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| UNBOUNDED SOLUTIONS OF CONSERVATIVE OSCILLATORS UNDER ROUGHLY PERIODIC PERTURBATIONS |
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Citation： |
Ding Tongren.UNBOUNDED SOLUTIONS OF CONSERVATIVE OSCILLATORS UNDER ROUGHLY PERIODIC PERTURBATIONS[J].Chinese Annals of Mathematics B,1984,5(4):687~694 |
| Page view： 797
Net amount： 794 |
Authors： |
Ding Tongren; |
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| Abstract： |
This note is concerned with the equation
$$\[\frac{{{d^2}x}}{{d{t^2}}} + g(x) = p(t)\begin{array}{*{20}{c}}
{}&{(1)}
\end{array}\]$$
where g(x) is a continuously differentiable function of a $\[x \in R\]$, $\[xg(x) > 0\]$ whenever $\[x \ne 0\]$, and
$\[g(x)/x\]$ tends to $\[\infty \]$ as \[\left| x \right| \to \infty \]. Let p(t) be a bounded function of $\[t \in R\]$. Define its norm by
$\[\left\| p \right\| = {\sup _{t \in R}}\left| {p(t)} \right|\]$
The study of this note leads to the following conclusion which improves a result due to
J. E. Littlewood,
For any given small constants $\[\alpha > 0,s > 0\]$, there is a continuous and roughly periodic(with respect to $\[\Omega (\alpha )\]$) function p(t) with $\[\left\| p \right\| < s\]$ such that the corresponding equation (1)
has at least one unbounded solution. |
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