Stability Analysis of a System of Sturm-Liouville and Langevin Fractional Differential Equations

Abstract

This thesis falls within the framework of studying fractional differential equations. This subject was inspired by the work of A. Berhail, N. Tabouche, M.M. Matar and J. Alzabut, article [9] on which Belaadi, and Benkamouche [7] based their work to study of the existence and uniqueness of the solution to a generalized system of Sturm-Liouville and Langevin type, using the Hilfer-Katugampola fractional derivative under an initial condition. Inspired by the results of these studies, we sought to complete the mathematical analysis by further exploring the aspect of stability. In this context, we focused on the stability analysis of the solutions of the studied system according to several classical and generalized notions of stability, namely Ulam-Hyers stability, generalized Ulam-Hyers stability, Ulam-Hyers-Rassias stability, and generalized Ulam-Hyers-Rassias stability. We used rigorous analytical techniques to establish sufficient conditions that guarantee the validity of each type of stability within the framework of the studied fractional system. At the end of this thesis, we presented a practical example to illustrate the theoretical aspects and highlight the effectiveness of the obtained results . Key words: Generalized Sturm-Liouville and Langevin system, Hilfer-Katugampola fractional derivative, Arzela-Ascoli theorem, Schauder fixed point theorem, Banach contraction principle, Ulam-Hyers stability, generalized Ulam-Hyers stability, Ulam-Hyers-Rassias stability, generalized Ulam-Hyers-Rassias stability.