Stability analysis of Runge-Kutta methods for unbounded retarded differential equations with piecewise continuous arguments

This paper deals with the asymptotical stability of the analytic solutions for the unbounded retarded differential equations with piecewise continuous arguments (EPCA) u′(t)=f(t,u(t),u([tN])),t⩾0,u(0)=u0, where N∈Z+. A sufficient condition that the differential equations are asymptotically stable is derived. This paper is also concerned with the stability analysis of the Runge-Kutta methods for equation u′(t)=au(t)+bu([tN]),t⩾0,u(0)=u0, where N∈Z+. The conditions that the numerical solutions preserve the stability of the analytic solutions are obtained and some numerical experiments are given.