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Perron conditions for pointwise and global exponential dichotomy of linear skew-product flows. (English) Zbl 1064.34035

This paper gives a characterization of exponential dichotomy for linear skew-product flows in terms of its admissibility, i.e., the existence of a unique solution \(f \in C_0(\mathbb{R},X)\) to \[ f(t) = \Phi(\sigma(\theta,s),t-s)f(s) + \int_s^t \Phi(\sigma(\theta,\tau),t-\tau)u(\tau) \,d\tau , \quad t \geq s, \] where \(\pi = (\Phi,\sigma)\) is the underlying skew-product flow and \(u \in C_0(\mathbb{R},X)\). The result is a continuation of earlier papers by the same authors [Bull. Belg. Math. Soc. - Simon Stevin 10, No. 1, 1–21 (2003; Zbl 1045.34022) and Monatsh. Math. 138, No. 2, 145–157 (2003; Zbl 1023.34043)].
The ideas follow a quite old tradition of characterizing exponential dichotomies for \(\dot{x} = A(t)x\) in terms of solutions of inhomogeneous equations \[ \dot{x} = A(t)x + f(t) , \qquad t \in \mathbb{R}, \] which goes back to the work of Perron.

MSC:

34D09 Dichotomy, trichotomy of solutions to ordinary differential equations
34D05 Asymptotic properties of solutions to ordinary differential equations
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