Unbounded solutions of conservative oscillators under roughly periodic perturbations (Q759888)

This note is concerned with the equation (1) \(d^ 2x/dt^ 2+g(x)=p(t)\), where g(x) is a continuously differentiable function of \(x\in {\mathbb{R}}\), \(xg(x)>0\) whenever \(x\neq 0\), and g(x)/x tends to \(\infty\) as \(| x| \to \infty\). Let p(t) be a bounded function of \(t\in {\mathbb{R}}\). Define its norm by \(\| p\| =\sup_{t\in {\mathbb{R}}}| p(t)|\). This note leads to the following conclusion which improves a result due to J. E. Littlewood. For any given small constants \(\alpha >0\) and \(\epsilon >0\), there is a continuous and roughly periodic (with respect to \(\Omega\) (\(\alpha)\)) function p(t) with \(\| p\| <\epsilon\) such that the corresponding equation (1) has at least one unbounded solution.