We will interpret (ADDE) as problem (6) with some nonlinear perturbation G:X→Y and use a variation-of-constants formula in X to obtain results about the perturbed problem, such as normal form coefficients for local bifurcations. As G maps X into Y, we would like to embed Y in a natural way into X. A naive approach would be to use a delta-function as an embedding. However, this embedding is not bounded, so the domain of A0 would not be preserved under perturbation. This is indeed the case, as the rule for extending a function beyond its original domain, i.e. φ˙(0)=Bφ(0), is incorporated in D(A0). Hence adding a perturbation to the rule for extension changes the domain of the generator. A way out is to embed this problem into a larger space. A natural choice would be Y×X, where we have a continuous embedding ℓ:Y→Y×{0}, and we can separate the extension and translation part of A0 into Y×{0} and {0}×X respectively.
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