## Periodicity and stability of solutions to neutral differential equations with $\varphi $-Lipschitz and infinite delays

### Volume 125 / 2020

#### Abstract

We prove the existence and uniqueness of a periodic solution to neutral function partial differential equations with infinite delay of the form $$ \frac {\partial Fu_t}{\partial t}= A(t)Fu_t +g(t,u_t)\quad \mbox {for } t \gt 0,\quad u(t)=\phi (t)\quad \hbox {for } t\le 0, $$ where the family $(A(t))_{t\ge 0}$ of linear partial differential operators is such that the mapping $t\mapsto A(t)$ is 1-periodic, and the nonlinear delay operator $g(t,u)$ is 1-periodic with respect to $t$ and $\varphi $-Lipschitz with respect to $u$ (i.e. $\|g(t,u)-g(t,v)\|\le \varphi (t)\|u-v\|$ for some function $\varphi $ in an admissible function space). Then we apply the results to study the existence, uniqueness and conditional stability of a periodic solution to the above equation when $(A(t))_{t\ge 0}$ generates an evolution family which has an exponential dichotomy. Moreover, we prove the existence of a local stable manifold around such a periodic solution.