Solutions to Partial Functional Differential Equations with Infinite Delays: Periodicity and Admissibility

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Abstract

Under some appropriate conditions, we prove the existence and uniqueness of periodic solutions to partial functional differential equations with infinite delay of the form \(\dot {u}=A(t)u+g(t,u_{t})\) on a Banach space X where A(t) is 1-periodic, and the nonlinear term g(t, ϕ) is 1-periodic with respect to t for each fixed ϕ in fading memory phase spaces, and is φ(t)-Lipschitz for φ belonging to an admissible function space. We then apply the attained results to study the existence, uniqueness, and conditional stability of periodic solutions to the above equation in the case that the family (A(t))t≥ 0 generates an evolution family having an exponential dichotomy. We also prove the existence of a local stable manifold near the periodic solution in that case.

Keywords

Partial functional differential equations Periodic solutions Admissibility of function spaces Conditional stability Local stable manifolds 

Mathematics Subject Classification (2010)

34K19 35B10 

Notes

Acknowledgements

We would like to thank the referee of this paper for his/her suggestions to improve the appearance of the paper.

Funding Information

This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.02-2017.303.

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Copyright information

© Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd. 2018

Authors and Affiliations

  1. 1.School of Applied Mathematics and InformaticsHanoi University of Science and TechnologyHanoiVietnam
  2. 2.Faculty of Natural SciencesThai Binh College of Education and TrainingThai BinhVietnam

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