On Lyapunov inequality in stability theory for Hill's equation on time scales (Q1840688)

Let \(T\) be a closed subset of \(\mathbb{R}\). If there exists a positive number \(\Lambda\) such that \(t+n\Lambda\in T\) for all \(t\in T\) and all integers \(n\), then \(T\) is called a periodic time scale with period \(\Lambda\). Consider the linear \(\Delta\)-differential equation \[ \bigl[p(t) y^\Delta(t) \bigr]^\Delta +q(t)y\bigl( \sigma(t)\bigr) =0,\;t\in T,\tag{1} \] where \(\sigma\) is the forward jump operator and \(p(t)>0\) and \(q(t)\) are real smooth \(\Lambda\)-periodic functions defined on \(T\). It is shown that if the nontrivial \(q=q(t)\leq 0\), then all solutions of (1) are unbounded on \(T\); and if \[ \int^\Lambda_0 q(t) \Delta t\geq 0,\;q\neq 0, \] and \[ \left[\max_{t \in[0,\rho (\Lambda)]} {\sigma(t)-t\over p(t)}+ \int^\Lambda_0 {\Delta t\over p(t)}\right] \int^\Lambda_0 \max \bigl\{ q(t), 0\bigr\}\Delta t\leq 4, \] where \(\rho\) is the backward jump operator, then all solutions of (1) are bounded on \(T\).