Existence and Ulam-Hyers stability for a singular Caputo-fractional problem with integral boundary conditions

Abstract

In this article, we study a singular fractional-order Caputo differential problem involving boundary conditions with linear operators. The analysis is carried out in an appropriate Banach space framework. First, by applying the Banach contraction principle, we establish a uniqueness result for the considered problem. Then, using Krasnosel'skii's fixed point theorem, we derive an existence theorem under suitable assumptions on the nonlinear term. In the second part of the paper, we investigate different types of stability for the obtained solutions, namely Ulam-Hyers stability and Ulam-Hyers-Rassias stability. Sufficient conditions ensuring these stability properties are presented and proved. The obtained theoretical results highlight the influence of the singular fractional operator and the associated boundary conditions. Finally, two illustrative examples are discussed in order to demonstrate the applicability and effectiveness of the established results.