Stability of higher-order nonlinear impulsive differential equations

For a higher-order nonlinear impulsive ordinary differential equation, we present the concepts of Hyers–Ulam stability, generalized Hyers–Ulam stability, Hyers–Ulam–Rassias stability, and generalized Hyers– Ulam–Rassias stability. Furthermore, we prove the generalized Hyers–Ulam–Rassias stability by using in- tegral inequality of Gronwall type for piecewise continuous functions. These results extend related con- tributions to the corresponding first-order impulsive ordinary differential equation. Hyers–Ulam stability, generalized Hyers–Ulam stability, and Hyers–Ulam–Rassias stability can be discussed by the same methods.