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Journal of Function Spaces
Volume 2019, Article ID 4631530, 14 pages
https://doi.org/10.1155/2019/4631530
Research Article

Inequalities of Lyapunov and Stolarsky Type for Choquet-Like Integrals with respect to Nonmonotonic Fuzzy Measures

1College of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, China
2Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000, China

Correspondence should be addressed to Zengtai Gong; moc.361@gnog-tz

Received 9 October 2018; Accepted 21 November 2018; Published 2 January 2019

Academic Editor: Calogero Vetro

Copyright © 2019 Ting Xie and Zengtai Gong. 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

The aim of this paper is to generalize the Choquet-like integral with respect to a nonmonotonic fuzzy measure for generalized real-valued functions and set-valued functions, which is based on the generalized pseudo-operations and --measures. Furthermore, the characterization theorem and transformation theorem for the integral are given. Finally, we study the Lyapunov type inequality and Stolarsky type inequality for the Choquet-like integral.

1. Introduction

The Choquet integral with respect to a fuzzy measure , which is monotone, does not require continuity and was proposed by Murofushi and Sugeno [1]. It was introduced by Choquet [2] in potential theory with the concept of capacity. The Choquet integral of a nonnegative single-valued measurable function is defined aswhere . To generate the Choquet integral to the generalized real valued measurable function, the symmetric Choquet integral, which was most early proposed by Šipoš [3] in 1979, and the asymmetric Choquet integral were introduced and later in [4, 5] had been given specific discussions. Schmeidler [6] established an integral representation theorem through the Choquet integral for functionals satisfying monotonicity and a weaker condition than additivity, namely, comonotonic additivity. However, violations of monotonicity in multiperiod models occur frequently, and nonmonotone set functions seem to be better suited [7, 8]. Furthermore, from the mathematical point of view, the monotonicity is inessential. We can construct measure theory without monotonicity [1, 9]. Aumann and Shapely [10] had investigated nonmonotonic fuzzy measures as games and this issue had been addressed by Murofushi et al. in [9], where a complete characterization of nonmonotonic Choquet integral was achieved; that is, they generalized the representation to the case of bounded variation functionals omitting the monotonicity condition.

Sugeno introduced another integral for any fuzzy measure and any nonnegative single-valued measurable function , nowadays called a Sugeno integral, as follows:where . Notice that when the fuzzy measure is with the usual additive, the Choquet integral is coincident with the Lebesgue integral. However, the Sugeno integral is not with the usual additive; thus it is not an extension of the Lebesgue integral.

Recently, pseudo-analysis is a research hotspot, and it presents a contemporary mathematical theory that is being successfully applied in many different areas of mathematics as well as in various practical problems [5, 1114]. In fact, in many problems with uncertainty as in the theory of probabilistic metric spaces, fuzzy logics, fuzzy sets, and fuzzy measures, we often work with many operations different from the usual addition and multiplication of reals, e.g., triangular norms, triangular conorms, pseudo-additions, and pseudo-multiplications. The triangular conorm decomposable measure was first introduced by Dubois and Prade [15] as a special important class of fuzzy measures. Furthermore, it could be transferred into the corresponding results of reals [5, 11, 1619], such as the addition operator, multiplication operator, differentiability, and integrability, by using Aczel’s representation [20, 21]. Gong and Xie [22] coincided the definition of -integrability with the definition of pseudo-integrability with respect to a decomposable measure in different papers, obtained Newton-Leibniz formula, and directly applied the results to the discussion of nonlinear differential equations. Sugeno and Murofushi [23] introduced an integral (briefly, SM integral) with respect to a pseudo-additive measure based on pseudo-operations. Note that the Choquet integral and the SM integral are extensions of the Lebesgue integral but not of the Sugeno integral and the SM integral does not cover some well-known integrals such as the Sugeno integral and the Choquet integral, in general. Mesiar [18] characterized the operations of pseudo-addition and pseudo-multiplication leading to integrals with properties similar to those of the Choquet and the Sugeno integral, respectively, and developed a type of integral based on the SM integral, the so-called Choquet-like integral, which generalized the concepts of some well-known integrals including both the Sugeno integral and the Choquet integral. However, as the basis for the pseudo-integrals, the definitions of the pseudo-operations and the relative measures have some differences. In fact, the pseudo-operations need to be continuous and valued in and the relative measures need to be continuous from below introduced in [18, 23, 24], while the pseudo-operations need not to be continuous and valued in , the relative measures need not to be continuous from below and the measurable function, and need not to be nonnegative in the relative integral in [5, 25]. In this paper, we generalized the Choquet-like integrals with respect to nonmonotonic measures based on the generalized definitions of pseudo-operations.

