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

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.