Advances in Difference Equations
Volume 2004 (2004), Issue 2, Pages 141-182
doi:10.1155/S1687183904308010
𝒞m-smoothness of invariant fiber bundles for dynamic equations on measure chains
1Department of Mathematics, University of Augsburg, Augsburg 86135, Germany
2Department of Mathematics, J. W. Goethe University, Robert-Mayer-Straße 10, Frankfurt 60325, Germany
Received 8 August 2003
Copyright © 2004 Christian Pötzsche and Stefan Siegmund. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
We present a new self-contained and rigorous proof of the smoothness of invariant fiber bundles for dynamic equations on measure chains or time scales. Here, an invariant fiber bundle is the generalization of an invariant manifold to the nonautonomous case. Our main result generalizes the “Hadamard-Perron theorem” to the time-dependent, infinite-dimensional, noninvertible, and parameter-dependent case, where the linear part is not necessarily hyperbolic with variable growth rates. As a key feature, our proof works without using complicated technical tools.