Non-trivial periodic solutions of a non-linear Hill's equation with positively homogeneous term (Q2496801)

Consider the differential equation \[ x''+ a(t)x= g(t,x)\tag{\(*\)} \] under the assumptions \((H_1)\) \(a\in L^\infty(\mathbb{R})\) is \(T\)-periodic and satisfies \(a(t)\geq 0\) for a.e. \(t\in [0,T]\). The Hill equation \(x''+ a(t)x= 0\) has no nontrivial \(T\)-periodic solution. \((H_2)\) \(g: \mathbb{R}\times\mathbb{R}\to \mathbb{R}\) is a Carathéodory function, \(T\)-periodic in the first variable, continuously differentiable with respect to the second variable and satisfies for \(r> 1\) \(g(t,\lambda x)= \lambda^r g(t,x)\) for a.e. \(t\in [0,T]\), \(\forall \lambda> 0\), \(x\in\mathbb{R}\), \(\lim_{x\to\infty} g(t,x)=+\infty\) uniformly for a.e. \(t\). The author proves by perturbation arguments and upper and lower solutions that \((*)\) has a nontrivial \(T\)-periodic solution under the hypotheses \((H_1)\) and \((H_2)\).