 00236 Mohamed S. ElBialy
 SubStable and WeakStable Manifolds
Associated with Finitely NonResonant Spectral Subspaces
For Maps of a Banach Space
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May 19, 00

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Abstract. In this work we study $C^{k,\gd}, 2 \leq k \leq \infty, 0\leq \gd \leq 1,$
maps of a Banach space near a fixed point.
We show the existence and uniqueness of a class of
$C^{k,\gd}$ local invariant submanifolds of
the stable manifold which correspond to a spectral subspace
satisfying a finite nonresonance
condition of order $L\leq k$
and an overriding condition of order $L\leq k$
(condition (3) of Theorem \ref{TH:1}).
We study the dependence of these invariant manifolds on a parameter that
lies in a Banach space.
We also show that a $C^{k,\gd}$
local weakstable manifold that satisfies these two conditions
is unique in the class of $C^{k,\gd}$ maps.
The uniqueness is due to the fact that our method does not
require a cutoff function.
An infinite dimensional Banach space does not always admit smooth
cutoff functions.
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