Stability analysis of continuous implicit Runge-Kutta methods for Volterra integro-differential systems with unbounded delays Original Research Article

In this paper we investigate conditions for the stability of solutions of variable-step continuous, implicit, Runge-Kutta-based methods for linear and nonlinear delay Volterra integro-differential equations (DVIDEs) with a “fading memory”. We present these methods as adaptations (based upon techniques in the treatment of Volterra integral equations) of corresponding collocation Runge-Kutta (CORK) methods for delay differential equations. The basic tool of our stability analysis, and a novel feature of the paper, is the introduction of and use of suitably-constructed discrete Lyapunov functionals, combined with a discrete difference inequality of unbounded order.