The Properties Of Multi-dimensional Fuzzy Integral And Generalized Fuzzy Integral

Fuzzy integral(Liu integral) has been widely applied in the dynamic fuzzy phenomenon. Liu integral, Sugeno integral and Choquet integral are fuzzy integrals with respect to different measure, this led to their common features. The operational structure of Choquet integral is selecting the maximum or minimum and Choquet integral’s is making multiplication and summation. Although Liu integral’s operational structure is also making multiplication and summation, the differences with them lies in its integral variable dosen’t accords with the normality. When try to make a further study for Liu integral, it is not completely identical with Choquet integral. But for Liu process is not differentiable, it is difficult to calculate fuzzy differential equation driven by Liu process and prove the existence and uniqueness of fuzzy differential equation driven by Liu process using the definition of Liu integral. In general, fuzzy integral can not be flexibly applied to various fields as usual function integration, so it is necessary to study the properties of fuzzy integral and such research is one of the currently hot topics. Many researches have been done concerning fuzzy integral, such as Lipschitz continuity and the existence and uniqueness theorems.Based on the Lipshitz and continuity of Liu progress and the properties of fuzzy differential equations, Liu integral was extended to multi-integral and the integral interval was extended to the case of infinite, at the same time, the integrand of fuzzy integral was extended to the case of unbounded function.So that the theory of fuzzy integral will be more richer and complete, and the calculating will be more convenient.This paper is structured as follows: First of all, introduced some basic concepts of Liu integral, secondly, the mean value theorem of a new kind of fuzzy integral was obtained, final, discussed generalized Liu integral.In summary, the innovation of this paper:Proved the properties of multi-dimensional Liu integral, which will make thecalculation of fuzzy integral more easier. Given the mean value theorem of Liu integral, so the studies of Liu integral have theoretical basis.Defined the generalized Liu integral, thus the range of Liu integral is more broad than before.