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The procedure is completely the same as that done for k = 2. 1 is proved. 1). ) is natural. But the hypothesis Lip(f) < So (or II Df II < So) for a small So > 0 is quite strong. 1), then we need a local center manifold. This can be obtained from the global center manifold of a modified equation by using the cut-off technique, and the hypothesis II Df II < So will be satisfied automatically since f(0) = 0 and Df (0) = 0. Let us discuss this in detail. We consider a cut-off function x: W' - R with the following properties: (i) X(x) E C"; (ii) 0

Then sup e'"tIi(t, x) -1(t, x)I < oo, tz0 if and only if 1 = Hcu(x). 9. 1) that do not lie on the center-unstable manifold. It says that any solution i(t, x), x E W", converges exponentially for t -* +00 to a uniquely determined solution i(t, HHu(x)) which is on the centerunstable manifold. 1) converges exponentially as t -p +oo to a uniquely determined solution on the center manifold. This gives the stability property of center manifolds. 12. 10. 1, that f E Cbk(P") for some k >- 1. 3 is Ck from ES into Ecu.

1(Rn), Lip(f) < 8c", and z(t) E C,. : F X Ii" X V8" - R" is defined by f(t; x, z(t)) = f(z(t, x) + z(t)) - f(z(t, x)). Proof. 1), then z satisfies the equation i=Az+f(t;x,z). 9) z(T))dT. 9) and applying irs, we have Tr,z(t) = eA,7rz(0) + fOteA(t-T)Trsf(T; x, z(T))dT. 10) Since z(t) E Cy we obtain limto_. eA(t-to)7rruz(to) = 0. 6). 6) holds. 11) we then have X(t,x) + z(t) =eAt(x+z(0)) + fo +z(T))dT. 1). 5. 6) which is uniformly continuous in (x, x) E 6W' X E, such that M,(x) = {x + z*(x, x,)(0) I X, E ES}.