Ulam stability and data dependence for fractional differential equations with Caputo derivative

Abstract

In this paper, Ulam stability and data dependence for fractional differential equations with Caputo fractional derivative of order $\alpha$ are studied. We present four types of Ulam stability results for the fractional differential equation in the case of $0<\alpha<1$ and $b=+\infty$ by virtue of the Henry-Gronwall inequality. Meanwhile, we give an interesting data dependence results for the fractional differential equation in the case of $1<\alpha<2$ and $b<+\infty$ by virtue of a generalized Henry-Gronwall inequality with mixed integral term. Finally, examples are given to illustrate our theory results