Existence and Ulam-Hyers stability results for nonlinear fractional Langevin equation with modified argument

Abstract

In this paper, we give the existence and uniqueness result for the fractional order Langevin equation with modified argument by using the Bielecki norm. After, we consider a special form of this equation (delayed form) and then apply Burton's method to this special form to prove that there is a unique solution under weaker conditions than the other result. Further, we derive the Ulam-Hyers stability for this equation. Finally, two examples are given to illustrate our main results.