As is well known, the set-valued function, besides being an important mathematical notion, has become an essential tool in several practical areas, especially in economic analysis [26]. The integration of set-valued functions has roots in Aumann’s research [27] based on the classical Lebesgue integral. By using the approach of Aumann, Jang et al. [28] defined Choquet integrals of set-valued mappings aswhere is a measurable set-valued mapping and denotes the family of Choquet measurable selection of . In the field of the pseudo-analysis, an approach to the problem of integration of set-valued functions from the pseudo-analysis’ point of view has been introduced in [29]. Similarly, we introduce the Choquet-like integrals for set-valued functions.

On the other hand, integral inequalities are an important aspect of the classical mathematical analysis [30]. Generally, any integral inequality can be a very strong tool for applications. For example, when we think of an integral operator as a predictive tool, then an integral inequality can be very important in measuring and dimensioning such process. Recently, Flores, Agahi, Pap, and Mesiar et al. generalized several classical integral inequalities to Sugeno integral and choquet integral, including Chebyshev type inequality [31, 32], Jensen type inequality [33, 34], Stolarsky type inequality [35, 36], Hölder type inequality [37], Minkowski type inequalities [38], Carlson type inequality [39], and Liapunov type inequality [40]. Pseudo-analysis would be an interesting topic to generalize an inequality from the frame work of the classical analysis to that of some integrals which contain the classical analysis as special cases. In fact, Jensen inequality was generalized into pseudo-integrals by Pap and Štrboja [41], where two cases of real semirings defined by pseudo-operations were considered. In the first case, the pseudo-operations (pseudo-addition and pseudo-multiplication) are defined by the monotone and continuous function . In this case, the pseudo-integral reduces to the -integral. In the second case, the semiring (, sup, ) is used, where the pseudo-addition is the idempotent operation sup and is generated, as in the first case. Chebyshev type inequalities for pseudo-integrals were investigated in [42] and Chebyshev’s inequality for Choquet-like integral was subsequently introduced in [43]. Daraby [44] obtained generalization of the Stolarsky type inequality for pseudo-integrals. Li et al. [45] investigated generalization of the Lyapunov type inequality for pseudo-integrals. Jensen and Chebyshev inequalities for pseudo-integrals of set-valued functions were proved in [46]. In 2015, Agahi and Mesiar [47] introduced Cauchy-Schwarz’s inequality for Choquet-like integrals. In 2017, Mihailović and Štrboja [48] proposed the generalized Minkowski type inequality for pseudo-integrals. Abbaszadeh et al. established a refinement of the Hadamard integral inequality [49] for -integrals in 2018 and Hölder type integral inequalities [50] for pseudo-integrals by means of the above two cases of real semirings in 2019. As a further study, we generalize some of these inequalities to the frame of the Choquet-like integral presented in this paper and prove the Lyapunov type inequality and Stolarsky type inequality for the Choquet-like integral.

To make our analysis possible, we recall some basic results of the pseudo-analysis and the Choquet integral in Section 2. Section 3 defines the Choquet-like integral with respect to a nonmonotonic fuzzy measure and gives the characterization theorem and transformation theorem for the integral. In addition, the Choquet-like integral of set-valued functions is also obtained. The Lyapunov type inequality and Stolarsky type inequality for the Choquet-like integral are investigated in Sections 4 and 5, respectively.

