Abstract and Applied Analysis

Volume 2014 (2014), Article ID 953893, 11 pages

http://dx.doi.org/10.1155/2014/953893

Research Article

Choquet Integral of Fuzzy-Number-Valued Functions: The Differentiability of the Primitive with respect to Fuzzy Measures and Choquet Integral Equations

Zengtai Gong,¹ Li Chen,1,2 and Gang Duan³

¹College of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, China

²Department of Mathematics, Lanzhou City University, Lanzhou 730070, China

³School of Traffic and Transportation, Lanzhou Jiaotong University, Lanzhou 730070, China

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

Academic Editor: Marco Donatelli

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

Abstract.

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

1. Introduction

The Choquet integral [1–4] with respect to a fuzzy measure was proposed by Murofushi and Sugeno. It was introduced by Choquet in potential theory with the concept of capacity. Then, it has been used for utility theory in the field of economic theory [5] and has been used for image processing, pattern recognition, information fusion, and data mining [4, 6–8] in the context of fuzzy measure theory [9–13].

The development of the theory of integral equations is closely linked to the study of mathematical physics problems. The integral equation has the extremely widespread application in the field of engineering and mechanics and so forth. The early history of integral equation goes back to the special integral equation studied by several mathematicians, such as Laplace, Fourier, Poisson, Abel, and Liouville in the late eighteenth and early nineteenth century. With the development of computing technology, the integral equation as one of the important foundations of engineering calculation has been widely and effectively used. Today, with physical problems becoming more and more complex, integral equation is becoming more and more useful.

Fuzzy integral and differential equations were discussed by many authors [14–16], which have been suggested as a way of modeling uncertain and incompletely specified systems. Sugeno has described carefully the representation of Choquet integral and Choquet integral equations of real-valued increasing functions, and some important conclusions have been obtained [17]. Unfortunately, it is not reasonable to assume that all data are real data before we elicit them from practical data. Sometimes, fuzzy data may exist, such as in pharmacological, financial, and sociological applications. Motivated by the above papers and related research works on this topic, the paper discusses the representation of Choquet integral and Choquet integral equations of increasing fuzzy-number-valued functions.

The rest of this study is organized as follows. In Section 2, we review some basic definitions of fuzzy measure and Choquet integrals of real-valued functions. Section 3 gives the definitions and the characterizations of the Choquet integrals of interval-valued functions and fuzzy-number-valued functions based on the nonadditive Sugeno measure. Furthermore, the operational schemes of above several classes of integrals on a discrete sets are investigated which enable us to calculate Choquet integrals in applications. Section 4 gives a representation of the Choquet integral of a nonnegative, continuous, and increasing fuzzy-number-valued function with respect to a fuzzy measure. In Section 5, in order to solve Choquet integral equations of fuzzy-number-valued functions, a concept of the Laplace transformation for the fuzzy-number-valued functions in the sense of Choquet integral is introduced. For distorted Lebesgue measures, it is shown that Choquet integral equations of fuzzy-number-valued functions can be solved by the Laplace transformation. In addition, an example is given to illustrate the main results at the end of the paper. The paper ends with conclusions in Section 6.

2. Preliminaries

In this section, we will introduce some basic definitions about fuzzy measures, Choquet integral, and fuzzy numbers.

Definition 1 (see [6, 7, 18–20]). Let be a nonempty set and a -algebra on . A fuzzy measure on is a set function satisfying the following conditions:

(1);(2), , implies ;(3)In , if , and , then ;(4)In , if , and , then .

is said to be lower semicontinuous if it satisfies the above conditions (1)–(3); is said to be upper semicontinuous if it satisfies the above conditions (1), (2), and (4); is said to be continuous if it satisfies the above conditions (1)–(4).

is said to be a nonadditive measure space.

One can see that a fuzzy measure is a normal monotone set function which vanishes at the empty set. Furthermore, a fuzzy measure on is said to be

(i)additive if for all disjoint subsets ;(ii)subadditive if for all disjoint subsets ;(iii)superadditive if for all disjoint subsets ;(iv)cardinality-based if for any depends only on the cardinality of ;(v)a fuzzy measure if its range is ;(vi)a possibility fuzzy measure focused on , denoted by , if if and only if , and otherwise;(vii)a necessity fuzzy measure focused on , denoted by , if if and only if , and otherwise.

Let be a measurable function with respect to . That is, satisfies the condition

for any .

