Abstract and Applied Analysis

Choquet Integral of Fuzzy-Number-Valued Functions: The Differentiability of the Primitive with respect to Fuzzy Measures and Choquet Integral Equations

Zengtai Gong, Li Chen, and Gang Duan

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Abstract

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.

Article information

Source
Abstr. Appl. Anal., Volume 2014 (2014), Article ID 953893, 11 pages.

Dates
First available in Project Euclid: 2 October 2014

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1412277042

Digital Object Identifier
doi:10.1155/2014/953893

Mathematical Reviews number (MathSciNet)
MR3224329

Zentralblatt MATH identifier
07023390

Citation

Gong, Zengtai; Chen, Li; Duan, Gang. Choquet Integral of Fuzzy-Number-Valued Functions: The Differentiability of the Primitive with respect to Fuzzy Measures and Choquet Integral Equations. Abstr. Appl. Anal. 2014 (2014), Article ID 953893, 11 pages. doi:10.1155/2014/953893. https://projecteuclid.org/euclid.aaa/1412277042


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