2. Preliminaries

In the paper, the following concepts and notations will be used. denotes the set of all real numbers, denotes the set of generalized real numbers, denotes the set of extended nonnegative real numbers, denotes the class of all the subsets of . denotes a nonempty set, is a -algebra on , and is a measurable space. Let be a set function; then is called a fuzzy measure ([51]) or a pre-measure ([3, 18]), if

(1) ,

(2) whenever , .

The triplet is called a fuzzy measure space. We say that is finite if . When is finite, we define the conjugate of by for all .

For measurable functions , , and , we have ([4, 5]) the following:

(i) The integralis called a symmetric Choquet integral, also called Šipoš integral.

(ii) Suppose . The integralis called a asymmetric Choquet integral.

In the case that, the right-hand side is and the Choquet integral is not defined.

Definition 1 (see [9]). A nonmonotonic fuzzy measure on is a real-valued set function satisfying .

We can represent the relation between the fuzzy measure in the original monotonic version and the nonmonotonic fuzzy measure in the nonmonotonic version as , ; if , the members of are turned out. Generally, for each nonmonotonic fuzzy measure on , if we define a set function on by , , then is a monotonic fuzzy measure. We denote the set of monotonic fuzzy measures on by FM, and the set of nonmonotonic fuzzy measures of bounded variation on by BV.

Definition 2 (see [10]). For a given real-valued set function , the total variation of on is defined byA real-valued set function is said to be of bounded variation if .
A finite monotonic fuzzy measure is of bounded variation since .

For every , we define ([9, 10])where and . We call the total variation, positive (or upper) variation, and negative (or lower) variation of on , respectively.

Lemma 3 (see [10]). Let BV. We have(1) FM.(2).(3).

The Choquet integral of a measurable function with respect to a nonmonotonic fuzzy measure is defined by ([9])whenever the integral in the right-hand side exists, whereA measurable function is called integrable if the Choquet integral of exists and its value is finite.

Sugeno-Murofushi ([23]) introduced an integral (briefly, SM integral) with respect to a pseudo-additive measure based on pseudo-operations, where the pseudo-operations and pseudo-additive measures were defined as follows, respectively:

(1) A binary operation is called a pseudo-addition if it satisfies (P1) (neutral element), (P2) (associativity), (P3) and (monotonicity), and (P4) and (continuity).

Another binary operation on is called a pseudo-multiplication corresponding to if it satisfies (M1) , (M2) , (M3) or , (M4) such that for any , (M5) and and , (M6) , and (M7) .

(2) A set function is said to be a pseudo-additive measure with respect to (-additive measure, for short) ([23]) or -decomposable measure in Klement and Weber’s paper ([24]) if satisfies the following conditions:(i),(ii) and ,(iii) and .

The triple is called a -measure space.

Later, Mesiar ([18]) developed Choquet-like integrals based on the SM integrals. More on Choquet-like integrals can be found in ([43, 47, 52]). H. Ichihashi and E. Pap et al. ([5, 25]) generalized the above pseudo-operations from to and introduced the relative measures and integrals based on the generalized pseudo-operations. Let be a closed real interval of and be a total order on .

Definition 4 (see [25]). A 2-place function is called a pseudo-addition if it satisfies the following conditions
(i) is commutative,
(ii) is nondecreasing in each place (with respect to ),
(iii) is associative,
(iv) There exist a zero element, denoted by , i.e., for all .
A pseudo-addition is said to be continuous if it is a continuous function in ; a pseudo-addition is called strict if it is continuous and strictly monotone. Pseudo-addition is idempotent if for any , holds.

Definition 5 (see [25]). A 2-place function is called a pseudo-multiplication if it satisfies the following conditions
(i) is commutative,
(ii) is nondecreasing in each place (with respect to ),
(iii) is associative,
(iv) There exists a unit element, denoted by , i.e., for all .
A pseudo-multiplication is said to be continuous if it is a continuous function in .

