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Periodic solutions of linear differential and integral equations. (English) Zbl 0834.34088

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 \(t\).

MSC:

34K13 Periodic solutions to functional-differential equations
45M15 Periodic solutions of integral equations
34C25 Periodic solutions to ordinary differential equations
45A05 Linear integral equations
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