On sequences of \(C^{k,\delta}_{b}\) maps which converge in the uniform \(C^{0}\)-norm. (English) Zbl 0947.37016
Summary: We study maps \(f\in C^{k,\delta}_{b}(U,Y)\) and give detailed estimates on \(\|D^{k}f(x)\|\), \(x\in U,\) in terms of \(\|f\|\) and \(\|f\|_{k,\delta}\). These estimates are used to prove a lemma by D. Henry for the case \(k\geq 2\). Here \(U\subset X\) is an open subset and \(X\) and \(Y\) are Banach spaces.
MSC:
| 37D10 | Invariant manifold theory for dynamical systems |
| 37C75 | Stability theory for smooth dynamical systems |
References:
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| [2] | M. S. ElBialy, Sub-stable and weak-stable manifolds associated with finitely non-resonant spectral subspaces, Mathematische Zeitschrift, to appear. · Zbl 1020.47046 |
| [3] | Daniel Henry, Geometric theory of semilinear parabolic equations, Lecture Notes in Mathematics, vol. 840, Springer-Verlag, Berlin-New York, 1981. · Zbl 0456.35001 |
| [4] | Ivar Stakgold, Daniel D. Joseph, and David H. Sattinger , Nonlinear problems in the physical sciences and biology, Lecture Notes in Mathematics, Vol. 322, Springer-Verlag, Berlin-New York, 1973. · Zbl 0254.00025 |
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