Ulam type stability analysis for generalized proportional fractional differential equations

The main aim of the current paper is to be appropriately defined several types of Ulam stability for non-linear fractional differential equation with generalized proportional fractional derivative of Riemann-Liouville type. In the new definitions, the initial values of the solutions of the given equation and the corresponding inequality could not coincide but they have to be closed enough. Some sufficient conditions for three types of Ulam stability for the studied equations are obtained, namely Ulam-Hyers stability, Ulam-Hyers-Rassias stability and generalized Ulam-Hyers-Rassias stability. Some of them are applied to a fractional generalization of a biological model.