Definition 2 (see [1]). Let be a nonadditive measure space and a measurable function on . The Choquet integral of a real-valued function is defined as

if both of Riemann integrals exist and at least one of them has finite value.

Let . Then Choquet integral of a nonnegative real-valued function is defined as

Since is nonincreasing with respect to , the Choquet integral of real-valued function with respect to exists. If , then is said to be -integrable with respect to on . Choquet integral has the following properties [1].

(1)If , then .(2)If , then .(3)Let be lower semicontinuous. If a.e. in , then .(4)Let be upper semicontinuous. If a.e. in , and there exists a -integrable function such that , then .

denotes the set of all interval numbers on , where . With the definition from Wu et al. [21], interval numbers should satisfy the following basic operations:

(1) denotes ;(2), ;(3), ;(4);(5)If , then .

A fuzzy subset of is a function . For each fuzzy set defined as above, we denote by , for any , its -level set. By supp we denote the support of ; that is, the set . By we denote the closure of supp; that is, . Let be the collection of all fuzzy sets of . We call a fuzzy number if it satisfies the following conditions [22]:

(1) is normal; that is, there exists such that ;(2) is fuzzy convex; that is, for any ;(3) is upper semicontinuous; that is, for any ;(4) is compact.

We denote the collection of all fuzzy numbers by .

is said to be a nonnegative fuzzy number if supp, . We can define the semiorder and the distance in space [22]. Let . Then   if   and for ; if for ; is the distance of and . Let . If , then .

denotes the collection of all nonnegative fuzzy numbers.

Lemma 3 (see [22]). Let . Then

(1) is a nonempty, bounded, and closed interval for each ;(2)if then ;(3)if and , then .

Conversely, there exists a for each , which satisfies the conditions (1)–(3); then there exists a unique fuzzy number such that and .

3. Choquet Integrals of Interval-Valued Functions and Fuzzy-Number-Valued Functions Based on Nonadditive Measures

In this section, we will give the definitions and the characterizations of the Choquet integrals of interval-valued functions and fuzzy-number-valued functions based on the nonadditive Sugeno measure. Furthermore, the operational schemes of above several classes of integrals on a discrete sets are investigated which enable us to calculate Choquet integrals in applications.

We first introduce the concept of the Choquet integrals for the interval-valued functions as follows.

Definition 4 (see [23]). An interval-valued function is said to be measurable if both and are measurable functions, where , is the left end point of interval and is the right end point of interval .

Interval-valued function is -integrally bounded if there exists a Choquet integrable function such that for every selection .

We denote . Let be a measure space,   , and let be a measurable set valued mapping. is said to be a measurable selection of if there exists a measurable mapping such that for every .

Definition 5. Let be a nonadditive measure space. Assume that is measurable, -integrally bounded interval-valued function, and . is said to be -integrable if

is a closed interval on , where

Theorem 6. Let be a nonadditive measure space. Suppose that is a fuzzy measure, , and is nonnegative measurable, -integrally bounded interval-valued function; then is -integrable on and

Proof. Since is nonnegative measurable on ,   and are measurable on . Thus and are two measurable selections of . On the other hand, is -integrally bounded; we have

By the properties of Choquet integral of real valued functions, we know that and are -integrable on . Let . That is to say, there exists a measurable selection of , such that . We can prove that . Indeed, notice that , and by the properties of Choquet integral of real-valued functions we have

This follows that

Thus,

Conversely, we can show that

For measurable selections , we get that

is measurable selection of for any . Therefore,

It implies that is a convex set. On the other hand, since and are two measurable selections of , and

we have

Hence,

That is, is -integrable.

From the above theorem, we know that interval-valued function is -integrable on if and exist and are bounded.

Next, we will introduce the concept of the Choquet integrals for the Fuzzy-number-valued functions as follows.

Fuzzy-number-valued function on is said to be measurable if and are measurable functions with respect to for any .

Fuzzy-number-valued function is said to be -integrally bounded if there exists a Choquet integrable function such that for every selection .

Definition 7. Let be a nonadditive measure space. Assume that , is measurable and -integrally bounded function. is said to be -integrable if

determines a unique fuzzy number , which is denoted by , where is a measurable selection of .

Theorem 8. Let be a nonadditive measurable space. Assume is a continuous fuzzy measure, , is measurable and -integrally bounded function; then is -integrable on if and only if   and are -integrable on and

for any .

Proof. For the necessity, since is measurable and -integrally bounded function, and are measurable for every and there exists a Choquet integrable function such that , , and then

That is, , are -integrable and

for any .

