## 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

#### 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

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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