A Note on Linear Impulsive Fractional Differential Equations

Abstract

Abstract. This paper deals with linear impulsive fractional differential equations involving the Caputo derivative with non-integer order q. We provide exact solutions of linear impulsive fractional differential equations with constant coefficient by mean of the MittagLeffler functions. Then we apply the exact solutions to improve impulsive integral inequalities with singularity.

Stability for Caputo Fractional Differential Systems

“…Lemma 7 (see [9,Lemma 3.2]). If one sets ℎ( ) ≡ in (23) with a constant , then the solution of (24) reduces to…”

“…We can obtain an upper bound of solutions for Caputo fractional differential equations via fractional Gronwall's inequality. The following result is adapted from Theorem 5.1 in [7] and Theorem 3.15 in [9].…”

“…Choi and Koo [8] improved on the monotone property of Lemma 1.7.3 in [5] for the case ( , ) = with a nonnegative real number . Choi et al [9] also investigated Mittag-Leffler stability of solutions of fractional differential equations by using the fractional comparison principle.…”