The higher-order nonlinear ordinary differential equation $$x^{\left(n\right)}+\lambda p\left(t\right)f\left(x\right)=0,t\ge a,$$ is considered and the problem of counting the number of zeros of bounded nonoscillatory solutions $x(t;\lambda )$ satisfying ${lim}_{t\to \infty }x(t;\lambda )=1$ is studied. The results can be applied to a singular eigenvalue problem.

In this paper we have considered completely the equation $$y^{\text{'}\text{'}\text{'}}+a\left(t\right)y^{\text{'}\text{'}}+b\left(t\right)y^{\text{'}}+c\left(t\right)y=0,(*)$$ where $a\in C^2([\sigma ,\infty ),R)$, $b\in C^1([\sigma ,\infty ),R)$, $c\in C\left(\right[\sigma ,\infty ),R)$ and $\sigma \in R$ such that $a\left(t\right)\le 0$, $b\left(t\right)\le 0$ and $c\left(t\right)\le 0$. It has been shown that the set of all oscillatory solutions of (*) forms a two-dimensional subspace of the solution space of (*) provided that (*) has an oscillatory solution. This answers a question raised by S. Ahmad and A.  C. Lazer earlier.

In this paper we consider the third-order nonlinear delay differential equation (*) $${\left(a\left(t\right){\left(x^{\text{'}\text{'}}\left(t\right)\right)}^{\gamma }\right)}^{\text{'}}+q\left(t\right)x^{\gamma }\left(\tau \left(t\right)\right)=0,t\ge t_0,$$ where $a\left(t\right)$, $q\left(t\right)$ are positive functions, $\gamma >0$ is a quotient of odd positive integers and the delay function $\tau \left(t\right)\le t$ satisfies ${lim}_{t\to infty}\tau \left(t\right)=infty$. We establish some sufficient conditions which ensure that (*) is oscillatory or the solutions converge to zero. Our results in the nondelay case extend and improve some known results and in the delay case the results can be applied to new classes of equations which are not covered by the known criteria....