The purpose of this paper is to study the existence of periodic solutions for the non-autonomous second order Hamiltonian system $$\left\{\begin{array}{c}\ddot{u}\left(t\right)=\nabla F(t,u\left(t\right)),\text{a.e.}t\in [0,T],\hfill \\ u\left(0\right)-u\left(T\right)=\dot{u}\left(0\right)-\dot{u}\left(T\right)=0.\hfill \end{array}\right.$$ Some new existence theorems are obtained by the least action principle.

This paper deals with the system of functional-differential equations $$\frac{dx\left(t\right)}{dt}=p\left(x\right)\left(t\right)+q\left(t\right),$$ where $p:C_{\omega }\left(R^n\right)\to L_{\omega }\left(R^n\right)$ is a linear bounded operator, $q\in L_{\omega }\left(R^n\right)$, $\omega >0$ and $C_{\omega }\left(R^n\right)$ and $L_{\omega }\left(R^n\right)$ are spaces of $n$-dimensional $\omega$-periodic vector functions with continuous and integrable on $[0,\omega ]$ components, respectively. Conditions which guarantee the existence of a unique $\omega$-periodic solution and continuous dependence of that solution on the right hand side of the system considered are established.

We consider a neutral type operator differential inclusion and apply the topological degree theory for condensing multivalued maps to justify the question of existence of its periodic solution. By using the averaging method, we apply the abstract result to an inclusion with a small parameter. As example, we consider a delay control system with the distributed control.

We investigate the nonautonomous periodic system of ODE’s of the form $\dot{x}=\overrightarrow{v}\left(x\right)+r_p\left(t\right)(\overrightarrow{w}\left(x\right)-\overrightarrow{v}\left(x\right))$, where $r_p\left(t\right)$ is a $2p$-periodic function defined by $r_p\left(t\right)=0$ for $t\in \langle 0,p\rangle$, $r_p\left(t\right)=1$ for $t\in (p,2p)$ and the vector fields $\overrightarrow{v}$ and $\overrightarrow{w}$ are related by an involutive diffeomorphism.