Abstract
LetU=(U(t, s)) t≥s≥O be an evolution family on the half-line of bounded linear operators on a Banach spaceX. We introduce operatorsG O,G X andI X on certain spaces ofX-valued continuous functions connected with the integral equation\(u(t) = U(t,s)u(s) + \int_s^t {U(t,\xi )f(\xi )d\xi }\), and we characterize exponential stability, exponential expansiveness and exponential dichotomy ofU by properties ofG O,G X andI X , respectively. This extends related results known for finite dimensional spaces and for evolution families on the whole line, respectively.
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This work was done while the first author was visiting the Department of Mathematics of the University of Tübingen. The support of the Alexander von Humboldt Foundation is gratefully acknowledged. The author wishes to thank R. Nagel and the Functional Analysis group in Tübingen for their kind hospitality and constant encouragement.
Support by Deutsche Forschungsgemeinschaft DFG is gratefully acknowledged.
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Nguyen Van Minh, Räbiger, F. & Schnaubelt, R. Exponential stability, exponential expansiveness, and exponential dichotomy of evolution equations on the half-line. Integr equ oper theory 32, 332–353 (1998). https://doi.org/10.1007/BF01203774
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DOI: https://doi.org/10.1007/BF01203774