## 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