Stability of BDF methods for nonlinear Volterra integral equations with delay (Q1609041)

The authors consider the nonlinear delay Volterra integral equation with constant delay \(\tau> 0\), \[ y(t)= g(t)+ \int^t_0 f(\xi, y(\xi), y(\xi- \tau)) d\xi,\quad t> 0, \] with \(y(t)= \phi(t)\), \(t\in [-\tau, 0]\). Here, \(f= f(s,y,z)\) (which does not depend on \(t\)) is subjet to certain non-classical (one-sided) Lipschitz conditions. Its solution is approximated by the backward differentiation formula (BDF) method on a uniform mesh, using an implementation based on \textit{B. Cahlon} [J. Comput. Appl. Math. 33, No. 1, 97-104 (1990; Zbl 0719.65092)]. The asymptotic stability analysis is based on Dahlquist's concept of \(G\)-stability which is extended to what the authors call \(G(c,p,q)\)-stability for the resulting discrete Volterra equation.