A second-order differential equation

Definitions and mathematical statement

Fix $p>0$, $T>0$, and a real continuous function $f$ on $\mathbb R$ satisfying $f(t+T)=f(t)$ and $f\not\equiv0$. Seek a twice continuously differentiable function $x:\mathbb R\to(0,\infty)$ such that \[ x(t+T)=x(t),\qquad x(t)^p x''(t)=f(t) \quad(t\in\mathbb R). \] Positivity is pointwise; by periodicity it also gives a positive minimum. No upper or lower amplitude bound is specified in advance. The problem is to give necessary and sufficient conditions on $f$, for each fixed $p,T$, for at least one such positive periodic solution to exist.

The conditions should be stated directly in terms of the forcing and should apply to arbitrary continuous periodic functions, including those with many sign changes. Merely restating the existence of a fixed point of a solution operator is not a characterization of the intended kind.

Two necessary conditions illustrate the issue. Periodicity gives $\int_0^T f(t)x(t)^{-p}\,dt=0$, and integration by parts yields \[ \int_0^T f(t)\,dt=-p\int_0^T x(t)^{p-1}x'(t)^2\,dt<0. \] The strict inequality follows because nonzero forcing excludes constant solutions. In particular $f$ must take both positive and negative values. These restrictions explain why nontrivial sign information is needed; the question asks for a complete criterion, without assuming that these obvious necessary conditions are sufficient.

Short English statement

Characterize the periodic forcings for which this singular second-order equation has an everywhere-positive periodic solution.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 4700009 (AMR-046-0009), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Gasull - Some open problems in low dimensional dynamical systems (2020), Problem environment 9: A second-order differential equation. Original source indexed by AMR-046-0009. https://arxiv.org/abs/2012.02524.