Existence results and the Ulam Stability for fractional differential equations with hybrid proportional-Caputo derivatives
Mohamed I. Abbas, A Kilbas, H Srivastava, J Trujillo, R Khalil, M Al, A Horani, M Yousef, Sababheh, H Batarfi, J Losada, J Nieto, W, M Bouaouid, K Hilali, S Melliani, M Bouaouid, M Hannabou, K Hilali, S Meng, Y Cui, W Zhong, L Wang, D Ding, X Zhang, J Cao, D Liang, D Anderson, D Ulness, F Jarad, Th, J Abdeljawad, D Baleanu, A Fernandez, A Akgl, D Hyers, Th, Rassias, M Abbas, M Abbas, K Liu, M Fekan, D O'regan, J Wang, J Vanterler Da, C Sousa, E Capelas De Oliveira, M Caputo, K Diethelm, N Ford
Abstract
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.