In this paper, we investigate a class of nonlinear fractional differential equations with integral boundary condition. By means of Krasnosel'skiȋ fixed point theorem and contraction mapping principle we prove the existence and uniqueness of solutions for a nonlinear system. By means of Bielecki-type metric and the Banach fixed point theorem we investigate the Ulam-Hyers and Ulam-Hyers-Rassias stability of nonlinear fractional differential equations. Besides, we discuss an example for illustration of the main work.

We prove the local existence and uniqueness of the strong solutions for a class of full non-Newtonian fluids in one space dimension with the hypotheses that the initial data are small in some sense and satisfy some compatibility conditions. The initial density need not be positive, which means that we allow the initial vacuum.

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