On Henstock Integral Of Fuzzy-Number-Valued Function In The Plane

In this paper, the integral of fuzzy-number-valued functions in the plane is discussed. Firstly, as the generalization ofδ-fine partitions, the fine partition in the plane which is named derivate base is proposed, and the Hen-stock integral of fuzzy-number-valued functions in the plane is defined in the sense of the derivate base above. In addition, the properties and the characteristic theorems of this kind of integral are discussed by means of studying abstract functions in Banach space. Nextly, the definition of Perron integral and Denjoy integral for fuzzy-number-valued functions in the plane are given, and by using them, we discuss the characterization theorems of the primitive functions for fuzzy Henstock integral in the sense of the special derivate bases. And we also study the absolute integrability and continuity of fuzzy-number-valued functions in the plane under consideration of different derivate bases. Finally, for the fuzzy Henstock integral in plane, two calculating methods are proposed: one is to calculate directly by using the method of approximation including quadrature rules and the error estimates such as the rectangular rule and Simpson's formular; another is to calculate by using the equivalent characteristic of fuzzy Henstock integrability, whose membership function could be obtained by solving nonlinear programming problem.