Existence results and the Ulam Stability for fractional differential equations with hybrid proportional-Caputo derivatives

In this paper, we study the Ulam-Hyers and the generalized Ulam-Hyers-Rassias stability for linear fractional differential equations with hybrid proportional-Caputo derivatives using the Laplace transform method. The existence and uniqueness of solutions for nonlinear fractional differential equations with hybrid proportional-Caputo derivatives are established by means of Schaefer’s fixed point theorem, Banach’s fixed point theorem and the generalized Gronwall’s inequality. Two examples are also given to illustrate the main results.