# American Institute of Mathematical Sciences

October  2017, 22(8): 3127-3144. doi: 10.3934/dcdsb.2017167

## Dichotomy and periodic solutions to partial functional differential equations

 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, Viet Nam 2 Thai Binh College of Education and Training, Cao dang Su pham Thai Binh, Chu Van An, Quang Trung, Thai Binh, Viet Nam

Received  March 2016 Revised  December 2016 Published  June 2017

Fund Project: The authors thank the referee of this paper for his/her comments, suggestions and corrections which help to improve the paper. The work of the first author is partly supported by the Vietnam Institute for Advanced Study in Mathematics (VIASM). This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) by Grant Number 101.02-2014.02

We establish a sufficient condition for existence and uniqueness of periodic solutions to partial functional differential equations of the form $\dot{u}=A(t)u+F(t)(u_t)+g(t,u_t)$ on a Banach space $X$ where the operator-valued functions $t\mapsto A(t)$ and $t\mapsto F(t)$ are $1$-periodic, the nonlinear operator $g(t,φ)$ is $1$-periodic with respect to $t$ for each fixed $φ∈ {\mathcal{C}}:=C([-r,0],X)$, and satisfying $\|g(t,φ_1)-g(t,φ_2)\|≤\varphi(t)\|φ_1-φ_2\|_C$ for $φ_1, φ_2∈ {\mathcal{C}}$ with $\varphi$ being a positive function such that $\sup_{t≥0}∈t_{t}^{t+1}\varphi(τ)dτ < ∞$. We then apply the 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.

Citation: Nguyen Thieu Huy, Ngo Quy Dang. Dichotomy and periodic solutions to partial functional differential equations. Discrete & Continuous Dynamical Systems - B, 2017, 22 (8) : 3127-3144. doi: 10.3934/dcdsb.2017167
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