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

In this paper asymptotic stability properties of -methods fordelay differential equations (DDEs) are considered with respectto the test equation y'(t) = ay(t) + by(t - ), t > 0, y(t)= g(t), - t 0, where > 0. First we examine extensivelythe instance where a, b and g(t) is a continuous real-valuedfunction; 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 papersdevoted to the study of the stability of -methods, using thetest equation (0.1). In those papers, conditions that are strongerthan necessary for the (asymptotic) stability of the zero solutionare assumed; for instance, a]+¦b¦ < 0, thatis the set of complex pairs (a, b) such that the zero solutionof (0.1) is asymptotically stable for every > 0. In thispaper we study, instead, the stability properties of -methodsfor equation (0.1) with an arbitrary but fixed value of .