Invariance of homotopy and an extension of a theorem by Habets-Metzen on periodic solutions of Duffing equations (Q5958903)

The author considers boundary value problems such as NEWLINE\[NEWLINE \begin{multlined} x''+f(t,x,x')x'+g^+(t,x,x')x^++g^-(t,x,x')x^-+h(t,x,x')=0,\\ x(0)=x(1),\quad x'(0)=x'(1), \end{multlined}\tag{1}NEWLINE\]NEWLINE where the functions \(f(t,x,x')\) and \(g^{\pm}(t,x,x')\) are Carathéodory functions that satisfy the conditions \(a(t)\leq f(t,x,x')\leq b(t)\), \(c^{\pm}(t)\leq g^{\pm}(t,x,x')\leq d^{\pm}(t)\) and \(|h(t,x,x')|\leq e(t)\) for some \(a\), \(b\), \(c^{\pm}\), \(d^{\pm}\), \(e\in L^1(0,1)\). The author aims to prove the existence of solutions to (1) if the problem NEWLINE\[NEWLINEx''+p(t)x'+q^+(t)x^++q^-(t)x^-=0, \qquad x(0)=x(1),\quad x'(0)=x'(1),NEWLINE\]NEWLINE has no nontrivial solutions for every \(p\), \(q^{\pm}\in L^1(0,1)\) such that \(p(t)\in [a(t),b(t)]\) and \(q^{\pm}(t)\in[c^{\pm}(t),d^{\pm}(t)]\). Such a result is false for the Dirichlet problem, might be true for the periodic one and would generalize several previous works. Unfortunately, the argument presented by the author uses an assumption, \(q^{\pm}\in {\mathcal F}_n(p)\), which he claims unnecessarily.