AI 中文总结
该研究针对具(μ,ν)-二分的非自治线性系统，采用格林函数方法构造Palmer型映射，在小性条件下得到Cʳ类光滑拓扑等价，并验证了无界扰动情形下假设的可实现性。
AI 中文摘要
我们研究正半轴上的非自治线性系统与一类拟线性扰动系统之间的光滑拓扑等价，其中该拟线性扰动的非线性部分不要求对状态变量全局有界。假设该线性方程容许(μ,ν)-二分，且构造通过与该二分关联的格林算子完成。扰动的通常全局有界性被替换为沿相关线性与非线性解的局部一致格林可积性条件。在涉及扰动的全局利普希茨常数与格林算子的小性条件下，我们首先构造Palmer型映射，其给出正半轴ℝ⁺上的连续拓扑等价。随后，我们对逐次变分方程施加格林可积性条件，并施加进一步的一阶小性条件以保证逆映射导数的可逆性。在这些假设下，该等价为Cʳ类。我们还提供了一类示例，其中扰动对空间变量无界，且所有假设均可直接验证。
英文摘要
We study the smooth topological equivalence between a nonautonomous linear system on the positive half-line and a quasilinear perturbation whose nonlinear part is not assumed to be globally bounded with respect to the state variable. The linear equation is assumed to admit a $(μ,ν)$--dichotomy, and the construction is carried out through the Green operator associated with this dichotomy. The usual global boundedness of the perturbation is replaced by locally uniform Green-integrability conditions along the relevant linear and nonlinear solutions. Under a smallness condition involving the global Lipschitz constant of the perturbation and the Green operator, we first construct Palmer-type maps which give a continuous topological equivalence on $\mathbb R^+$. We then impose Green-integrability conditions on the successive variational equations and a further first-order smallness condition which guarantees the invertibility of the derivative of the inverse map. Under these assumptions, the equivalence is of class $C^r$. We also provide a class of examples for which the perturbation is unbounded with respect to the space variable and all the hypotheses can be verified directly.
