Nonuniform nonautonomous Markus–Yamabe conjecture

Consider the nonautonomous nonlinear system

$$\dot{x}=f(t,x),$$

where $f\colon\mathbb{R}_{0}^{+}\times\mathbb{R}^{n}\to\mathbb{R}^{n}$. Assume that $f$ satisfies the following conditions: it is continuous on $\mathbb{R}_{0}^{+}\times\mathbb{R}^{n}$ and $C^1$ with respect to $x$, its forward solutions are defined on $[t_0,+\infty)$ for every $t_0\geq 0$, $f(t,x)=0$ whenever $x=0$, and, for every piecewise continuous function $t\mapsto\omega(t)$, the linear system

$$\dot{\vartheta}=Jf(t,\omega(t))\vartheta$$

where $Jf(t,\cdot)$ is the Jacobian matrix of $f(t,\cdot)$, has a $(Ke^{\varepsilon s},\gamma)$-nonuniform exponential dichotomy spectrum satisfying

$$\Sigma^{+}(Jf(t,\omega(t)))\subset(-\infty,0).$$

Nonuniform nonautonomous Markus–Yamabe conjecture. Under these conditions, the trivial solution of the nonlinear system is globally nonuniformly asymptotically stable.

This is a nonautonomous, nonuniform analogue of the Markus–Yamabe global stability problem, replacing pointwise Jacobian conditions by negativity of the relevant nonuniform exponential dichotomy spectrum. The supplied text states the implication as a conjecture, while the material provided gives no resolution status.