We consider the second order self-adjoint differential equation (1) (r(t)y’(t))’ + p(t)y(t) = 0 on an interval I, where r, p are continuous functions and r is positive on I. The aim of this paper is to show one possibility to write equation (1) in the same form but with positive coefficients, say r₁, p₁ and to derive a sufficient condition for equation (1) to be oscillatory in the case p is nonnegative and ${\int }^{\infty }[1/r\left(t\right)]dt$ converges.

We study solutions tending to nonzero constants for the third order differential equation with the damping term $${\left(a_1\left(t\right){\left(a_2\left(t\right)x^{\text{'}}\left(t\right)\right)}^{\text{'}}\right)}^{\text{'}}+q\left(t\right)x^{\text{'}}\left(t\right)+r\left(t\right)f\left(x\left(\varphi \left(t\right)\right)\right)=0$$ in the case when the corresponding second order differential equation is oscillatory.

Sufficient conditions are given which guarantee that the linear transformation converting a given linear Hamiltonian system into another system of the same form transforms principal (antiprincipal) solutions into principal (antiprincipal) solutions.

In this paper two sequences of oscillation criteria for the self-adjoint second order differential equation ${\left(r\left(t\right)u^{\text{'}}\left(t\right)\right)}^{\text{'}}+p\left(t\right)u\left(t\right)=0$ are derived. One of them deals with the case ${\int }^{\infty }\frac{\mathrm{d}t}{r\left(t\right)}=\infty$, and the other with the case ${\int }^{\infty }\frac{\mathrm{d}t}{r\left(t\right)}<\infty$.

The aim of this paper is to deduce oscillatory and asymptotic behavior of the solutions of the ordinary differential equation L_nu(t)+p(t)u(t)=0.

The equation to be considered is $$L_{n}y\left(t\right)+p\left(t\right)y\left(\tau \left(t\right)\right)=0.$$ The aim of this paper is to derive sufficient conditions for property (A) of this equation.

The aim of this paper is to deduce oscillatory and asymptotic behavior of delay differential equation $$L_{n}u\left(t\right)-p\left(t\right)u\left(\tau \left(t\right)\right)=0,$$ from the oscillation of a set of the first order delay equations.