For example, the usual addition +, , and -conorm are pseudo-additions; the usual multiplication , , and -norm are pseudo-multiplications.

The structure is called a semiring, where is a distributive pseudo-multiplication (corresponding to ); i.e., it is positively nondecreasing ( and There are three basic classes of semirings with continuous (up to some points) pseudo-operations. The first class contains semirings with idempotent pseudo-addition and nonidempotent pseudo-multiplication. Semirings with strict pseudo-operations defined by monotone and continuous generator function , i.e., -semirings, form the second class. In this paper, we generalize -semirings to generalized -semirings. Semirings with both idempotent operations belong to the third class. More on this structure can be found in [5, 1114].

Total order on is closely connected to the choice of the pseudo-addition. If is an idempotent operation, total order is induced by ; if is given by generalized generator , total order is given by . Additionally, and .

Let us suppose that the interval is endowed with metric. A function is a pseudo-metric if it satisfies the conditions (i) for all , (ii) all , and (iii) for all .

Definition 6 (see [5]). Let be a -algebra of subsets of . A set function is said to be a --measure if
(i) ,
(ii) , where is a sequence of pairwise disjoint sets from .

A pseudo-integral based on --measure is defined as ([5])

(i) for elementary function , where , , and is the pseudo-characteristic function of given by

(ii) for bounded measurable function ,where is a sequence of elementary functions such that uniformly while and is previously mentioned metric.

(iii) for function on some arbitrary subset of is given by

Notice that is also denoted by .

Lemma 7 (Aczel’s theorem, [20, 21]). If is continuous and strictly increasing in , then there exists a monotone function such that andwhere is called a generator of .

Obviously, the pseudo-addition is strict. And the pseudo-multiplication with the generator of strict pseudo-addition is defined asThe pseudo-operations with the generator are also called -operations.

It is not difficult to obtain the -power operation

Corollary 8. Let be a --measure; is continuous and strictly increasing in , and then there exists a monotone function such that and

Proof. According to Lemma 7 and by induction, it is not difficult to obtainLet , we have (16).

Example 9. Let ; is said to be a Sugeno measure ([53]), denoted by , if
(i) is normal, i.e., ,
(ii) satisfies the - rule, i.e.,where , , and , .
Then the sugeno measure is a --measure, and the generating function for pseudo-addition isthenObviously, if , then we have ; if , then we haveBy induction, we obtainMoreover, notice that if , then the - rule is -+-additive, i.e., -additive, and is the probability measure; if and , we havewhen , is said to be -additive; when , it is the addition formula of Probability.

Definition 10 (see [54]). Let be a strictly monotone real-valued function defined on such that . The generalized generated pseudo-addition and the generalized generated pseudo-multiplication are given bywhere is pseudo-inverse function for function .

Remark 11. For nondecreasing function , where and are closed subintervals of the generalized real line , the pseudo-inverse is . If is nonincreasing function, its pseudo-inverse is . More on this subject can be found in ([55]).

The semiring is called a generalized -semiring, where pseudo-operations defined by the generalized generator function .

Proposition 12. Let be a --measure and be generalized generated by a generator .
(i) If the generating function is either strictly increasing right-continuous or strictly decreasing left-continuous function such that , then(ii) If the generating function is either strictly increasing left-continuous or strictly decreasing right-continuous function such that , then(iii) If the generating function is a monotone bijection, then

Proof. For strictly increasing right-continuous or strictly decreasing left-continuous generating function that fulfills condition or , respectively, holds , for all . According to Definition 10 and by induction, we haveProof for (ii) is similar and based on , for all . In (iii) pseudo-inverse coincides with inverse, which gives us (28).

Remark 13. If the generalized generator is a monotone bijection, then the pseudo-inverse coincides with the inverse. We haveThe integral is said to be -integral ([11, 16, 17, 19, 22]). For the sake of brevity, we denote .
In addition, the generalized -power operation isWe give the definition of comonotonic, which is similar to the definition of comonotonic [6] or compatible ([51]) in real-analysis. Let and be generalized real-valued bounded measurable functions on . We say and are comonotonic, denoted by , if for . We denote by the set of bounded measurable functions on . Let be a functional defined on .

