Existence results for Langevin equations involving generalized Liouville–Caputo fractional derivatives with non-local boundary conditions

The main objective of the present paper is to establish the existence and uniqueness (EU) results for nonlinear fractional Langevin equation involving Liouville-Caputo generalized fractional derivative (GFD) of different order with non-local boundary conditions. The existence solution is obtained by using Krasnoselskii's fixed point theorem, and the uniqueness result is obtained by using the Banach contraction mapping principle. An example is introduced to validate the effectiveness of the results. The results are novel and provide an extension to some of the findings known in the literature.