Abstract
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.
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Barreira, L., Valls, C. Stability Theory of General Contractions for Delay Equations. Integr. Equ. Oper. Theory 68, 413–426 (2010). https://doi.org/10.1007/s00020-010-1834-3
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DOI: https://doi.org/10.1007/s00020-010-1834-3