Definition 14. (1) is said to be comonotonically -additive if .
(2) is said to be positively -homogeneous if , .
(3) is said to be monotonic if .

The total variation of is defined by ([9]) is said to be of bounded variation if . Note that if is monotonic, then and hence is of bounded variation.

For every pair of and of functions in for which , we define by

3. Choquet-Like Integral with respect to a Nonmonotonic Fuzzy Measure

In this section, we introduce the Choquet-like integrals based on and with respect to (w.r.t) nonmonotonic fuzzy measures for generalized real-valued functions and set-valued functions. In addition, the characterization theorem and transformation theorem for the integrals are given.

3.1. Choquet-Like Integral with respect to a Nonmonotonic Fuzzy Measure for Generalized Real-Valued Functions

Definition 15. Let be a nonmonotonic fuzzy measure, be a --measure satisfying for and for , and be a given pseudo-addition and corresponding a pseudo-multiplication . Let be a -measurable function and be a measurable set. Then the integral of with respect to the nonmonotonic fuzzy measure over defined bywherewill be called a Choquet-like integral if it is
(1) monotone; i.e., implies ,
(2) comonotone -additive; i.e., implies ,
(3) positively -homogeneous; i.e., for ,
(4) coincident; i.e., is -additive if and only if is -additive; if is continuous from below, -additive and nonnegative, and is nonnegative, then .
Instead of , we shall write . If the Choquet-like integral of a measurable function exists and its value is finite, we say is Choquet-like integrable, denoted by , i.e., .

Remark 16. If , , the Choquet-like integral coincides with the Choquet integral w.r.t. a nonmonotonic fuzzy measure introduced in ([9]); if , , is monotone, and is nonnegative, the Choquet-like integral coincides with the Sugeno integral; if are continuous, is continuous from below and nonnegative, and is nonnegative, then the Choquet-like integral is coincident with the Choquet-like integral introduced by Mesiar ([18]).

Proposition 17. Let be Choquet-like integrable. If , then for any real numbers , we have

Proof. Since is Choquet-like integrable and , according to Definition 15, and are comonotone -additive and positively -homogeneous; that is, for , we have

Corollary 18. Let be Choquet-like integrable. If , then for every fuzzy measure on , we havewhere is the ordinary Choquet integral w.r.t fuzzy measure .

This result was proved with use of the representation theory of fuzzy measures by Murofushi-Sugeno in [51].

Theorem 19. Let be a functional defined on , BV, and . If , , then is comonotonically -additive, positively -homogeneous, and of bounded variation.

Proof. Since , according to Definition 15, is comonotonically -additive, positively -homogeneous. Then we prove is of bounded variation. It follows from the definition of the total variation that and . Since BV, we have , and FM, thus, we obtainthat is, is of bounded variation. Therefore, is of bounded variation.

Theorem 20. Let and be generalized generated by a generator . If is a monotone bijection, then the Choquet-like integral of a measurable function over a measurable set w.r.t. a nonmonotonic fuzzy measure can be represented aswhere , is pseudo-inverse function for function , denotes the Choquet integral w.r.t a nonmonotonic fuzzy measure, and the right-hand side integral is the Lebesgue integral.

Proof. Let and be a --measure. Since is a monotone bijection, , for disjoint sets , we haveBy induction, it is easy to see that , , is a +-decomposable measure, i.e., a -additive measure. Consequently, is a -additive measure on Borel subsets of such that for each and for each . According to Definition 15, we haveSince is a monotone bijection, we see thatwhere the integral on the right-hand side is the Lebesgue integral on . Let be a function on identical with the first derivative of the generalized generator in those points, where this derivative exists (recall that is a strictly monotone function). Thenwhere is the common Lebesgue measure on and the substitution is used. Therefore,

Remark 21. If and is monotone, then the integral coincides with the symmetric Choquet integral. Moreover, when is nonnegative, the integral coincides with the original Choquet integral.
The theorem shows that Choquet-like integral w.r.t a nonmonotonic fuzzy measure can be transformed into the Choquet integral w.r.t a nonmonotonic fuzzy measure and the Lebesgue integral.
Note that the Choquet-like based on the -operations will be called a -Choquet-like integral and we denote .

