In this paper, we investigated the sharp estimate for the condition of the given interval which guarantees for the unique solution of a Reimman-Liouville-type fractional differential equations with boundary conditions. The method of analysis is obtained by the principle of contraction mapping through using the maximum value of the integral of the Green's function. Besides, we also concluded a sharper lower bound of the eigenvalues for an eigenvalue problem. Finally, two examples are presented to clarify the principle results.