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Abstract and Applied Analysis
Volume 2014, Article ID 953893, 11 pages
http://dx.doi.org/10.1155/2014/953893
Research Article

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

1College of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, China
2Department of Mathematics, Lanzhou City University, Lanzhou 730070, China
3School of Traffic and Transportation, Lanzhou Jiaotong University, Lanzhou 730070, China

Received 25 January 2014; Accepted 22 May 2014; Published 9 June 2014

Academic Editor: Marco Donatelli

Copyright © 2014 Zengtai Gong et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.