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.

The unstable properties of the linear nonautonomous delay system $x^{\text{'}}\left(t\right)=A\left(t\right)x\left(t\right)+B\left(t\right)x(t-r\left(t\right))$, with nonconstant delay $r\left(t\right)$, are studied. It is assumed that the linear system $y^{\text{'}}\left(t\right)=(A\left(t\right)+B\left(t\right))y\left(t\right)$ is unstable, the instability being characterized by a nonstable manifold defined from a dichotomy to this linear system. The delay $r\left(t\right)$ is assumed to be continuous and bounded. Two kinds of results are given, those concerning conditions that do not include the properties of the delay function $r\left(t\right)$ and the results depending on the asymptotic properties of the...

Our aim in this paper is to obtain sufficient conditions under which for every $\xi \in R^n$ there exists a solution $x$ of the functional differential equation $\dot{x}\left(t\right)={\int }_c^t\left[d_{s}Q(t,s)\right]f(t,x\left(s\right)),t\in [t_0,T]$ such that $li{m}_{t\to T-}x\left(t\right)=\xi$.