Attractive invariant fiber bundles

Abstract

This work on implicit nonautonomous difference equations (iterations) is devoted to an abstract technical result on the existence of attractive invariant manifolds. We investigate their existence, smoothness and their asymptotic phase property using invariant foliations. Such results lay the basic foundation of further investigations on discretization methods of evolutionary differential equations.

Keywords:

• Invariant fiber bundles
• Inertial manifold
• Attractive invariant manifold
• Asymptotic phase
• Discretization
• Nonautonomous

Keywords:

• 2000 Mathematics Subject Classifications: 37D10
• 39A11
• 37B55
• 37C60
• 65M06
• 65Q05

Acknowledgement

This article was written while the author was a postdoctoral associate at the School of Mathematics, University of Minnesota in Minneapolis MN. Its hospitality, the financial support of the Deutsche Forschungsgemeinschaft, as well as the mathematical expertise of Prof. George R. Sell are strongly appreciated.

Notes

† To apply this result from integration theory, one has to write the infinite sum as an integral over piecewise-constant functions and use the Lipschitz estimate on H_n , which is implied by (2.13), to get an integrable majorant.