On Lr-norm-based derivatives and fuzzy Henstock-Kurzweil integrals with an application - Yabin Shao; Yubing Li; Zengtai Gong (Alexandria Engineering Journal, 2023)
The fundamental theorem of calculus not only plays an important role in the study of differential equation theory, but also in many practical problems solving. In order to enrich its content, in present paper, we extend fuzzy derivatives and fuzzy Henstock-Kurzweil integrals. Firstly, we discuss some properties of Lr-norm-based derivatives for fuzzy Henstock-Kurzweil integrable function (or briefly FLr-derivative). Then, we give some relationships with well-known fuzzy derivatives. After that, we also define a kind of fuzzy Henstock-Kurzweil integral based on FLr-derivative (or briefly Lr-FHK-integral). In addition, we give some important properties and convergence theorems for the Lr-FHK-integral. Meanwhile, we obtain a sufficient and necessary condition for integrability of this integral. As the application, we discuss the fuzzy Fourier series.