Applied Mathematics and Computation 298 (2017) 336–350 Contents lists available at ScienceDirect Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc Fractional differential equations with a constant delay: Stability and asymptotics of solutions Jan ˇ Cermák a,∗ , Zuzana Došlá b , Tomáš Kisela a a Institute of Mathematics, Brno University of Technology, Technická 2, 616 69 Brno, Czech Republic b Department of Mathematics and Statistics, Faculty of Science, Masaryk University, Kotlᡠrská 2, 611 37 Brno, Czech Republic a r t i c l e i n f o MSC: 34K37 34K20 34K25 Keywords: Delay differential equation Fractional-order derivative Stability Asymptotic behaviour a b s t r a c t The paper discusses stability and asymptotic properties of a fractional-order differential equation involving both delayed as well as non-delayed terms. As the main results, ex- plicit necessary and sufficient conditions guaranteeing asymptotic stability of the zero so- lution are presented, including asymptotic formulae for all solutions. The studied equation represents a basic test equation for numerical analysis of delay differential equations of fractional type. Therefore, the knowledge of optimal stability conditions is crucial, among others, for numerical stability investigations of such equations. Theoretical conclusions are supported by comments and comparisons distinguishing behaviour of a fractional-order delay equation from its integer-order pattern. © 2016 Elsevier Inc. All rights reserved. 1. Introduction We investigate stability and asymptotic properties of the fractional delay differential equation D α y(t ) = a y(t ) + b y(t − τ ), t > 0 (1) with real coefficients a, b, a positive real lag τ and the fractional Caputo derivative operator D α (0 < α < 1 is assumed to be a real number). Letting α → 1 from the left, D α y(t) becomes y  (t) and (1) is reduced to the classical delay differential equation y  (t ) = a y(t ) + b y(t − τ ), t > 0 , (2) studied frequently due to its theoretical as well as practical importance (see, e.g. [12]). This equation serves, among others, as the basic test equation for stability analysis of various numerical discretizations of delay differential equations (see, e.g. [1,9]). In this connection, stability conditions for (2) are traditionally required in the optimal form, i.e. as the necessary and sufficient ones. There are known two types of such conditions for asymptotic stability of the zero solution of (2) that we recall in the following two assertions (see [12]). As it is customary, by asymptotic stability of the zero solution of (2) we understand the property that any solution y of (2) is eventually tending to the zero solution. ∗ Corresponding author. E-mail addresses: cermak.j@fme.vutbr.cz (J. ˇ Cermák), dosla@math.muni.cz (Z. Došlá), kisela@fme.vutbr.cz (T. Kisela). http://dx.doi.org/10.1016/j.amc.2016.11.016 0096-3003/© 2016 Elsevier Inc. All rights reserved.