On the Hadamard-Perron lemma (Q2487437)

Let \(S(\cdot):H\to H\) be a continuous mapping defined on the Banach space \(H\) and \(0\) be its fixed-point. Let two projection operators \(P_{+},P_{-}:H\to H\), a bounded linear operator \(L:H\to H\) and a continuous mapping \(R(h)=S(h)-Lh\) be defined and satisfy the inequalities \(1^0\). \(P_{+}+P_{-}=I,\;\|P_{+}\|=\|P_{-}\|=1,\) \(2^0\). \(L(P_{+}H)=P_{+}H,\;L(P_{-}H)\subset P_{-}H\), \(3^0\). \(\|Lx\|\geq (1+\delta_{+})\|x\| \quad \forall x \in P_{+}H,\;\delta_{+}\geq 0\), \(4^0\). \(\|Ly\|\leq (1-\delta_{-})\|y\| \quad \forall y \in P_{-}H,\;\delta_{-}\geq 0\), \(5^0\). \(\|R(h_1)-R(h_2)\|<\theta \left(\max\{\|h_1\|,\|h_2\|\}\right)\|h_1-h_2\|,\; \forall h_i \in H\), with a continuous positive function \(\theta(\cdot):\;\theta(0)=0,\; \max_{h\in \mathcal{O}}\theta(\|h\|)<{1}/ {2}\), where \(x,y\) and \(h_i\) are arbitrary elements in a neighborhood of zero \(\mathcal{O}\subset H\). If in the conditions \(3^0\) and \(4^0\) the norm of the operator \(L\) is bounded away from unity, then the point \(0\) is hyperbolic; otherwise it is nonhyperbolic. If the inequality \(\|R(h_1)-R(h_2)\|<\widehat{\theta} \|h_1-h_2\|,\;h_i \in H,\;\widehat{\theta}<\frac{\delta}{2},\;\delta=\delta_+=\delta_->0\) is true, the following generalization of the Hadamard-Perron lemma is proved: (1) There exists a stable invariant manifold \(\mathcal{M}^-\) and an unstable invariant manifold \(\mathcal{M}^+\). (2) Under the action of the operator \(S(\cdot)\), an arbitrary point \(h\) is attracted to the manifold \(\mathcal{M}^+\) with an exponential velocity: \[ \text{dist}(\mathcal{M}^+,\;S^n(h))\leq C_1(1-\delta+2\widehat{\theta})^n\text{dist}(\mathcal{M}^+,h). \] The points \(m^+\) for which the operator \(S^{-n}(m^+)\) exists and \(S^{-n}(m^+)\subset \mathcal{O}\) for all \(n>0\) belong to the manifold \(\mathcal{M}^+\), and the estimate \(\|S^{-n}(m^+)\|\leq C_2(1-\delta+2\widehat{\theta})^n\|m^+\|\) holds. (3) Under the action of the \(S^{-1}(\cdot)\), an arbitrary point \(h\) is attracted to the manifold \(\mathcal{M}^-\) with an exponential velocity: \[ \text{dist}(\mathcal{M}^-,S^{-n}(h))\leq C_3(1-\delta+2\widehat{\theta})^n\text{dist}(\mathcal{M}^-,h). \] The points \(m^-\) for which the trajectory satisfied \(S^{n}(m^-)\subset \mathcal{O}\) for all \(n>0\) belong to \(\mathcal{M}^-\), and the following estimate holds: \(\|S^{n}(m^-)\|\leq C_4(1-\delta+2\widehat{\theta})^n\|m^-\|\).