Stability of higher-order nonlinear impulsive differential equations

Abstract

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 integral inequality of Grönwall type for piecewise continuous functions. These results extend related contributions 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.