Oscillation of second-order nonlinear differential equations with damping term

E. M. Elabbasy, Mansoura University, Mansoura, Egypt
W. W. Elhaddad, Mansoura University, Mansoura, Egypt

E. J. Qualitative Theory of Diff. Equ., No. 25. (2007), pp. 1-19.

Abstract: We present new oscillation criteria for the second order nonlinear differential equation with damping term of the form
\begin{equation*}
\left( r(t)\psi (x)f(\dot{x})\right) ^{\cdot }+p(t)\varphi \left(g(x),r(t)\psi(x)f(\dot{x})\right)+q(t)g(x)=0,
\end{equation*}
where $p$, $q$, $r:[t_{o},\infty )\rightarrow \mathbf{R}$ and\ $\psi $, $g $, $f:\mathbf{R}\rightarrow \mathbf{R}$ are continuous, $r(t)>0$,\ $p(t)\geq 0$ and $\psi (x)>0$, $xg(x)>0$ for $x\neq 0$, $uf(u)>0$ for $u\neq0 $. Our results generalize and extend some known oscillation criteria in the literature. The relevance of our results is illustrated with a number of examples.

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