Oscillation of differential equations with non-monotone retarded arguments
2016 ◽
Vol 19
(1)
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pp. 98-104
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Keyword(s):
Consider the first-order retarded differential equation $$\begin{eqnarray}x^{\prime }(t)+p(t)x({\it\tau}(t))=0,\quad t\geqslant t_{0},\end{eqnarray}$$ where $p(t)\geqslant 0$ and ${\it\tau}(t)$ is a function of positive real numbers such that ${\it\tau}(t)\leqslant t$ for $t\geqslant t_{0}$, and $\lim _{t\rightarrow \infty }{\it\tau}(t)=\infty$. Under the assumption that the retarded argument is non-monotone, a new oscillation criterion, involving $\liminf$, is established when the well-known oscillation condition $$\begin{eqnarray}\liminf _{t\rightarrow \infty }\int _{{\it\tau}(t)}^{t}p(s)\,ds>\frac{1}{e}\end{eqnarray}$$ is not satisfied. An example illustrating the result is also given.
1963 ◽
Vol 3
(2)
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pp. 185-201
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Keyword(s):
1978 ◽
Vol 26
(3)
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pp. 323-329
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1998 ◽
Vol 41
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pp. 207-213
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1974 ◽
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(1)
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pp. 95-101
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Vol 43
(1)
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pp. 147-152
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1979 ◽
Vol 22
(4)
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pp. 403-412
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1999 ◽
Vol 59
(2)
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pp. 305-314
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2000 ◽
Vol 130
(3)
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pp. 517-525
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