The HenstockStieltjes integral for fuzzy-number-valued functions ?
In this paper, we firstly define and discuss the HenstockStieltjes integral for fuzzy-number-valued functions which is an extension of the usual fuzzy RiemannStieltjes integral. In addition, several necessary and sufficient conditions of the integrability for fuzzy-number-valued functions are given by means of the HenstockStieltjes integral of real-valued functions and Henstock integral of fuzzy-number-valued functions. Secondly, the continuity and the differentiability of the primitive for the fuzzy HenstockStieltjes integral are discussed. We find that there exists a fuzzy-number-valued function which is fuzzy HenstockStieltjes integrable, but whose primitive is not ?-differentiable almost everywhere. Thirdly, we introduce some quadrature rules for the fuzzy HenstockStieltjes integral by giving error bounds for the mappings of bounded variation and of Lipschitz type. We also consider the generalization of classical quadrature rules, such as midpoint-type, trapezoidal and Simpsons quadrature. Finally, we propose the concept of weak equi-integrability for sequences of fuzzy HenstockStieltjes integrable functions. Under this concept, we prove two convergence theorems for sequences of the fuzzy HenstockStieltjes integrable functions. At the same time, the formula of integration by parts is also studied.
differentiability convergence theorem fuzzy number fuzzy henstock integral stieltjes integral numerical calculate