Integral conditions for exponential dichotomy: a nonlinear approach
This paper gives necessary and sufficient conditions for linear skew-product semiflows to be exponentially dichotomic. The mentioned skew-product semiflows \((\Phi,\sigma)\) are defined on the product \(\Theta\times X\) of a metric space \((\Theta,d)\) and a Banach space \(X\); applications to infinite-dimensional evolutionary equations are indicated. In this framework a characterization of exponential dichotomies is derived in terms of related results on uniform exponential stability due to \textit{S. Rolewicz} [J. Math. Anal. Appl. 115, 434--441 (1986; Zbl 0597.34064)]. The latter involve certain integrability conditions on \(N(\|\Phi(\theta,\tau)x\|)\), where \(\Phi\) is the evolution operator and \(N\) denotes a continuous nondecreasing positive definite function. An application to evolution families is given and corresponding results for discrete skew-product flows can be found in [\textit{B. Sasu}, Discrete Dyn. Nat. Soc. 2009, Article ID 324273, 18 p. (2009; Zbl 1178.39029)].
scientific article; zbMATH DE number 5702055
scientific article