The Integrals, Numerical Integrals And Differentials Of Fuzzy-number-valued Functions On Infinite Interval

In this paper, the concept of the fuzzy Henstock integral on infinite interval for the fuzzy-number-valued functions is proposed in order to calculate the expectation of fuzzy random variables and the needs of development of fuzzy integrals. Two calculating methods are proposed: one is to calculate directly by the fuzzy Henstock integral proposed in this paper, including quadrature rules and the error estimates such as the midpoint-type rule, trapezoidal-type rule, Simpson's formula,δ—fine quadrature rules and their error estimates; another is to calculate by using the equivalent characteristic of fuzzy Henstock integrability, whose the membership function could be obtained by solving a nonlinear programming problem. At the same time, we define the F-differentiability of fuzzy-number-valued function by using its support function, the results point that it is weaker than that Puri and Ralescu introduced. And it is proved that the primitives of Henstock integrable functions are differentiable almost everywhere in the sense of F-differentiability.