For the sufficiency, let be a Fuzzy-number-valued function. Note that

we need only to prove that the interval family

determines a unique fuzzy number. Indeed, the interval family

satisfies the conditions of Lemma 3.

(1) is a measurable fuzzy-number-valued function; for each , we have , and therefore

(2)Since for , that is,

we have

(3)For each , , that is, , . It is easy to see that , , are integrable, and by the continuity of , then

In conclusion, there exists a unique fuzzy number such that

Furthermore, we get that is -integrable on and

In the last part of the section, we will investigate the operational schemes of above several classes of integrals on a discrete set.

Let be a discrete set. Then we will give a new scheme to calculate the value of the Choquet integral.

Theorem 9 (see [8]). Let be a real-valued function on . Then Choquet integral of with respect to a fuzzy measure on is given by

or equivalently, by

where is a permutation of such that , , , , and  .

Theorem 10. Let be an interval-valued function on . Then Choquet integral of with respect to a fuzzy measure on is given by

where is a permutation of such that , , , , and .

Proof. Since is an interval-valued function on , in view of Theorem 6, we have

Note that and are real-valued function on , respectively. By Theorem 9 we get

where is a permutation of such that , , , , and .

Theorem 11. Let be a fuzzy-number-valued function on . Then Choquet integral of with respect to a fuzzy measure on is given by

where is a permutation of such that , , , , and .

Proof. Since is a fuzzy-number-valued on , in view of Theorem 8, we have

for any . By the semiorder in space (i.e., let . Then if and and Theorem 10, there is a sequence on such that , , , , , and is a permutation of . Consequently,

for any . Therefore,

4. The Representation of Choquet Integral of Fuzzy-Number-Valued Functions

Sugeno has described carefully the representation of Choquet integral of real-valued increasing functions, and some important conclusions have been obtained [17]. Motivated by this, we will discuss the representation of Choquet integral of fuzzy-number-valued functions in this section.

Definition 12. Fuzzy-number-valued function is said to be continuous on if for every there are

where .

Definition 13. Fuzzy-number-valued function is said to be increasing on if for every there are and , where and .

Let be a class of measurable, nonnegative, continuous, and increasing fuzzy-number-valued functions.

Let be a Lebesgue measure for , .

Definition 14 (see [17]). Let be a continuous and increasing function and . A fuzzy measure , a distorted Lebesgue measure, is defined by .

Definition 15 (see [24]). A fuzzy-number-valued function is differentiable at if there exist fuzzy number-valued functions such that

for every , where is the fuzzy derivative of .

Note that is induced from the Lebesgue measure by a monotone transformation, where . Apparently it loses additivity, unless , but reserves monotonicity. In what follows we assume that is differentiable.

In this section, we consider the calculation of Choquet integrals. Let be a general fuzzy measure and consider for a closed interval ; then is decreasing for and increasing for . Throughout the paper, we assume that the functions , and are continuously differentiable. We also assume that is continuously differentiable with respect to on for every . In addition, we require the regularity condition that holds for every . We write , where we note that for . If then where . First we consider a case that is strictly increasing.

Fuzzy-number-valued function is said to be -integrally bounded if there exists a Lebesgue integrable function such that for every section .

Definition 16 (see [22]). Let fuzzy-number-valued functions be measurable and -integrally bounded. is called Kaleva-integrable if

determines a unique fuzzy number , which is denoted by , where is a measurable selection.

Lemma 17 (see [22]). Let be measurable and -integrally bounded fuzzy-number-valued function; then is Kaleva integrable on if and only if   and are -integrable on and

for any .

Theorem 18. Let . Then is Kaleva integrable on and

Proof. Since , for every we get that and are measurable, nonnegative, continuous, and increasing real-valued functions on . It follows that and are -integrable on . is -integrable on as it is continuously differentiable with respect to on for every . In view of Lemma 17, is Kaleva integrable on and

Lemma 19 (see [17]). Let be a real strictly increasing function. Then the Choquet integral of with respect to on is represented as

In particular, for

Lemma 20 (see [17]). Let be a constant real-valued function: , . Then

In particular, for

Lemma 21 (see [17]). Let be a measurable, nonnegative, continuous, and increasing real-valued function. Then the Choquet integral of with respect to on is represented as

In particular, for

Theorem 22. Let be a strictly increasing fuzzy-number-valued function. Then the Choquet integral of   with respect to on is represented as

In particular, for

Proof. By Theorem 8, we have

for any . Since is strictly increasing, and are strictly increasing real-valued functions, respectively. In view of Lemma 19, we have

It follows that

On the other hand,

Therefore,

By the arbitrary of , we have

For , we note that ; hence,

Theorem 23. Let be a constant fuzzy-valued function; that is, ,   . Then

In particular, for

Proof. Note that , where and are constant real-valued functions, respectively. In view of Lemma 20, we have

The rest of the proof follows in exactly the same way as that of Theorem 22. Hence,

For , we obtain . It follows that

where .

