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Oscillatory and asymptotic behaviour of a nonlinear second order neutral differential equation. (English) Zbl 1150.34026
The authors study the oscillation and asymptotic behaviour of solutions of second order neutral delay differential equations. They first consider the equation $$ \bigl [r(t)\bigl (y(t)\bigr )- p(t)y(t-\tau )\bigr )'\bigr ]' + q(t)G\Bigl (y\bigl ( h(t)\bigr )\Bigr )= 0, $$ where   $q,h\in C\bigl ([0,\infty ), \Bbb R\bigr )$ such that $q(t)\ge 0$, $\,r\in C^{(1)}\bigl ([0,\infty )$, $(0,\infty )\bigr )$, $\, p\in C\bigl ([0,\infty ), \Bbb R\bigr )$, $G\in C(\Bbb R,\Bbb R)$ and $\tau \in \Bbb R^+$. The authors derive certain sufficient conditions for every solution of the above differential equation to be oscillatory or tending to zero. Finally, they consider the non-homogeneous form of the above equation, where zero on the right hand side is replaced by a function $f(t)\in C\bigl ([0,\infty ), \Bbb R\bigr )$ and find necessary conditions for every solution of this equation to be oscillatory or tending to zero.

MSC:
34K11Oscillation theory of functional-differential equations
34K25Asymptotic theory of functional-differential equations
34K40Neutral functional-differential equations
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Full Text: DOI EuDML
References:
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