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.

In this paper, we discuss the conditions for a center for the generalized Liénard system $$\frac{\mathrm{d}x}{\mathrm{d}t}=\varphi \left(y\right)-F\left(x\right),\frac{\mathrm{d}y}{\mathrm{d}t}=-g\left(x\right),$$ or $$\frac{\mathrm{d}x}{\mathrm{d}t}=\psi \left(y\right),\frac{\mathrm{dy}}{\mathrm{d}t}=-f\left(x\right)h\left(y\right)-g\left(x\right),$$ with $f\left(x\right)$, $g\left(x\right)$, $\varphi \left(y\right)$, $\psi \left(y\right)$, $h\left(y\right)ℝ\to ℝ$, $F\left(x\right)={\int }_0^{x}f\left(x\right)\mathrm{d}x$, and $xg\left(x\right)>0$ for $x\ne 0$. By using a different technique, that is, by introducing auxiliary systems and using the differential inquality theorem, we are able to generalize and improve some results in [1], [2].

We study the vector $p$-Laplacian $$\left\}\begin{array}{c}-\left(\right|u^{\text{'}}{{|}^{p-2}{u}^{\text{'}})}^{\text{'}}=\nabla F(t,u)\text{a.e.}t\in [0,T],u\left(0\right)=u\left(T\right),u^{\text{'}}\left(0\right)=u^{\text{'}}\left(T\right),1<p<\infty .\hfill \end{array}\right.(*)$$ We prove that there exists a sequence $\left(u_n\right)$ of solutions of ($*$) such that $u_n$ is a critical point of $\varphi$ and another sequence $\left(u_n^{*}\right)$ of solutions of $(*)$ such that $u_n^{*}$ is a local minimum point of $\varphi$, where $\varphi$ is a functional defined below.

We study the existence of one-signed periodic solutions of the equations $$\begin{array}{ccc}& x^{\text{'}\text{'}}\left(t\right)-a^2\left(t\right)x\left(t\right)+\mu f(t,x\left(t\right),x^{\text{'}}\left(t\right))=0,\hfill & \hfill x^{\text{'}\text{'}}\left(t\right)+a^2\left(t\right)x\left(t\right)=\mu f(t,x\left(t\right),x^{\text{'}}\left(t\right)),\end{array}$$ where $\mu >0$, $a:(-\infty ,+\infty )\to (0,\infty )$ is continuous and 1-periodic, $f$ is a continuous and 1-periodic in the first variable and may take values of different signs. The Krasnosielski fixed point theorem on cone is used.