Sign-changing first derivative of positive solutions of forced second-order nonlinear differential equations (Q2343694)

Consider the scalar equation \[ (r(t)x')'+ p(t) x'+ q(t)x+ f(t,x)= e(t),\quad t\geq t_0\tag{\(*\)} \] under the assumptions \(p,e\in C([t_0,\infty), \mathbb{R})\), \(r,q\in C^1([t_0,\infty), \mathbb{R})\), \(f(t,x) x\geq 0\,\forall t\geq t_0\), \(x> 0\), and \(e\) is oscillatory, that is, \((-1)^ne(t)\geq 0\) on \((a_n,b_n)\) \(\forall n\in\mathbb{N}\). The author derives sufficient conditions on \(r\), \(p\), \(q\) such that every positive solution of \((*)\) has a sign-changing first derivative.