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In this paper, we study existence of solutions to a Cauchy problem for nonlinear ordinary differential equations involving two Caputo fractional derivatives. The existence and uniqueness of solutions are obtained by using monotonicity, continuity and explicit estimation of Mittag-Leffler functions via fixed point theorems. Further, we present Ulam-Hyers stability results by using direct analysis methods. Finally, examples are given to illustrate our theoretical results.