Asymptotic stability, contractivity and dissipativity of one-leg θ-method for non-autonomous delay functional differential equations

This paper focuses on asymptotic stability, contractivity and dissipativity of non-autonomous nonlinear delay functional differential equations with bounded lag, and the corresponding dynamical properties of one-leg θ-method. Sufficient conditions for these delay functional differential equations to be dissipative, asymptotically stable and contractive are established. One-leg θ-method is constructed to solve such equations numerically. An important result on the growth of solution of a class of difference inequalities with variable coefficients is obtained. Finally, it is proved that the one-leg θ-method is asymptotically stable, contractive and dissipative if and only if θ = 1. Numerical examples are given to confirm our theoretical results.