Periodic solutions of dissipative functional differential equations with infinite delay

We consider periodic, infinite delay differential equations. We investigate dissipativeness for these equations. Massat proved that dissipative, periodic, infinite delay equations have a periodic solution. For our purpose we need a weaker dissipativeness, so we prove Massat's theorem from this weak dissipativeness in an elementary way. Then we extend a theorem of Pliss giving a necessary and sufficient condition for this weak dissipativeness. We also present a theorem using Liapunov functionals to show the weak dissipativeness and hence the existence of a periodic solution.