Periodic solutions of dissipative functional differential equations with infinite delay (Q1362545)

We consider \(T\)-periodic infinite delay differential equations. We investigate dissipativeness for these equations, which is a weaker condition than uniform ultimate boundedness. The later condition is usually used together with uniform boundedness to prove the existence of a \(T\)-periodic solution. Massat proved that dissipative \(T\)-periodic infinite delay equations have a \(T\)-periodic solution. For our purpose, we need a weaker dissipativeness, so we prove Massat's theorem for this weak dissipativeness in an elementary way. Then we extend a theorem of Pliss giving a necessary and sufficient condition for weak dissipativeness. We also present a theorem using Lyapunov functionals to show weak dissipativeness and hence the existence of a \(T\)-periodic solution.