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.
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Pötzsche, C., Siegmund, S. ʗm-smoothness of invariant fiber bundles for dynamic equations on measure chains. Adv Differ Equ 2004, 572930 (2004). https://doi.org/10.1155/S1687183904308010