On the solution and Ulam-Hyers-Rassias stability of a Caputo fractional boundary value problem.

Abstract

In this paper, we investigate a class of boundary value problems involving Caputo fractional derivative $ {{}^C\mathcal{D}^{\alpha}_{a}} $ of order $ \alpha \in (2, 3) $, and the usual derivative, of the form $ \begin{equation*} ({{}^C\mathcal{D}^{\alpha}_{a}}x)(t)+p(t)x'(t)+q(t)x(t) = g(t), \quad a\leq t\leq b, \end{equation*} $ for an unknown $ x $ with $ x(a) = x'(a) = x(b) = 0 $, and $ p, \; q, \; g\in C^2([a, b]) $. The proposed method uses certain integral inequalities, Banach's Contraction Principle and Krasnoselskii's Fixed Point Theorem to identify conditions that guarantee the existence and uniqueness of the solution (for the problem under study) and that allow the deduction of Ulam-Hyers and Ulam-Hyers-Rassias stabilities.