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Smoothness of invariant manifolds. (English) Zbl 0762.93041

This article deals with the smoothness properties of invariant manifolds for discrete and continuous systems of type \[ x^ \Delta=Ax+g(x)\quad (t\in\mathbb{T}), \] where \(A\) is a linear operator in a Banach space \(X\) for which \(I+hA\) is invertible. The spectrum \(\sigma(A)\) is divided into three parts \(\sigma^ -=\{\lambda\in\sigma(A): \rho_ h(1,\lambda)\leq\underset\leftarrow a\}\), \(\sigma^ 0=\{\lambda\in\sigma(A): \underset\leftarrow b\leq\rho_ h(1,\lambda)\leq b{_} \}\), \(\sigma^ +=\{\lambda\in\sigma(A): \rho_ h(1,\lambda)\geq\underset\rightarrow a\}\) \((\rho_ h(1,\lambda)=h^{- 1}[| 1+\lambda h|-1]\) if \(h>0\) and \(\rho_ h(1,\lambda)=\text{Re }\lambda\) if \(h=0)\) with \(\underset\leftarrow a<\underset\leftarrow b\leq\underset\rightarrow b<\underset\rightarrow a\) satisfying the gap condition, \(g: X\to X\), \(g(0=0\), \(Dg(0)=0\) belongs to \({\mathcal C}^{k,n}\), \(x^ \Delta\) is \(h^{-1}(x(t+h)-x(t))\) in the discrete case when \(\mathbb{T}=h\mathbb{Z}\) and is \(x'\) in the continuous case when \(\mathbb{T}=\mathbb{R}\), \(h=0\). The main result is an existence theorem on invariant manifolds of the class \({\mathcal C}^{k,n}\).
Reviewer: P.Zabreiko (Minsk)

MSC:

93C10 Nonlinear systems in control theory
34D99 Stability theory for ordinary differential equations
39A99 Difference equations
47H30 Particular nonlinear operators (superposition, Hammerstein, Nemytskiĭ, Uryson, etc.)
47H10 Fixed-point theorems
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