Controlled convergence theorems for infinite dimension Henstock integrals of fuzzy valued functions based on weak equi-integrability

In this paper, we give the concept of weak Henstock equi-integrability for a sequence of fuzzy-number-valued functions. Under this notion, we investigate a new version of the Henstock's Lemma of fuzzy-number-valued functions. Thus, it is possible to discuss the controlled convergence theorems of fuzzy Henstock integral in sense of Vitali covering. Moreover, we prove that a uniform version of Sklyarenko's and Lusin's integrability condition of fuzzy Henstock integrals together with pointwise convergence of a sequence of integrable functions is sufficient for a convergence theorem of fuzzy Henstock integrals. As the applications of the controlled convergence theorem, we discuss the existence theorems of generalized solution for a class of discontinuous fuzzy differential equations.