The Henstock–Stieltjes integral for fuzzy-number-valued functions

In this paper, we firstly define and discuss the Henstock–Stieltjes integral for fuzzy-number-valued functions which is an extension of the usual fuzzy Riemann–Stieltjes integral. In addition, several necessary and sufficient conditions of the integrability for fuzzy-number-valued functions are given by means of the Henstock–Stieltjes 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 Henstock–Stieltjes integral are discussed. We find that there exists a fuzzy-number-valued function which is fuzzy Henstock–Stieltjes integrable, but whose primitive is not α-differentiable almost everywhere. Thirdly, we introduce some quadrature rules for the fuzzy Henstock–Stieltjes 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 Simpson’s quadrature. Finally, we propose the concept of weak equi-integrability for sequences of fuzzy Henstock–Stieltjes integrable functions. Under this concept, we prove two convergence theorems for sequences of the fuzzy Henstock–Stieltjes integrable functions. At the same time, the formula of integration by parts is also studied.