Criterions for detecting the existence of the exponential dichotomies in the asymptotic behavior of the solutions of variational equations
Let \(X\) be a Banach space, \(\Theta\) a metric space, and \((E(\mathbb R_+, X),F(\mathbb R_+, X))\) a pair of vector-valued Schäffer function spaces. Write \(\alpha(t,E)=\inf\{\alpha>0\mid \int^t_0|f(t)|\,ds\leq \alpha\|f\|_{E(\mathbb R_+,\mathbb R)}\), \(\forall f\in E(\mathbb R_+,\mathbb R)\}\), and \(\beta(t,F)=\|\chi_{[0,t]}\|_{F(\mathbb R_+, \mathbb R)}\). The autors show that if \((E(\mathbb R_+, X),F(\mathbb R_+, X))\) is uniformly admissible to an exponentially bounded, strongly continuous cocycle \(\{\pmb{\Phi}(\theta,t)\}_{\theta\in\Theta, t\geq 0}\) (over a semiflow \(\{\sigma(\theta,t)\}_{\theta\in\Theta, t\geq 0}\)) and \(\lim_{t\to\infty}\alpha(t,E)\beta(t,F)=\infty\), then there exist a family of projectors \(\{P(\theta)\}_\Theta\) and positive constants \(N_1\), \(N_2\), \(\nu_1\), \(\nu_2\) such that, for \(t\geq 0\), \(\theta\in\Theta\), \[ \|\pmb{\Phi}(\theta,t)x\|\leq N_1e^{-\nu_1 t}\|x\| \quad (x\in \operatorname{Im} P(\theta)); \qquad \|\pmb{\Phi}(\theta,t)x\|\geq N_2e^{\nu_2 t}\|x\| \quad (x\in \operatorname{Ker} P(\theta)). \]
scientific article; zbMATH DE number 5655695