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On the basic solutions to the generalized fuzzy integral equation. (English) Zbl 0929.45001

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.

MSC:

45A05 Linear integral equations
28E10 Fuzzy measure theory
26E50 Fuzzy real analysis
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References:

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