Fractional Langevin type delay equations with two fractional derivatives

In this article, a delayed Mittag-Leffler type function is introduced. With the help of delayed Mittag-Leffler type functions, we give an explicit solution formula to linear non-homogeneous fractional delayed Langevin equations containing two Riemann-Liouville fractional derivatives. Estimation for delayed Mittag-Leffler type function in terms of exponential function is given. Existence uniqueness and Ulam-Hyers stability of solutions are obtained in terms of weighted norms. (C) 2020 Elsevier Ltd. All rights reserved.