Local stable manifold of Langevin differential equations with two fractional derivatives

In this paper, we investigate the existence of local center stable manifolds of Langevin differential equations with two Caputo fractional derivatives in the two-dimensional case. We adopt the idea of the existence of a local center stable manifold by considering a fixed point of a suitable Lyapunov-Perron operator. A local center stable manifold theorem is given after deriving some necessary integral estimates involving well-known Mittag-Leffler functions.