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

The aim of this paper is to generalize the Choquet-like integral with respect to a nonmonotonic fuzzy measure for generalized realvalued functions and set-valued functions, which is based on the generalized pseudo-operations and σ-⊕-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.


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 as where   = { ∈  | () ⩾ }.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,[11][12][13][14].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 pseudomultiplications.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,[16][17][18][19], 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 pseudomultiplication 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 pseudooperations and the relative measures have some differences.In fact, the pseudo-operations need to be continuous and valued in [0, +∞] 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 as where  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 pseudooperations were considered.In the first case, the pseudooperations (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.

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, A is a -algebra on , and (, A) is a measurable space.Let  : A → [0, ∞] be a set function; then  is called a fuzzy measure ( [51]) or a pre-measure ( [3,18]), if (1) (0) = 0, (2) () ⩽ () whenever  ⊂ , ,  ∈ A.
The triplet (, A, ) is called a fuzzy measure space.We say that  is finite if () < +∞.When  is finite, we define the conjugate   of  by   () = () − (  ) for all  ∈ A.
In the case that, the right-hand side is ∞ − ∞ and the Choquet integral is not defined.
We can represent the relation between the fuzzy measure  in the original monotonic version and the nonmonotonic fuzzy measure  in the nonmonotonic version as () = max{() |  ⊆ }, ∀ ⊆ ; if () = (), the members of  \  are turned out.Generally, for each nonmonotonic fuzzy measure  on (, A), if we define a set function  on (, A) by () = sup{() |  ⊆ ,  ∈ A}, ∀ ∈ A, then  is a monotonic fuzzy measure.We denote the set of monotonic fuzzy measures on (, A) by FM(, A), and the set of nonmonotonic fuzzy measures of bounded variation on (, A) by BV(, A).Definition 2 (see [10]).For a given real-valued set function  : A → , the total variation () of  on  is defined by A real-valued set function  is said to be of bounded variation if () < +∞.
Lemma 3 (see [10]).Let  ∈ BV(, A).We have ( 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, where A measurable function  is called integrable if the Choquet integral of  exists and its value is finite.) 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 Another binary operation ⊙ on  + is called a pseudo- (2) A set function μ : A → [0, ∞] is said to be a pseudoadditive measure with respect to ⊕ (⊕-additive measure, for short) ( [23]) or ⊕-decomposable measure in Klement and Weber's paper ( [24]) if μ satisfies the following conditions: The triple (, A, μ) is called a ⊕-measure space.
A pseudo-integral based on -⊕-measure  is defined as (ii) for bounded measurable function where   is a sequence of elementary functions such that (  , ()) → 0 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  : where  is called a generator of ⊕.
Obviously, the pseudo-addition ⊕ is strict.And the pseudo-multiplication with the generator  of strict pseudoaddition ⊕ is defined as The 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  : [, ] → [0, ∞] such that (0) = 0 and Proof.According to Lemma 7 and by induction, it is not difficult to obtain Let  → ∞, we have (16).
Then the sugeno measure g  is a -⊕-measure, and the generating function for pseudo-addition ⊕ is then Obviously, if  = 0, then we have By induction, we obtain Moreover, notice that if  = 0, then the - rule is -+additive, i.e., -additive, and g  is the probability measure; if  ̸ = 0 and  = 2, we have when  ̸ = −1, g  is said to be -additive; when  = −1, it is the addition formula of Probability.
Remark 13.If the generalized generator is a monotone bijection, then the pseudo-inverse coincides with the inverse.We have The integral is said to be -integral ([11, 16, 17, 19, 22]).For the sake of brevity, we denote () = .
In addition, the generalized -power operation is We 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 (, A) the set of bounded measurable functions on (, A).Let  be a functional defined on (, A).
The total variation () of  is defined by ( [9]) is said to be of bounded variation if () < ∞.Note that if  is monotonic, then (1) = (1) and hence  is of bounded variation.
For every pair of  and  of functions in (, A) for which  ≤ , we define   (, ) by

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 setvalued functions.In addition, the characterization theorem and transformation theorem for the integrals are given.

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 ([0, ]) =  for  ∈ [0, ∞] and ([, 0]) = − for  ∈ [−∞, 0], and ⊕ be a given pseudoaddition and corresponding a pseudo-multiplication ⊙.Let  :  →  be a A-measurable function and  ∈ A be a measurable set.Then the integral of  with respect to the nonmonotonic fuzzy measure  over  defined by where Proof.Since ,  is Choquet-like integrable and  ∼ , according to Definition 15, () ∫ ⊕,⊙  and () ∫ ⊕,⊙  are comonotone ⊕-additive and positively ⊙-homogeneous; that is, for ,  > 0, we have Corollary 18.Let ,  be Choquet-like integrable.If  ∼ , then for every fuzzy measure  on A, we have where () ∫ 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 20.Let ⊕ and ⊙ be generalized generated by a generator .If  is a monotone bijection, then the Choquetlike integral of a measurable function  :  →  over a measurable set  ∈ A w.r.t. a nonmonotonic fuzzy measure  can be represented as  ( Since  is a monotone bijection, we see that (  ∩ ) ) , (43) where the integral on the right-hand side is the Lebesgue identical with the first derivative of the generalized generator  in those points, where this derivative exists (recall that  is a strictly monotone function).Then where  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,  (−1) () = , according to Theorem 20, we have Since 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 (, A) and if () = (1  ), ∀ ∈ A, then  is positively homogeneous and  =   .
Note that if ⊕ = sup 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].
Since ℎ ∈  1  , ℎ 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 setvalued function  is

Lyapunov Type Inequality for the Choquet-Like Integral
In this section, we discuss the Lyapunov and Stolarsky type inequality for the Choquet-like integral based on the semiring ([0, 1], ⊕, ⊙).Without loss generality, suppose that 0 ≺ 1.

Stolarsky Type Inequality for the Choquet-Like Integral
The classical Stolarsky integral inequality provides the inequality ( [57]) where  is a nonmonotonic fuzzy measure.
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 [, ].