On interior points of a set of equations with singular periodic solutions (Q945978)

The number of solutions of period \(\omega\) to equations of the form \[ \dot x=x^n+p_1(t)x^{n-1} +\dots + p_n(t), \] where \(p_j(t)\) are real-valued \(\omega\)-periodic functions satisfying aLipschitz condition, was studied by V. A. Pliss. If we fix \(n\) and \(\omega\), identify the above equation with a vector function \(P: \mathbb R \to \mathbb R^n\), where \(P(t)=(p_1(t), \dots , p_n(t))\), and introduce the metric \[ \rho(P,Q)=\max\{| p_j(t)-q_j(t)|:t\in \mathbb R,\quad j=1,\dots n\}, \] then the set of the above equations becomes a metric space \(X_n(\omega)\). Denote the set of all equations of the space \(X_n(\omega)\) having singular periodic solutions by \(B_n(\omega)\). This set is known to be closed. A. P. Begun investigated the problem of existence of interior points in \(B_n(\omega)\) and found the non-existence for \(n\geq 3\). The authors establish a sufficient condition for an equation from \(X_2(\omega)\) to be an interior point of the set \(B_2(\omega)\) and give an example satisfying this condition.