Periodic solutions of partial functional differential equations

Abstract:

In this paper we study the existence of periodic solutions to the partial functional differential equation \begin{equation*} \left \{ \begin {array}{l} \frac {dy(t)}{dt}=By(t)+\hat {L}(y_{t})+f(t,y_{t}), \;\forall t\geq 0,\\ y_{0}=\varphi \in C_{B}. \end{array} \right . \end{equation*} where $B: Y \rightarrow Y$ is a Hille-Yosida operator on a Banach space $Y$. For $C_{B}≔\{\varphi \in C([-r,0];Y): \varphi (0)\in \overline {D(B)}\}$, $y_{t}\in C_{B}$ is defined by $y_{t}(\theta )=y(t+\theta )$, $\theta \in [-r,0]$, $\hat {L}: C_{B}\rightarrow Y$ is a bounded linear operator, and $f:\mathbb {R}\times C_{B}\rightarrow Y$ is a continuous map and is $T$-periodic in the time variable $t$. Sufficient conditions on $B$, $\hat {L}$ and $f(t,y_{t})$ are given to ensure the existence of $T$-periodic solutions. The results then are applied to establish the existence of periodic solutions in a reaction-diffusion equation with time delay and the diffusive Nicholson’s blowflies equation.