In this paper, relying on the Banach contraction mapping principle to discuss the existence of solution for a multi-order nonlinear fractional differential equations on the infinite interval $[0,+\mathrm{\infty })$. Moreover the stability of Ulam-Hyers-Rassias and Ulam-Hyers to the initial value problem are obtained. An example that can illustrate the conclusions of this paper have been given at the end.