Stability analysis of a class of Langevin equations in the frame of generalized Caputo fractional operator with nonlocal boundary conditions

The main objective of the present paper is to establish the existence and uniqueness of solutions for the fractional Langevin equation involving the ϕ-Caputo fractional operator with nonlocal boundary conditions. The Langevin differential equation can accurately depict many physical phenomena and help researchers effectively represent anomalous diffusion. This paper considers a fractional Langevin differential equation, including the ϕ-Caputo fractional operator. Furthermore, some novel boundaries are selected to be considered as a problem. The existence of the solution is obtained by applying a fixed point theorem, and the uniqueness of the solution is obtained by using the Banach contraction mapping principle to the considered problem. Moreover, we discuss the Hyres-Ulam stability result. The manuscript is concluded with an illustrative example to corroborate the reported results. This study extends and generalizes various results in the literature and provides new insights into the qualitative behavior of fractional differential systems.