THE DISCRETE LYAPUNOV FUNCTION FOR SCALAR DIFFERENTIAL DELAY EQUATIONS

The discrete Lyapunov function is introduced to determine whether or not a solution of a scalar differential delay equation is a super-exponential solution (small solution). Furthermore, for a solution of periodic systems, the relations between the exponential decay rate, the limit of the Lyapunov function, and the asymptotic exponential expansions are discussed. For an application of our results, we show that homoclinic orbits of scalar differential delay equations will never correspond to super-exponential solutions. © 1990.