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 asymptotic properties of solutions of the equation $u^{\text{'}\text{'}\text{'}}\left(t\right)=p_1\left(t\right)u\left({\tau }_1\left(t\right)\right)+p_2\left(t\right)u^{\text{'}}\left({\tau }_2\left(t\right)\right)$, are investigated where $p_i:[a,+\infty [\to R(i=1,2)$ are locally summable functions, ${\tau }_i:[a,+\infty [\to R(i=1,2)$ measurable ones and ${\tau }_i\left(t\right)\ge t(i=1,2)$. In particular, it is proved that if $p_1\left(t\right)\le 0$, $p_2^2\left(t\right)\le \alpha \left(t\right)\left|p_1\left(t\right)\right|$, $${\int }_a^{+\infty }{[{\tau }_1\left(t\right)-t]}^2{p}_1\left(t\right)dt<+\infty \text{and}{\int }_a^{+\infty }\alpha \left(t\right)dt<+\infty ,$$ then each solution with the first derivative vanishing at infinity is of the Kneser type and a set of all such solutions forms a one-dimensional linear space.

Sufficient conditions are established for the oscillation of proper solutions of the system $$\begin{array}{cc}\hfill u_1^{\text{'}}\left(t\right)& =p\left(t\right)u_2\left(\sigma \left(t\right)\right),\hfill \\ \hfill u_2^{\text{'}}\left(t\right)& =-q\left(t\right)u_1\left(\tau \left(t\right)\right),\hfill \end{array}$$ where $p,q:R_{+}\to R_{+}$ are locally summable functions, while $\tau$ and $\sigma :R_{+}\to R_{+}$ are continuous and continuously differentiable functions, respectively, and $\underset{t\to +\infty }{lim}\tau \left(t\right)=+\infty$, $\underset{t\to +\infty }{lim}\sigma \left(t\right)=+\infty$.

The existence of positive periodic solutions for a kind of Rayleigh equation with a deviating argument $$x^{\text{'}\text{'}}\left(t\right)+f\left(x^{\text{'}}\left(t\right)\right)+g(t,x(t-\tau \left(t\right)))=p\left(t\right)$$ is studied. Using the coincidence degree theory, some sufficient conditions on the existence of positive periodic solutions are obtained.