On the basic solutions to the generalized fuzzy integral equation
type: Item
instance of: scholarly article
title: On the basic solutions to the generalized fuzzy integral equation
zbMATH Open document ID: 0929.45001
DOI: 10.1016/S0165-0114(96)00297-7
author: Congxin Wu
author: Haiyan Wang
author: Shiji Song
published in: Fuzzy Sets and Systems
publication date: 2000-01-23T00:00:00Z
review text: Fuzzy measures and fuzzy integrals were introduced and examined by M. Sugeno [cf. \textit{M. M. Gupta, G. N. Saridis} and \textit{B. R. Gaines} (eds.), Fuzzy automata and decision processes, North-Holland, Amsterdam, 89-102 (1977; Zbl 0378.68035)]. Generalized fuzzy integrals based on generalized triangular norms were introduced by \textit{C. Wu, S. Wang}, and \textit{M. Ma} [Fuzzy Sets Syst. 57, No. 2, 219-226 (1993; Zbl 0786.28016)]. The authors of the present paper consider a condition for a function \(f\) with a constant fuzzy integral (e.g. all probability distribution functions fulfil a similar condition with the Lebesgue-Stieltjes integral). A condition of the form \(\sup_{\alpha>0} \min(\alpha,\mu(\{x\in A: \min(f(x),h(x)) > \alpha\})) = \beta\) with given constant \(\beta\), measure \(\mu\), measurable set \(A\), and measurable function \(h\) is rather not connected with the theory of integral equations. Constant and step functions \(f\) fulfilling such conditions are described.
Mathematics Subject Classification ID: 45A05
Mathematics Subject Classification ID: 28E10
Mathematics Subject Classification ID: 26E50
zbMATH DE Number: 1305162
zbMATH Keywords: fuzzy measure
zbMATH Keywords: fuzzy integral
zbMATH Keywords: generalized t-norm
zbMATH Keywords: generalized fuzzy integral equation
zbMATH Keywords: basic solutions
reviewed by: Józef Drewniak
MaRDI profile type: Publication
cites work: Generalized fuzzy integrals. III: Convergence theorems
cites work: Generalized fuzzy integrals. I: Fundamental concepts
cites work: On fuzzy integral inequalities and fuzzy expectation
cites work: The autocontinuity of set function and the fuzzy integral
description: scientific article; zbMATH DE number 1305162