Elsevier

Fuzzy Sets and Systems

Volume 304, 1 December 2016, Pages 110-130
Fuzzy Sets and Systems

Inequalities of Hölder and Minkowski type for pseudo-integrals with respect to interval-valued ⊕-measures

https://doi.org/10.1016/j.fss.2015.11.014Get rights and content

Abstract

In the present paper, the Hölder and Minkowski type of inequality for the pseudo-integral of a real-valued function with respect to the interval-valued pseudo-additive measure is proven. Two cases of semirings are considered. In the first case, pseudo-operations (pseudo-addition and pseudo-multiplication) are set by a strictly monotone continuous function. In the second case, the pseudo-addition is the idempotent operation sup, and pseudo-multiplication is specified by a strictly monotone continuous function, as in the first case. Trivial and nontrivial examples of interval-valued pseudo-additive measures are provided, as well as Hölder and Minkowski type of inequalities with respect to those measures.

Introduction

In mathematical analysis, inequalities between integrals, such as Hölder and Minkowski inequalities, are important tools for studying Lp(m) spaces. Let (X,Σ,m) be a space with a nonnegative measure m. From Hölder inequality follows that if fLp(m) and gLq(m), where p,q(1,) and 1p+1q=1, then fgL1(m). Minkowski inequality implies that if f,gLp(m), where p[1,], then f+gLp(m).

One generalization of mathematical analysis is pseudo-analysis. It is obtained by the replacement of the field of real numbers with a real interval [a,b][,] and by the replacement of the operations of addition and multiplication with some new operations, so-called pseudo-addition and pseudo-multiplication. In pseudo-analysis field, instead of (additive) measures, non-additive monotone set-functions with additional property, so-called pseudo-additivity are considered. Those set-functions are called pseudo-additive measures. Based on pseudo-operations and pseudo-additive measures, pseudo-integrals are obtained [18], [19], [21]. There are various relations and applications of pseudo-analysis to purely mathematical, as well as applied mathematical sciences. For example, pseudo-analysis has been used in solving non-linear partial differential equations [20] and game theory [13], [14]. The inequalities of Hölder and Minkowski type in pseudo-analysis surrounding have been studied in [2], [3].

The investigation of upper and lower probabilities is performed in [6]. For a measurable space (X,Σ), two set functions P and P_ are considered. Both P and P_ map ∅ to 0 and Ω to 1, P_ has the property of super-additivity, P has the property of sub-additivity and P_(A)+P(XA)=1, for all AΣ. Set-valued functions P and P_ are called lower and upper probability functions, respectively.

Upper and lower probabilities can also be introduced using multi-valued mappings. Let (Ω,Σ,P) be a probability space, (X,Σ) a measurable space and Γ:ΩX a multi-valued mapping. For AΣ and sets A=Γ(A)={ωΩ:Γ(ω)A} and A=Γ(A)={ωΩ:Γ(ω)A}, upper probability induced by multi-valued mapping Γ on Σ is defined by P(A)=P(A) and lower probability is P(A)=P(A), for every AΣ. For the class S(Γ) of the measurable selections of multi-valued mapping Γ and the class P(Γ)={PU:US(Γ)} of probability distributions induced on Σ by measurable selections, in the paper [17], authors investigate the conditions under which the equality P(Γ)(A)=[P(A),P(A)] holds, i.e. P(Γ)(A) can be represented as the interval. Also, it is shown when P(A)=maxP(Γ)(A) and P(A)=minP(Γ)(A).

Non-additive interval-valued measures and integrals of real-valued functions with respect to this type of measures are studied in [7], [10], [12].

One generalization of pseudo-additive measures can be constructed using the family M of pseudo-additive measures, such that (M,) is a dense linear order with endpoints, where ⪯ is a total order [7]. The obtained pseudo-additive measure is an interval-valued measure, i.e. it has the form [μl,μr] and it has the property of pseudo-additivity. This type of measure, the pseudo-integral of a real-valued function with respect to it, and the inequalities of Jensen and Chebyshev type for this type of measures are investigated in [7].

Choquet integral with respect to an interval-valued capacity is considered in [12]. Some convergence theorems for interval-valued capacity functionals and interval-valued probability measures are proven. Choquet integral of a bounded real-valued continuous function with respect to the interval-valued upper semi-continuous capacity has the similar form as the pseudo-integral of a real-valued function with respect to the interval-valued pseudo-additive measure considered in the present paper.

