Analysis of solutions of fuzzy differential equations under the generalized derivative

The generalized derivative represents the broadest notion of the Hukuhara-type derivative of a fuzzy number-valued function in the literature. It exists for a wide class of fuzzy processes, since the generalized difference exists for any pair of fuzzy numbers. Despite its historical significance, few papers provide theoretical results on the g-derivative, primarily because of its complex analytical behavior. On the other hand, analyzing solutions to fuzzy differential equations from a comparative Hukuhara-type perspective allows establishing features of FDEs whose solutions are exclusively g-differentiable. The study begins by providing a corrected version of the fundamental theorem of calculus via the g-differentiability and Aumann integrability of a fuzzy function. Sufficient conditions over the field of a fuzzy initial value problem for the gH -differentiability of the solutions are presented. Lastly, results on fuzzy processes derived from solutions of FDEs that are g-differentiable, but not gH, and not even gH -differentiable, are given. Examples of population dynamics governed by the Malthusian and Logistic models are provided to illustrate different scenarios for the presented analysis.