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

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

130 Applying p_O* and

130 Applying p_O* and putting t = 0 we deduce, using the exponential dichotomy tlP_°'z(0)ll < Ke-(~-+0"llP_z(s)l I. As we assume that the difference of the two solutions lies in the space BC~(R; X) we conclude that P_°*z(0) = 0. Similarly one proves that the X+ component of the difference of two solutions necessarily vanishes at t = 0. We conclude that (i) holds. If P0°*h = 0 then we obtain the estimate e-'71~lll~h(t)ll

131 Definition 6.1 Let E and F be Banach spaces. Let f be a locally Lipschitz mapping from E into F. We say that f has vanishing Lipschitz constant at the origin if there exists 6o positive and a continuous mapping L : [0,60] --* R+ such that L(O) = 0 and if Ilzll, tIYH -< 6 then IIf(x) - f(Y)tl -< L(6)ilx - vii- Lemma 6.2 The mapping rmod is globally Lipschitz continuous with vanishing Lipschitz con- stant at the origin. Lemma 6.3 Let E and F be Banach spaces and let f be a globally Lipschitz continuous function, with Lipschitz constant L. Let f be the substitution operator from BC~(R; E) into BC~(R; F) defined by (](h))(s) = f(h(s)). Then ] is globally Lipschitz continuous with the same Lipschitz constant. Proof. The result follows from the estimate llf(h) - f(g)ll, = sup e-nldllf(h(s)) - f(g(s))l) < sER sup e-nl'lLIlh(s) - g(s)ll = Lllh - gun. stir Corollary 6.4 If we define ~,nod as above, then this mapping is globally Lipschitz continuous with a constant L,m~,) = Lr(26) which is vanishing at the origin. 6.2 A Lipschitz center manifold We define the mapping ~" from BC~(R; X) x Xo into BCn(R; X) by Choose 5 in (6.2) small enough such that .~'(u, ¢) = T(.)¢ + IC~rnod(U). (6.3) I L,.,,~,) ItX:ll < ~. This can be done uniformly for r/in a compact interval of (0,min{7+,-7-}). If tt I1¢11 < ~--ff, then ~'(., ¢) leaves the ball with radius rt in BC~(R; X) invariant. Moreover, ~-(., ~) is Lips- chitz continuous with Lipschitz constant ½. We thus obtain the following Theorem 6.5 If ~ and R are chosen as above, then there exists a Lipschitz continuous map- ping u* from B~rr(Xo ) into BR(BC-~(R; X)) such that u = u*(~b) is the unique solution of the equation = Y(u, ¢). D

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