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

In this paper asymptotic stability properties of Theta-methods for delay differential equations (DDEs) are considered with respect to the test equation [GRAPHICS] where tau > 0. First we examine extensively the instance where a, b epsilon R and g(t) is a continuous real-valued function; then we investigate the more general case of a, b epsilon 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 Theta-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 tau > 0. In this paper we study, instead, the stability properties of Theta-methods for equation (0.1) with an arbitrary but fixed value of tau.