A new Lyapunov matrix based approach to the robustness analysis developed in recent years for linear time delay systems is nontrivially extended to address some problems of nonlinear analysis. In particular, the asymptotic stability condition for a quasilinear system is derived, as well as attraction region for a class of nonlinear time delay systems with exponentially stable linear approximation is estimated. A peculiarity of the approach is that the negative definiteness condition for the derivative of the nominal functional along the solutions of a nonlinear system is replaced with just negativeness of an “infinite” part of the integral of this derivative.