Abstract
Consider the delay differential equation x 0 (t) = f(xt) with the history xt : (−∞, 0] → Rn of x at ‘time’ t defined by xt(s) = x(t + s). In order not to lose any possible entire solution of any example we work in the Frechet space C 1 ((−∞, 0], Rn with the topology of uniform convergence of maps and their derivatives on compact sets. A previously obtained result, designed for the application to examples with unbounded state-dependent delay, says that for maps f which are slightly better than continuously differentiable the delay differential equation defines a continuous semiflow on a continuously differentiable submanifold X ⊂ C 1 of codimension n, with all time-t-maps continuously differentiable. Here continuously differentiable for maps in Frechet spaces is understood in the sense of Michal and Bastiani. It implies that f is of locally bounded delay in a certain sense. Using this property – and a related further mild smoothness hypothesis on f – we construct stable, unstable, and center manifolds of the semiflow at stationary points, by means of transversality and embeddings.
Highlights
Let us briefly recall the motivation for working in the Fréchet space C1, and not in a smaller Banach space of continuously differentiable maps (−∞, 0] → Rn : we did not want to exclude any possible continuously differentiable map satisfying (1.1) on some interval, neither by growth conditions at −∞ nor by integrability conditions
[18, Proposition 1.1] says that property follows from continuous differentiability in the sense of Michal and Bastiani [1,12]. The latter notion means for a continuous map g : U → G, U ⊂ F open, F and G Fréchet spaces, that all directional derivatives
In the present paper we find local invariant manifolds at a stationary point φ ∈ X ⊂ C1 of the semiflow S, for f continuously differentiable (MB), with property (e), and satisfying a further mild smoothness assumption (d) which requires that a map induced by f via property is continuously differentiable (F)
Summary
Let U be a set of maps (−∞, 0] → Rn and let a map f : U → Rn be given. A solution of the delay differential equation x (t) = f (xt). [18, Proposition 1.1] says that property (lbd) follows from continuous differentiability in the sense of Michal and Bastiani [1,12] The latter notion means for a continuous map g : U → G, U ⊂ F open, F and G Fréchet spaces, that all directional derivatives. For each k ∈ N0 the topology on the Fréchet space Ck of k times continuously differentiable maps For reals a < b and k ∈ N0 let Ck([a, b], Rn) denote the Banach space of k times continuously differentiable maps [a, b] → Rn, with the norm given by k.
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