Wu, Congxin; Song, Shiji; Wang, Haiyan On the basic solutions to the generalized fuzzy integral equation. (English) Zbl 0929.45001 Fuzzy Sets Syst. 95, No. 2, 255-260 (1998). Fuzzy measures and fuzzy integrals were introduced and examined by M. Sugeno [cf. M. M. Gupta, G. N. Saridis and 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 C. Wu, S. Wang, and 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. Reviewer: J.Drewniak (Katowice) Cited in 10 Documents MSC: 45A05 Linear integral equations 28E10 Fuzzy measure theory 26E50 Fuzzy real analysis Keywords:fuzzy measure; fuzzy integral; generalized t-norm; generalized fuzzy integral equation; basic solutions Citations:Zbl 0378.68035; Zbl 0786.28016 PDFBibTeX XMLCite \textit{C. Wu} et al., Fuzzy Sets Syst. 95, No. 2, 255--260 (1998; Zbl 0929.45001) Full Text: DOI References: [1] Congxin, W.; Ming, M.; Shifi, S.; Shaotai, Z., Generalized fuzzy integrals: Part 3, convergent theorems, Fuzzy Sets and Systems, 70, 75-87 (1995) [2] Congxin, W.; Shuli, W.; Ming, M., On generalized fuzzy integrals: Part 1, fundamental concepts, Fuzzy Sets and Systems, 57, 219-226 (1993) · Zbl 0786.28016 [3] Di Nola, A.; Venter, G. S., On fuzzy integral inequalities and fuzzy expectation, J. Math. Anal. Appl., 125, 589-599 (1987) · Zbl 0654.28014 [4] Zhenyuan, W., The autocontinuity of set function and fuzzy integral, J. Math. Anal. Appl., 99, 195-218 (1984) · Zbl 0581.28003 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.