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SN~ (~6) lff 2It 3k_~ , 5 ",,x_J {b)

SN~ (~6) lff 2It 3k_~ , 5 ",,x_J {b)

132 Definition 6.6

132 Definition 6.6 (Lipschitz center manifold) We define the center manifold as the map- ping from Brat(X0) into X given by C(¢) = u'(¢)(0). We end with a trivial but nevertheless important observation: Remark 6.7 Although u*(¢) may grow exponentially, this does not happen in the hyperbolic directions; indeed it follows easily that R I1(Z- p0O*)u*(¢)ll < ~. We will use the above results to deduce the smoothness of the mapping f,,~od defined in (6.2). We let V ~ = {h e B6~(R;X)I II(I- P0e*)hll0 < oo}, (6.4) with the norm Ilhllv. = IIPo°*hll. + II(I - Po°')hll0. Provided with this norm V ~ becomes a Banach space. Lemma 6.8 Let 71 and 72 be positive constants such that 0 < k~ll < r~. Let Ilhllv., -- < 5 6 ° Then ~,,od : Vm -.. BC'n(R; X ®*) is Ck-smooth at h. Proof. Because Ilhllv., _< ~ it follows that (compare (6.2)) (~,oa)(h)(s) -- r(h(s)), (llP°°'(~Ks))") . As both r and P~o* are smooth mappings the result follows from the next lemma. For a proof of this lemma we refer to [VvG87] [] Lemma 6.9 Let E and F be Banaeh spaces and let .f be a Gk-smooth mapping .from E into F. If h is a mapping .from R into E then we define the mapping ](h) .from R into F by ](h)(s) = ,f(h(s)). For 1 < l < k, multilinear mappings ~t(h) are defined as .follows. I'f gl, ...,gt are mappings .from R into E then ~t(h)(gl, ...,gt) is the mapping .from R into F defined by ~l(h)(gl,..., gt)(s) = Dt,f(h(s))(gl(s), ...,gt(s)). Finally, we set ~°(h) = ](h). Let 71 and ~ be positive constants such that k~h < 72. The mapping ] .from BCm(R; E) into BC'n(R; F) is Ck-smooth. Moreover, .for 1 < l < k the identity holds. D t] = ~t

133 6.3 Contractions on embedded Banach spaces Let Yo, Y, ]I1 and A be Banach spaces with norms denoted by {[-11o. II" II, II" }I1 and l" I and such that Yo is continuously embedded in Y, and Y is continuously embedded in Y1. We denote the embedding operators by Jo : Yo --* Y and J : Y -, Y1. We will consider a fixed point equation: Y = f(Y, ~) (6.5) where f : Y x A ~ Y satisfies the following hypotheses: H1 Jf : Y x A ~ ]I1 has a continuous partial derivative and for all (Y, ~) E Y x A we have Dy(Jy) : Y x A -~ £(Y, Y:) D,(Jf)(v, = JfOl(v, X) = f l)(v, for some f0) : y x A ~ £(Y) and fO) : y x A ~ £(Y1), H2 fo : Yo x h ~ Y, (Vo,)0 ~ fo(Yo, )t) := f(Jovo, )~) has a continuous partial derivative D:~fo : Yo x A -* £(A; Y), H8 There exists some ~ E [0,1) such that Vy, ~ E Y and VA E A and [If(Y, A) - f(~, A)][ _< ~[]y - ~[[ IlfO)(v, A)II < ~, ]lf~l)Cy, A)II < It follows from H3 that for each A E A (6.5) has a unique solution V = V*(A) E Y. We make a last assumption: I14 V*(A) = doy~(A) for some continuous V~ : A -* Y0. The hypotheses allow us to consider in £(A, Y) the equation A = fOl(y*() 0 , A)A + hAfo(y;()O, A) (6.6) Because of 1-I3 this equation has for each ,X a unique solution A*(A) E £(A ; Y). We will show that A*(A) is, if suitably looked at, the derivative of y*(,X). Theorem 6.10 Assume that H1-H4 hold. Then the solution map y* : A -~ Y of (6.5) is Lipschitz continuous and y~ = dY* : A ---, Y1 is of class C 1 with DV~()O = JA*(,~), VA e A. For the proof of this theorem we refer to [VvG87, Theorem 3].

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