Periodic solutions of partial functional differential equations

In this paper we study the existence of periodic solutions to the partial functional differential equation { d y ( t ) d t = B y ( t ) + L ^ ( y t ) + f ( t , y t ) , ∀ t ≥ 0 , y 0 = φ ∈ C B . \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 → Y B: Y \rightarrow Y is a Hille-Yosida operator on a Banach space Y Y . For C B ≔ { φ ∈ C ( [ − r , 0 ] ; Y ) : φ ( 0 ) ∈ D ( B ) ¯ } C_{B}≔\{\varphi \in C([-r,0];Y): \varphi (0)\in \overline {D(B)}\} , y t ∈ C B y_{t}\in C_{B} is defined by y t ( θ ) = y ( t + θ ) y_{t}(\theta )=y(t+\theta ) , θ ∈ [ − r , 0 ] \theta \in [-r,0] , L ^ : C B → Y \hat {L}: C_{B}\rightarrow Y is a bounded linear operator, and f : R × C B → Y f:\mathbb {R}\times C_{B}\rightarrow Y is a continuous map and is T T -periodic in the time variable t t . Sufficient conditions on B B , L ^ \hat {L} and f ( t , y t ) f(t,y_{t}) are given to ensure the existence of T 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.