Induced delay equations.

For the family of nonautonomous delay equations, we show that the generator of the evolution semigroup obtained from any such equation gives rise to an autonomous delay equation on a higher-dimensional space. We call it an induced delay equation. More significantly, we show that this equation can be used to study some important properties of the original dynamics in four main directions: the characterization of the hyperbolicity of the original delay equation via the hyperbolicity of the induced delay equation; the description of the spectral properties and of the corresponding Lyapunov exponents; the equivalence between the hyperbolicity of the induced delay equation and an admissibility property for its nonlinear perturbations; and, finally, the robustness of hyperbolicity under sufficiently small linear perturbations. The proofs of some of these results depend on a new variation of constants formula for the nonlinear perturbations of the induced delay equation that is also established in our work. [ABSTRACT FROM AUTHOR]