Delay dependent stability regions of Theta-methods for delay differential equations

In this paper asymptotic stability properties of Θ-methods for delay differential equations (DDEs) are considered with respect to the test equation y'(t) = ay(t) + by(t - τ), t > 0, y(t) = g(t), -τ ⩽ t ⩽ 0, where τ > 0. First we examine extensively the instance where a, b ∈ ℜ and g(t) is a continuous real-valued function; then we investigate the more general case of a, b ∈ C and g(t) a continuous complex-valued function.The last decade has seen a relatively large number of papers devoted to the study of the stability of Θ-methods, using the test equation (0.1). In those papers, conditions that are stronger than necessary for the (asymptotic) stability of the zero solution are assumed; for instance, ℜ[a]+¦b¦ < 0, that is the set of complex pairs (a, b) such that the zero solution of (0.1) is asymptotically stable for every τ > 0. In this paper we study, instead, the stability properties of Θ-methods for equation (0.1) with an arbitrary but fixed value of τ.