On a class of Langevin equations in the frame of Caputo function-dependent-kernel fractional derivatives with antiperiodic boundary conditions

In this manuscript, we consider a class of nonlinear Langevin equations involving two different fractional orders in the frame of Caputo fractional derivative with respect to another monotonic function [??] with antiperiodic boundary conditions. The existence and uniqueness results are proved for the suggested problem. Our approach is relying on properties of [??]-Caputo's derivative, and implementation of Krasnoselskii's and Banach's fixed point theorem. At last, we discuss the UlamHyers stability criteria for a nonlinear fractional Langevin equation. Some examples justifying the results gained are provided. The results are novel and provide extensions to some of the findings known in the literature. Keywords: [??]-Caputo-type fractional Langevin equation; existence and U-H stability; fixed point theorem Mathematics Subject Classification: 26A33, 34A60