Étude Qualitative de quelques Problèmes à Valeurs Initiales pour des Equations Différentielles d’ordre Fractionnaire

Abstract

The present work is basically devoted to study the existence and uniqueness of solutions for certain classes of nonlinear fractional differential equations with initial conditions via the ψ−Caputo fractional derivative with order α ∈ (0;1). To achieve these goals, we first transform our main problems into an equivalent fractional integral equation. After that, based on some fixed point theorems namely, Banach’s fixed point theorem combined with the ψ- fractional Bielecki norm, Schauder’s fixed point theorem, and Krasnoselskii’s fixed point theorem, the existence, and uniqueness of solutions to the proposed problems are discussed theoretically. Furthermore, examples are presented to illustrate our theoretical findings.