So-called p-additive set-valued measures and the set-valued fuzzy measure defined by set-valued fuzzy integrals are studied in [10]. The fuzzy integral used in this paper is Sugeno's integral. The concept of the set-valued g-measure is introduced and Radon–Nikodym Theorem is shown.

Our first results on Hölder and Minkowski type inequalities based on g-integrals and some elementary examples were presented at the 12th IEEE International Symposium on Intelligent Systems and Informatics, Subotica, Serbia. The paper [15] deals with Chebyshev, Hölder and Minkowski inequalities based only on g-integrals (as just one special case of pseudo-integrals) with respect to the interval-valued pseudo-additive measure.

The rest of the paper is organized as follows. Section 2 contains necessary facts on pseudo-operations, semirings, pseudo-additive measures and pseudo-integrals, as well as certain facts from [18], [19], [21]. This section also contains the definition of the interval-valued pseudo-additive measure and the construction of the pseudo-integral of a real-valued function with respect to it [7], [15]. It is shown that, by using a sequence of probability measures and a strictly monotone function g, it is possible to construct a nontrivial interval-valued pseudo-additive measure. In the third section, the notion of pseudo-power is extended to the pseudo-power of a set. Since the research is focused on closed intervals and g-semiring, some basic properties of the pseudo-power of intervals are proven and illustrative examples are provided. In the fourth section the main results of this paper, namely Hölder and Minkowski type of inequalities for the pseudo-integral of a real-valued function with respect to the interval-valued pseudo-additive measure, are proven. In the special case, when the interval-valued measure [μ,μ]={μ}, the results coincide with the results from [3]. This section also contains certain illustrative examples, for Hölder and Minkowski type of inequalities, with respect to the interval-valued pseudo-additive measure constructed in one of the examples from Section 2. Concluding remarks are presented in the final section.

Section snippets

Preliminary notions

This section contains some basic notions, such as pseudo-operations and a semiring, that are the core of the presented results. Pseudo-operations are extended on the space of nonempty subsets of [a,b]. This section also contains the definition of a ⊕-measure and an interval-valued ⊕-measure. The notion of the pseudo-integral of a real-valued function with respect to the ⊕-measure, as well as the pseudo-integral of a real-valued function with respect to the interval-valued ⊕-measure, is given

Pseudo-power on sets

For αR+, the pseudo-power x(α) can be extended to the pseudo-power A(α) of a set A[a,b]+ in the following manner.

Definition 3.1

For A[a,b]+ and αR+ we defineA(α)={x(α):xA}.

If pseudo-multiplication ⊙ is an idempotent operation (=sup or =inf) then x(α)=x and A(α)=A.

In the work that follows, we will only consider the non-idempotent pseudo-multiplication given by a generator g.

Theorem 3.2

For n,mN and xy=g1(g(x)g(y)), where g is a generating function, the following holds

  • i)

    [c,d](n)={x(n):cxd}=[c(n),d(n)],

  • ii)

    [c,d](1n)={x

The inequalities of Hölder and Minkowski type based on the interval-valued measure μM

This section contains the main results of this paper, i.e. Hölder and Minkowski type inequalities based on the interval-valued ⊕-measure (6).

Let ([a,b],,) be a semiring such that pseudo-multiplication is given by some generator g. Let u,v:X[a,b]+ be measurable pseudo-integrable functions, i.e., pseudo-integrals Xudμ and Xvdμ exist as a finite value in the sense of the observed semiring for all μM.

Conclusion

Some of the recent research by a number of authors is focused on inequalities for the integrals based on non-additive measures (see [1], [2], [3], [4], [5], [11], [22], [23], etc.). Since integrals with respect to ⊕-measure where =sup or =inf can be obtained as limits of families of g-integrals [16], the focus of this research was on g-semirings and the corresponding pseudo-integral of real-valued function with respect to interval-valued ⊕-measure. For the future work we propose Hölder and

Acknowledgements

The authors are partially supported by the Ministry of Science and Technological Development of the Republic of Serbia through grants III41103, TR36001, TR32035, OI174009 and TR36030.

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