Stability Theory of General Contractions for Delay Equations

We study the persistence of the asymptotic stability of delay equations both under linear and nonlinear perturbations. Namely, we consider nonautonomous linear delay equations v′ = L(t)v _t with a nonuniform exponential contraction. Our main objective is to establish the persistence of the nonuniform exponential stability of the zero solution both under nonautonomous linear perturbations, i.e., for the equation v′ = (L(t) + M(t))v _t , thus discussing the so-called robustness problem, and under a large class of nonlinear perturbations, namely for the equation v′ = L(t)v _t + f(t, v _t ). In addition, we consider general contractions e ^−λρ(t) determined by an increasing function ρ that includes the usual exponential behavior with ρ(t) = t as a very special case. We also obtain corresponding results in the case of discrete time.