Langevin equation with nonlocal boundary conditions involving a ψ-Caputo fractional operators of different orders

This paper studies Langevin equation with nonlocal boundary conditions involving a psi-Caputo fractional operators of different orders. By the aid of fixed point techniques of Krasnoselskii and Banach, we derive new results on existence and uniqueness of the problem at hand. Further, a new psi-fractional Gronwall inequality and psi-fractional integration by parts are employed to prove UlamHyers and Ulam-Hyers-Rassias stability for the solutions. Examples are provided to demonstrate the advantage of our major results. The proposed results here are more general than the existing results in the literature which can be obtained as particular cases.