A center-stable manifold theorem for differential equations in Banach spaces (Q1803952)

The author shows existence (and other properties) of stable center manifolds for the equation \(Z'(t)=Az(t)+f(z(t))\) in a Banach space \(E\) under the following assumptions: (a) \(E=F_ 1\oplus F_ 2\) (topological) with the \(F_ 1\) \(A\)-invariant, and if \(A_ 1\) is the restriction of \(A\) to \(F_ j\), \(A_ 1\) and \(-A_ 2\) generate strongly continuous semigroups, the latter of negative exponential growth. (b) \(f\) is sufficiently smooth with \(f(0)=Df(0)=0\), (c) there is a spectral gap condition and \(F_ 1\) has the \(C^ k\) extension property. There are no applications. The author states that this paper provides the ``mathematical apparatus'' for his paper with \textit{J.-P. Eckmann} [ibid., 221-248 (1993)].