A proof of Levin's half-period theorem (Q2468724)

This paper considers the Hill equation \[ \ddot x+p(t)x=0, \quad p(t)\neq 0, \quad p(t+1)=p(t), \tag \(*\) \] where the control \(p(t)\) is a continuous function. It is well known that if each nontrivial solution of \((*)\) has infinitely many zeros on the time axis and the distance between arbitrary consecutive zeros exceeds 1, then one says \((*)\) belongs to the zero stability zone. Let \(\Pi(a)= \{p(t)\): \(\int_t^{t+\frac12} p(s)\,ds=a\), \(a>0\}\) denotes the set of controls implies that \((*)\) belongs to the zero stability zone. A control \(p_0\in\Pi(a_0)\) and the constant \(a_0\) are considered to be an extremal pair if \[ \max_t \int_t^{t+\frac12} p_0(s)\,ds=a_0 \] and for \(a_0<a<a_0+ \varepsilon\), where \(\varepsilon\) is an arbitrarily small positive constant, there exists a control \(p_\varepsilon(t)\) satisfying the relation \[ \max_t \int_t^{t+\frac12} p_\varepsilon(s)\,ds= a \] and such that \((*)\) does not belong to the zero stability zone. In this paper, by using the method of the solved extremal problem \(a_0=\sup a\), \(p\in\Pi(a)\), the author gives a new proof of the Levin's half-period theorem: If \(\int_t^{t+\frac12}p(s)\,ds\leq 4\) for each \(t\), then \((*)\) belongs to the zero stability zone.