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

Ting Xie,Zengtai Gong

doi:10.1155/2019/4631530

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 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.

Highlights

The Choquet integral with respect to a fuzzy measure λ, which is monotone, does not require continuity and was proposed by Murofushi and Sugeno [1]
To generate the Choquet integral to the generalized real valued measurable function, the symmetric Choquet integral, which was most early proposed by Sipos [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

Summary

Introduction

The Choquet integral with respect to a fuzzy measure λ, which is monotone, does not require continuity and was proposed by Murofushi and Sugeno [1]. 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 λ and any nonnegative single-valued measurable function f, nowadays called a Sugeno integral, as follows:. The Lyapunov type inequality and Stolarsky type inequality for the Choquet-like integral are investigated in Sections 4 and 5, respectively

Preliminaries

Choquet-Like Integral with respect to a Nonmonotonic Fuzzy Measure

Choquet-Like Integral with respect to a Nonmonotonic

Lyapunov Type Inequality for the Choquet-Like Integral

Stolarsky Type Inequality for the Choquet-Like Integral