Solvability and stability of a class of fractional Langevin differential equations using the Laplace transform method

In this paper, we investigate the solvability and stability of a particular class of fractional Langevin equations under anti-periodic boundary conditions. By employing the Krasnoselskii fixed point theorem, the Banach contraction mapping theorem, and specific properties of the Mittag-Leffler function, we establish more relaxed sufficient conditions for the existence, uniqueness, and stability of solutions, extending previous research in the field. Additionally, we present numerical examples to illustrate the effectiveness and advantages of our results. Moreover, we add a real-life problem for the fractional Mathieu equation (tempered fractional model of the linearly damped ion motion with octopole) as an application.