Presentation of solutions of impulsive fractional Langevin equations and existence results

In this paper, a class of impulsive fractional Langevin equations is firstly offered. Formula of solutions involving Mittag-Leffler functions and impulsive terms of such equations are successively derived by studying the corresponding linear Langevin equations with two different fractional derivatives. Meanwhile, existence results of solutions are established by utilizing boundedness, continuity, monotonicity and nonnegative of Mittag-Leffler functions and fixed point methods. Further, other existence results of nonlinear impulsive problems are also presented. Finally, an example is given to illustrate our theoretical results.