On the properties of fuzzy differential equations under cross operations

This manuscript proposes a new framework for dealing with nonlinear fuzzy differential equations when the nonlinearity is given in terms of fuzzy arithmetic operations of product and division, here called ?-cross operations. To this end, the fuzzy environment considered is given by finite-dimensional Banach spaces of fuzzy numbers, and the fuzzy functions involved are related to a calculus theory given in terms of the ?-differentiability. The notion of power hedges with respect to the ?-cross operations is introduced, and several algebraic properties are presented, such as the fuzzy product and division rules. Lastly, a study on polynomial fuzzy differential equations with an application to the fuzzy Abel equation is provided.