Existence, uniqueness, approximation of solutions and $\mathbb E_{\alpha}$--Ulam stability results for a class of nonlinear fractional differential equations involving $\psi$--Caputo derivative with initial conditions

Choukri Derbazi, Zidane Baitiche, Mouffak Benchohra, Gaston N’Guérékata

The main purpose of this paper is to study the existence, uniqueness, $\mathbb E_{\alpha}$--Ulam stability results, and other properties of solutions for certain classes of nonlinear fractional differential equations involving the $\psi$--Caputo derivative with initial conditions. Modern tools of functional analysis are applied to obtain the main results. More precisely using Weissinger's fixed point theorem and Schaefer's fixed point theorem the existence and uniqueness results of solutions are proven in the bounded domain. While the well known Banach fixed point theorem coupled with Bielecki type norm are used with the end goal to establish sufficient conditions for existence and uniqueness results on unbounded domains. Meanwhile, the monotone iterative technique combined with the method of upper and lower solutions is used to prove the existence and uniqueness of extremal solutions. Furthermore, by means of new generalizations of Gronwall's inequality, different kinds of $\mathbb E_{\alpha}$--Ulam stability of the proposed problem are studied. Finally, as applications of the theoretical results, some examples are given to illustrate the feasibility and correctness of the main results.