2011 / March Volume 6 No.1
Oscillatory and asymptotic behavior of $\frac{dx}{dt}+Q(t)G(x(t-\sigma(t)))=f(t)$
| Published Date |
2011 / March
|
|---|---|
| Title | Oscillatory and asymptotic behavior of $\frac{dx}{dt}+Q(t)G(x(t-\sigma(t)))=f(t)$ |
| Author | |
| Keyword | |
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| Pagination | 97-113 |
| Abstract | Consider the
equation
$$\frac{dx}{dt}+Q(t)G(x(t-\sigma(t)))=f(t)\qquad{(*)}$$
where $\textit{f},\sigma,\textit{Q}\in C
([0,\infty),[0,\infty)),G\in C(R,R),G(-x)=-G(x),xG(x)> o $ for $ x
\neq 0,$ $\textit{G}$ is non-decreasing, $t >\sigma(t)$, $\sigma(t)$
is decreasing and $ t-\sigma(t)\rightarrow\infty$ as
$t\rightarrow\infty$.
When$ f(t)\equiv 0$, a sufficient condition in terms of the constants
\begin{eqnarray*}
k&=&\liminf_{t\rightarrow\infty}{\large\int_{t-\sigma(t)}^{t}{Q(s)ds}}\{\hbox{and}} \qquad
L &=&\limsup_{t\rightarrow{\infty}}{\large\int_{t-\sigma(t)}^{t}{Q(s)ds}}
\end{eqnarray*}
is established for all solutions of $(*)$ to be oscillatory.The
present results improve the earlier results of the literature by
both weakening the conditions
and considering a general non linear and non-homogeneous differential equation.
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| AMS Subject Classification |
34K11,34C10
|
| Received |
2010-10-05
|
| Accepted |
2010-10-05
|