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.

The differential equation of the form ${\left(q\left(t\right)k\left(u\right){\left(u^{\text{'}}\right)}^a\right)}^{\text{'}}=f\left(t\right)h\left(u\right)u^{\text{'}}$, a ∈ (0,∞), is considered and solutions u with u(0) = 0 and (u(t))² + (u’(t))² > 0 on (0,∞) are studied. Theorems about existence, uniqueness, boundedness and dependence of solutions on a parameter are given.

In this paper we offer criteria for property (B) and additional asymptotic behavior of solutions of the $n$-th order delay differential equations $${\left(r\left(t\right){\left[x^{(n-1)}\left(t\right)\right]}^{\gamma }\right)}^{\text{'}}=q\left(t\right)f\left(x\left(\tau \left(t\right)\right)\right).$$ Obtained results essentially use new comparison theorems, that permit to reduce the problem of the oscillation of the n-th order equation to the the oscillation of a set of certain the first order equations. So that established comparison principles essentially simplify the examination of studied equations. Both cases ${\int }^{\infty }{r}^{-1/\gamma }\left(t\right)t=\infty$ and ${\int }^{\infty }{r}^{-1/\gamma }\left(t\right)t<\infty$ are discussed.

Integral criteria are established for $dim{V}_i\left(p\right)=0$ and $dim{V}_i\left(p\right)=1,i\in \{0,1\}$, where $V_i\left(p\right)$ is the space of solutions $u$ of the equation $$u^{\text{'}\text{'}}+p\left(t\right)u=0$$ satisfying the condition $${\int }^{+\infty }\frac{u^2\left(s\right)}{s^i}ds<+\infty .$$