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Periodic solutions and their conditional stability for partial neutral functional differential equations

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Abstract

We prove the existence and uniqueness of the periodic solution as well as its conditional stability for the semilinear partial neutral function differential equation of the form

$$\begin{aligned} \frac{\partial Fu_t}{\partial t}= A(t)Fu_t +g(t,u_t), t\in (0,\infty );\;u_0=\phi . \end{aligned}$$

Here the operator-valued function \(t\mapsto A(t)\) is T-periodic; the nonlinear delay operator \(g(t,\psi )\) is T-periodic with respect to t and Lipschitz continuous with respect to \(\psi \). In our strategy, on the one hand, we prove a Massera-type theorem for the linearized neutral equation, then pass to the semilinear equation using fixed-point arguments and Neumann series. On the other hand, our abstract results fit perfectly with the case that the family \((A(t))_{t\ge 0}\) generates an evolution family having exponential dichotomy. In such a case, we can apply our abstract results to show the existence, uniqueness, and conditional stability of periodic solution to the above equation.

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Acknowledgements

We thank the referee and the editor very much for their corrections and suggestions which helped to improve the paper. 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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Authors and Affiliations

  1. School of Applied Mathematics and Informatics, Hanoi University of Science and Technology, Vien Toan ung dung va Tin hoc, Dai hoc Bach Khoa Hanoi, 1 Dai Co Viet, Hanoi, Vietnam

    Thieu Huy Nguyen & Thi Ngoc Ha Vu

  2. Department of Basic Science, Hungyen University of Technology and Education, Khoai Chau, Dan Tien, Hungyen, Vietnam

    Thi Loan Nguyen

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  1. Thieu Huy Nguyen
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  2. Thi Loan Nguyen
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  3. Thi Ngoc Ha Vu
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Correspondence to Thieu Huy Nguyen.

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Nguyen, T.H., Nguyen, T.L. & Vu, T.N.H. Periodic solutions and their conditional stability for partial neutral functional differential equations. J. Evol. Equ. 19, 1091–1110 (2019). https://doi.org/10.1007/s00028-019-00511-x

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