^{1}

^{2}

^{1}

^{1}

^{2}

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

The Choquet integral with respect to a fuzzy measure

Sugeno introduced another integral for any fuzzy measure

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 [

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 [

On the other hand, integral inequalities are an important aspect of the classical mathematical analysis [

To make our analysis possible, we recall some basic results of the pseudo-analysis and the Choquet integral in Section

In the paper, the following concepts and notations will be used.

(1)

(2)

The triplet

For measurable functions

(i) The integral

(ii) Suppose

In the case that, the right-hand side is

A nonmonotonic fuzzy measure on

We can represent the relation between the fuzzy measure

For a given real-valued set function

A finite monotonic fuzzy measure

For every

Let

The Choquet integral of a measurable function

Sugeno-Murofushi ([

(1) A binary operation

Another binary operation

(2) A set function

The triple

Later, Mesiar ([

A 2-place function

(i)

(ii)

(iii)

(iv) There exist a zero element, denoted by

A pseudo-addition is said to be continuous if it is a continuous function in

A 2-place function

(i)

(ii)

(iii)

(iv) There exists a unit element, denoted by

A pseudo-multiplication is said to be continuous if it is a continuous function in

For example, the usual addition +,

The structure

Total order

Let us suppose that the interval

Let

(i)

(ii)

A pseudo-integral based on

(i) for elementary function

(ii) for bounded measurable function

(iii) for function

Notice that

If

Obviously, the pseudo-addition

It is not difficult to obtain the

Let

According to Lemma

Let

(i)

(ii)

Then the sugeno measure

By induction, we obtain

Moreover, notice that if

Let

For nondecreasing function

The semiring

Let

(i) If the generating function

(ii) If the generating function

(iii) If the generating function

For

Proof for (ii) is similar and based on

If the generalized generator is a monotone bijection, then the pseudo-inverse coincides with the inverse. We have

In addition, the generalized

We give the definition of comonotonic, which is similar to the definition of comonotonic [

(1)

(2)

(3)

The total variation

For every pair of

In this section, we introduce the Choquet-like integrals based on

Let

will be called a Choquet-like integral if it is

(1) monotone; i.e.,

(2) comonotone

(3) positively

(4) coincident; i.e.,

Instead of

If

Let

Since

Let

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

Let

Since

Let

Let

If

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

Let

Since

If

Let

(a) If

(b)

(c)

(a)

(b)

(c)

If

A set-valued mapping is a mapping

Let

For a set-valued function

Let

A set-valued function

(i)

(ii)

(iii)

Note that if

If

Let

For example, if

Let

(1) If

(2) If

(3) If

(1) Suppose that

The statements (2) and (3) follow directly from Definition

Let

By the definition of pseudo-integral, we have

Set

Let

If

If (

In this section, we discuss the Lyapunov and Stolarsky type inequality for the Choquet-like integral based on the semiring

According to probability theory, the classical Liapunov inequality provides the inequality ([

Let the generalized generator

Using the classical Lyapunov inequality, then we obtain

If

Suppose that

According to Theorem

The classical Stolarsky integral inequality provides the inequality ([

Let the generalized generator

Using the classical Stolarsky inequality and then we obtain

Let

Suppose that

No data were used to support this study.

The authors declare that they have no conflicts of interest.

This research is supported by the National Natural Science Foundation of China (61763044).