| Abstract: |
| We are concerned with the behavior of solutions to the following two classes
of nonlinear second-order differential equations with a damping term:
\begin{equation}
(r(t)x^{\prime})^{\prime}+p(t)x^{\prime}+q(t)x+f(t,x)=e(t),\qquad t\geq
t_{0},\label{EqMain}%
\end{equation}
and%
\begin{equation}
(r(t)x^{\prime})^{\prime}+p(t)x^{\prime}+q(t)g(x)=e(t),\qquad t\geq
t_{0},\label{EqMain2}%
\end{equation}
where $e(t)$ is a continuous non-homogeneous term, and nonlinear terms
$f(t,x)$ and $g(x)$ satisfy, respectively, conditions
\begin{equation}
f(t,u)u\geq0\qquad\text{and\qquad}f(t,u)=-f(t,-u)\qquad\text{for all }t\geq
t_{0}\text{ and }u\in\mathbb{R},\label{ftuu}%
\end{equation}
and%
\begin{equation}
g^{\prime}(u)\geq K>0\qquad\text{and\qquad}g(u)=-g(-u)\qquad\text{for all
}u\in\mathbb{R}.\label{ftuu2}%
\end{equation}
Assumptions (\ref{ftuu}) and (\ref{ftuu2}) hold, for instance, for
Emden-Fowler differential equations where $f(t,u)=g(u)=|u|^{\nu}%
\mathrm{sgn}(u)$, $\nu>0$.
We start by providing sufficient conditions for the first derivative of a
solution $x(t)$ of equation (\ref{EqMain}) (or equation (\ref{EqMain2})) to
change sign at least once in a given interval (in a given infinite sequence of
intervals). These conditions imply global non-monotone behavior of solutions.
Recall that a function $h\in C^{1}(t_{0},\infty)$ is called \emph{non-monotone
(weakly oscillatory) on} $(t_{0},\infty)$ if there exists a sequence of points
$\left\{ s_{n}\right\} _{n\in\mathbb{N}}\in(t_{0},\infty),$ $s_{n}%
\rightarrow\infty$ as $n\rightarrow\infty,$ such that $h^{\prime}(t)$ changes
sign at $t=s_{n}$ for all $n\in\mathbb{N}.$ We also discuss how oscillation
criteria can be turned into non-monotonicity tests. |
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