Periodic solutions of linear differential and integral equations (Q1901107)

The paper deals with the functional differential equation (FDE) \(x'= F(t, x_t)\) and the functional equation (FE) \(x(t)= F(t, x_t)\), where \(F(t, \varphi)\) is periodic in \(t\) with period \(T\) and convex in \(\varphi\). The main result on (FDE) is that if there is a bounded solution, then there is a \(T\)-periodic solution no matter the delay is. The same result holds for (FE) under a Lipschitz type condition on \(F(t, \varphi)\) with respect to \(t\).