Asymptotic stability of multistep methods for nonlinear delay differential equations

This paper is concerned with the numerical solution of delay differential equations. The emphasis is on the nonlinear stability of multistep methods. It is shown that every A-stable linear multistep method with piecewise constant or linear interpolation can preserve the asymptotic stability of a class of nonlinear systems, which is an extension of the well known GP-stability result to the nonlinear case. As a comparison, it is also proved that strict stability at infinity is necessary to the asymptotic stability of one-leg methods for non-autonomous equations.