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.Keywords.Ulam stability; Hybrid proportional-Caputo derivatives; Laplace transform method; Schaefer's fixed point theorem.d dt E (t), where κ P and κ d are the proportional control parameter and the derivative control parameter, respectively.The function E is the error between the state variable and the process variable.This control law enables Ding et al. [12] to present the control of complex networks models.