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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

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Listed:
  • Zengtai Gong
  • Li Chen
  • Gang Duan

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.

Suggested Citation

  • Zengtai Gong & Li Chen & Gang Duan, 2014. "Choquet Integral of Fuzzy‐Number‐Valued Functions: The Differentiability of the Primitive with respect to Fuzzy Measures and Choquet Integral Equations," Abstract and Applied Analysis, John Wiley & Sons, vol. 2014(1).
    • Handle: RePEc:wly:jnlaaa:v:2014:y:2014:i:1:n:953893
    DOI: 10.1155/2014/953893
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    References listed on IDEAS

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    1. Schmeidler, David, 1989. "Subjective Probability and Expected Utility without Additivity," Econometrica, Econometric Society, vol. 57(3), pages 571-587, May.
    2. Abbasbandy, S. & Babolian, E. & Alavi, M., 2007. "Numerical method for solving linear Fredholm fuzzy integral equations of the second kind," Chaos, Solitons & Fractals, Elsevier, vol. 31(1), pages 138-146.
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