Fréchet differentiability in Fréchet spaces, and differential equations with unbounded variable delay

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We introduce and discuss Fréchet differentiability for maps between Fréchet spaces. For delay differential equations x'(t)=f(xₜ) we construct a continuous semiflow of continuously differentiable solution operators x₀ xₜ, t≥0, on submanifolds of the Fréchet space C¹((-∞,0],Rⁿ), and establish local invariant manifolds at stationary points by means of transversality and embedding properties. The results apply to examples with unbounded but locally bounded delay.