Abstract
We propose a new approach toward the existence and uniqueness of periodic solutions to linear and semilinear evolution equations. Our approach is based on the connection of the conditional stability of evolution families (i.e., stability only in a subspace of the Banach space containing the initial data) with the choice of the initial data from which emanates the periodic solution. We also give applications to exponentially dichotomic evolution families as well as to nonautonomous damped wave equations.
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Acknowledgements
Parts of this work were done when the first author was visiting TU Bergakademie Freiberg, Germany. Support by the German Academic Exchange Service (DAAD) is gratefully acknowledged. This work is financially supported by Vietnam National Foundation for Science and Technology Development (Nafosted). The work of the second author is financially supported by Vietnam Ministry of Education and Training.
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Nguyen, T.H., Vu, T.N.H. Conditional stability and periodicity of solutions to evolution equations. J. Evol. Equ. 21, 3797–3812 (2021). https://doi.org/10.1007/s00028-021-00707-0
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DOI: https://doi.org/10.1007/s00028-021-00707-0
Keywords
- Evolution families
- Conditional \(\varphi \)-stability
- Periodic solutions
- Exponential dichotomy
- Nonautonomous damped wave equations