Volume 2018

# Existence and stability results for Langevin equations with Hilfer fractional derivative-by S. Harikrishnan, K. Kanagarajan and E. M. Elsayed

Abstract: In this paper, we study two Hilfer fractional derivatives of the form
D^{\alpha_1, \beta} \left(D^{\alpha_2, \beta} + \lambda \right) x(t) = f(t, x(t)), \ t\in J:=(a,b],
\label{ee}$$I^{1-\gamma} x(a) = x_a, \gamma = (\alpha_1+\alpha_2)(1-\beta) + \beta,$$

where $\ D^{\alpha_1, \beta}, D^{\alpha_2, \beta}$ is two Hilfer fractional derivatives, $\alpha_1$ and $\alpha_2$ are the order of derivatives, $\beta$ determines to the type of initial condition used in the problem. In this problem existence of solution, Ulam-Hyers stability and generalized Ulam-Hyers-Rassias stability are analyzed.