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Smooth stable manifolds in Banach spaces

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Part of the Lecture Notes in Mathematics book series (LNM,volume 1926)

We establish in this chapter the existence of smooth stable manifolds for semiflows defined by nonautonomous differential equations in a Banach space. One can obtain unstable manifolds simply by reversing the time. We also establish the exponential decay on the stable manifold of the derivatives of the semiflow with respect to the initial condition (see (6.8) and (6.9)). We are not aware of any similar result in the literature even in the case of uniform exponential dichotomies. Our approach to the proof of the stable manifold theorem consists again in using the differential equation and the invariance of the stable manifold under the dynamics to conclude that it must be the graph of a function satisfying a certain fixed point problem.

Keywords

  • Banach Space
  • Unique Function
  • Unstable Manifold
  • Point Problem
  • Stable Manifold

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© 2008 Springer-Verlag Berlin Heidelberg

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(2008). Smooth stable manifolds in Banach spaces. In: Stability of Nonautonomous Differential Equations. Lecture Notes in Mathematics, vol 1926. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-74775-8_6

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