Proposition 22. Let and be nonmonotonic fuzzy measures and and be generalized generated by a generator . If is a monotone bijection, then for any real numbers and , we have

Proof. Since is a monotone bijection, , according to Theorem 20, we haveSince the Choquet integral is linear with respect to the nonmonotonic fuzzy measure, we obtain

Lemma 23 (see [6]). If is a continuous, comonotonically additive functional on and if , , then is positively homogeneous and .

Theorem 24. Let be a functional defined on . Then the following conditions are equivalent to one another:
(a) If , , then BV and .
(b) is comonotonically -additive, positively -homogeneous, and of bounded variation.
(c) is comonotonically -additive and uniformly continuous.

Proof. (a)(b). It can be easily obtained by Theorem 19.
(b)(c). Let be a positively -homogeneous and comonotonically -additive functional of bounded variation and . If we write , then obviously , and hence it follows thatTherefore, is uniformly continuous.
(c)(a). Let , . Since the uniform continuity implies the continuity, it follows from Theorem 20 and Lemma 23 that . We shall prove that BV. Assume that is not of bounded variation. Then for each positive integer there is a finite sequence such that andWe now put and , whereand then it is easy to see that butand thus, is not uniformly continuous. This contradicts the fact that is uniformly continuous. Therefore, is of bounded variation.

Corollary 25. If is monotonic, then a functional on is represented as a -Choquet-like integral with respect to the monotonic fuzzy measure if and only if is a monotonic and comonotonically -additive.

3.2. Choquet-Like Integral with respect to a Nonmonotonic Fuzzy Measure for Set-Valued Functions

A set-valued mapping is a mapping , and it is said to be measurable if for every , where is the Borel algebra of . Let be measurable set-valued mappings and be a nonmonotonic fuzzy measure on . If , then we say equals almost everywhere, denoted by a.e.

Definition 26. Let be a set-valued function and . Then the Choquet-like integral of on is defined bywhere is the family of Choquet-like integrable selections of , i.e.,Specially, when coincides with Lebesgue integral and is nonnegative, the set-valued Choquet-like integral is the classical Aumann’s integral.
For a set-valued function , we say that it is pseudo-integrable on some if .

Let ; we say if for all there exists such that and for all there exists such that .

Definition 27. A set-valued function is said to be Choquet-like integrably bounded if there is a function such that
(i) , for the idempotent pseudo-addition,
(ii) , for the pseudo-addition given by an increasing generalized generator ,
(iii) , for the pseudo-addition given by a decreasing generalized generator .
Note that if and nonnegative, i.e., is idempotent, then the definition of Choquet-like integrably bounded is coincident with the definition of Choquet integrably bounded proposed in [28].

Theorem 28. If is a Choquet-like integrably bounded set-valued function, then is Choquet-like integrable.

Proof. Let be a Choquet-like integrably bounded set-valued function; that is, let us suppose that the function from Definition 27 exists. If on -a.e., the set (53) is obviously not empty. If , let be a selection of , i.e., -a.e. on . It can be easily shown that holds almost everywhere. According to Definition 15, we haveSince , is a Choquet-like integrable function, thus, the function is also Choquet-like integrable and the set (53) is not empty.
For example, if and be generalized generated by a generator , then the Choquet-like integral of some set-valued function is

Proposition 29. Let be pseudo-integrable set-valued function, and Choquet-like integrably bounded set-valued functions and let .
(1) If , , then .
(2) If , then .
(3) If , then .

Proof. (1) Suppose that . By Definition 27, there exists such that . According to the definition of pseudo-integral, . Since , we have , thus, ; according