Theorem 24. Let . Then the Choquet integral of with respect to on is represented as

In particular, for

Proof. The theorem has been proved when is a strictly increasing or constant fuzzy-number-valued function by Theorem 22 or Theorem 23. Without loss of generality, we consider a continuous and increasing function such that

where and are strictly increasing, is constant, and .

(i)For

Note that is strictly increasing on ; from Theorem 22, we obtain

(ii)For

In view of the construction of on , we have

Since is strictly increasing on , and is constant on , by Theorems 22 and 23,

(iii)For

Since is strictly increasing on is constant on , and is strictly increasing on , by Theorems 22 and 23, we have

For , we obtain that

Remark 25. Note that Theorem 24 holds only for a continuous and increasing , but in the case , it holds for any .

Remark 26. Let us consider the representation of the Choquet integral for a continuous case shown in Theorem 24 in relation with a discrete case. Now is transformed into the discrete set . Since is an increasing fuzzy-number-valued function, we have , in Theorem 11. For the sake of simplicity, let , and . It follows that

5. Choquet Integral Equations of Fuzzy-Number-Valued Functions

In this section, let us consider a Choquet integral equation based on Theorem 24 as shown below. Given continuous and increasing fuzzy-number-valued functions with , let us find continuous and increasing fuzzy-number-valued functions such that

which is expressed as

Definition 27. Let be a real-valued function defined on . The following

is said to be the Laplace transformation of if the infinite integral is convergent with respect to the value of parameter . We denote its Laplace transformation as and the inverse Laplace transformation as .

Definition 28. Let be a fuzzy-number-valued function defined on . is said to be convergent with respect to the parameter if for every fixed , exists.

Definition 29. Let be a fuzzy-number-valued function defined on . The following

is said to be the Laplace transformation of if the Kaleva integral is convergent with respect to the value of parameter . We denote its Laplace transformation as and the inverse Laplace transformation as .

Remark 30. In view of Theorem 8, we have

for all .

Theorem 31. Let . Then

where , , and .

Proof. Since , it follows easily from the condition that

for all .

Notice that

Following the same argument, we can prove that

Hence,

Theorem 32. For continuous and increasing fuzzy-valued functions with ,

where , , and .

Proof. It follows easily from Theorem 31 that

It is obvious that

where , , and .

The Dirac delta function is defined by the properties

and ; that is, the function has unit area.

The Laplace transform of the Dirac Delta function is shown as follows:

So

Example 33. Let and let the membership function of be

Then by Theorem 32 we can solve the Choquet integral equation

Indeed, let ; then . The -cut of is represented by interval: ; that is , . In view of Remark 30, we have . Then

Furthermore,

Therefore, for every

Then we have the membership function of as

6. Conclusions and Remarks

In this paper, we have considered the Choquet integrals of fuzzy-number-valued functions based on the nonnegative real line. We have discussed the Choquet integrals of interval-valued functions and fuzzy-number-valued functions based on nonadditive Sugeno measures and showed the representation theorem of them, respectively. We also gave the operational schemes of above several classes of integrals on discrete sets. Then we have given a representation of the Choquet integral of a nonnegative, continuous, and increasing fuzzy-number-valued function with respect to a fuzzy measure. In addition, in order to solve Choquet integral equations of fuzzy-number-valued function, a concept of the Laplace transformation for calculation has been introduced. For distorted Lebesgue measures, it was shown that Choquet integral equations of fuzzy-number-valued function could be solved by the Laplace transformation. Finally, we have given an example to illustrate the main result at the end of the paper.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The work is supported by the Natural Scientific Funds of China (61262022), the Natural Scientific Fund of Gansu Province of China (1208RJZA251), Lanzhou Jiaotong University Young Scientific Research Fund Project (2011020), the Fundamental Research Funds for Gansu Province Universities (no. 213060), and the Scientific Research Project of Northwest Normal University (no. NWNU-KJCXGC-03-61).

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