Mohamed S. ElBialy
Sub-Stable and Weak-Stable Manifolds
Associated with Finitely Non-Resonant Spectral Subspaces
For Maps of a Banach Space
(192K, LaTeX 2e)

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 sub-manifolds of
the stable manifold which correspond to a spectral subspace
satisfying  a finite non-resonance
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 weak-stable 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 cut-off function.
An infinite dimensional Banach space does not always admit smooth
cut-off functions.