Ulam Type Stabilities of n-th Order Linear Differential Equations Using Gronwall's Inequality

In this paper, the stabilities in the sense of Hyers-Ulam, Hyers-Ulam-Rassias, Mittag-Leffler-Hyers-Ulam and Mittag-Leffler-Hyers-Ulam-Rassias of n-th order linear differential equations are investigated. The main results are proved by applying generalized replacement lemma and generalized Gronwall's inequality. The obtained results include many previous results and give improvements. In particular, the results have improved in two ways: no initial conditions for differential equations and no conditions for a coefficient. Most of the results are limited to the case where the interval is bounded, but a result for the unbounded case is obtained.