On two-order fractional boundary value problem with generalized Riemann-Liouville derivative

Abstract

In this paper we focus our study on the existence, uniqueness and Hyers-Ulam stability for the following problem involving generalized Riemann-Liouville operators: \begin{equation*} \mathcal{D}_{0^+}^{\rho_1,\Psi} \Big(\mathcal{D}_{0^+}^{\rho_2,\Psi} + \nu \Big) \mathrm{u}(\mathfrak{t}) = \mathfrak{f}(\mathfrak{t}, \mathrm{u}(\mathfrak{t})). \end{equation*} It is well known that the existence of solutions to the fractional boundary value problem is equivalent to the existence of solutions to some integral equation. Then it is sufficient to show that the integral equation has only one fixed point. To prove the uniqueness result, we use Banach fixed point Theorem, while for the existence result, we apply two classical fixed point theorems due to Krasnoselskii and Leray-Scauder. Then we continue by studying the Hyers-Ulam stability of solutions which is a very important aspect and attracted the attention of many authors. We adapt some sufficient conditions to obtain stability results of the Hyers-Ulam type.