the entropy of a discrete Choquetcapacityrdquo Information Sciences vol 172 no 1-2 pp 131ndash1532005

[19] E Pap Null-Additive Set Functions vol 337 of Mathematicsand Its Applications KluwerAcademic Publishers LondonUK1994

[20] M Sugeno Theory of fuzzy integrals and its applications [PhDdissertation] Tokyo Institute of Technology Tokyo Japan 1974

[21] C X Wu D L Zhang and C M Guo ldquoFuzzy number fuzzymeasures and fuzzy integralsrdquo Fuzzy Sets and Systems vol 98no 3 pp 355ndash360 1998

[22] C X Wu and M Ma The Foundament of Fuzzy AnalysisNational Defense Press Beijing China 1991 (Chinese)

[23] R Yang Z Y Wang P A Heng and K S Leung ldquoFuzzynumbers and fuzzification of the Choquet integralrdquo Fuzzy Setsand Systems vol 153 no 1 pp 95ndash113 2005

[24] S Seikkala ldquoOn the fuzzy initial value problemrdquo Fuzzy Sets andSystems vol 24 no 3 pp 319ndash330 1987

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of