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The neutral differential equation (1.1) $$\frac{{\mathrm{d}}^n}{\mathrm{d}{t}^n}[x\left(t\right)+x(t-\tau )]+\sigma F(t,x\left(g\left(t\right)\right))=0,$$ is considered under the following conditions: $n\ge 2$, $\tau >0$, $\sigma =\pm 1$, $F(t,u)$ is nonnegative on $[t_0,\infty )\times (0,\infty )$ and is nondecreasing in $u\in (0,\infty )$, and $limg\left(t\right)=\infty$ as $t\to \infty$. It is shown that equation (1.1) has a solution $x\left(t\right)$ such that (1.2) $$\underset{t\to \infty }{lim}\frac{x\left(t\right)}{t^k}\text{exists}\text{and}\text{is}\text{a}\text{positive}\text{finite}\text{value}\text{if}\text{and}\text{only}\text{if}{\int }_{t_0}^{\infty }{t}^{n-k-1}F(t,c{\left[g\left(t\right)\right]}^k)\mathrm{d}t<\infty \text{for}\text{some}c>0.$$ Here, $k$ is an integer with $0\le k\le n-1$. To prove the existence of a solution $x\left(t\right)$ satisfying (1.2), the Schauder-Tychonoff fixed point theorem is used.

We consider the half-linear differential equation of the form $$\left(p\left(t\right)\right|x^{\text{'}}{|}^{\alpha }\mathrm{sgn}{x}^{\text{'}}{)}^{\text{'}}+q\left(t\right){\left|x\right|}^{\alpha }\mathrm{sgn}x=0,t\ge t_0,$$ under the assumption that $p{\left(t\right)}^{-1/\alpha }$ is integrable on $[t_0,\infty )$. It is shown that if a certain condition is satisfied, then the above equation has a pair of nonoscillatory solutions with specific asymptotic behavior as $t\to \infty$.

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.

We consider linear differential equations of the form $${\left(p\left(t\right)x^{\text{'}}\right)}^{\text{'}}+\lambda q\left(t\right)x=0(p\left(t\right)>0,q\left(t\right)>0)\left(\mathrm{A}\right)$$ on an infinite interval $[a,\infty )$ and study the problem of finding those values of $\lambda$ for which () has principal solutions $x_0(t;\lambda )$ vanishing at $t=a$. This problem may well be called a singular eigenvalue problem, since requiring $x_0(t;\lambda )$ to be a principal solution can be considered as a boundary condition at $t=\infty$. Similarly to the regular eigenvalue problems for () on compact intervals, we can prove a theorem asserting that there exists a sequence $\left\{{\lambda }_n\right\}$ of eigenvalues such...