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Choquet integral of fuzzy-number-valued functions: the differentiability of the primitive with respect to fuzzy measures and Choquet integral equations. (English) Zbl 1474.28029

Summary: This paper deals with the Choquet integral of fuzzy-number-valued functions based on the nonnegative real line. We firstly give the definitions and the characterizations of the Choquet integrals of interval-valued functions and fuzzy-number-valued functions based on the nonadditive measure. Furthermore, the operational schemes of above several classes of integrals on a discrete set are investigated which enable us to calculate Choquet integrals in some applications. Secondly, we give a representation of the Choquet integral of a nonnegative, continuous, and increasing fuzzy-number-valued function with respect to a fuzzy measure. In addition, in order to solve Choquet integral equations of fuzzy-number-valued functions, a concept of the Laplace transformation for the fuzzy-number-valued functions in the sense of Choquet integral is introduced. For distorted Lebesgue measures, it is shown that Choquet integral equations of fuzzy-number-valued functions can be solved by the Laplace transformation. Finally, an example is given to illustrate the main results at the end of the paper.

MSC:

28E10 Fuzzy measure theory
26E50 Fuzzy real analysis
45N05 Abstract integral equations, integral equations in abstract spaces

Software:

kappalab
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Full Text: DOI

References:

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