Nonautonomous weak Markus–Yamabe conjecture

Let $f\colon\mathbb{R}^{+}\times\mathbb{R}^{n}\to\mathbb{R}^{n}$ be as in the system $\dot{x}=f(t,x)$, and assume that it satisfies (G1) and (G3$^{*}$). Here (G3$^{*}$) requires that for every bounded piecewise continuous function $t\mapsto\omega(t)$, the linear system $\dot{\vartheta}=Jf(t,\omega(t))\vartheta$ has nonuniform exponential dichotomy spectrum contained in $(-\infty,0)$. Nonautonomous weak Markus–Yamabe conjecture. The family of maps $t\mapsto F_t$, where $F_t(x)=f(t,x)$, is pseudo partially injective. This weak conjecture is introduced to connect the bounded nonuniform nonautonomous Markus–Yamabe problem with a parametrized form of the Jacobian Conjecture. The source does not provide a resolution.