Solvability of Langevin equations with two Hadamard fractional derivatives via Mittag-Leffler functions

In this paper we discuss the solvability of Langevin equations with two Hadamard fractional derivatives. The method of this discussion is to study the solutions of the equivalent Volterra integral equation in terms of Mittag-Leffler functions. The existence and uniqueness results are established by using Schauder's fixed point theorem and Banach's fixed point theorem, respectively. An example is given to illustrate the main results.

Automated summary

In this paper, the solvability of Langevin equations with two Hadamard fractional derivatives is discussed. The solutions of the equivalent Volterra integral equation in terms of Mittag-Leffler functions are studied. The existence and uniqueness of solutions are established using Schauder's fixed point theorem and Banach's fixed point theorem, respectively. An example is provided to illustrate the main results.