When studying the behavior of autonomous ordinary differential equations under time‐dependent perturbations vanishing for $t\to \pm \infty$, their equilibria generically persist locally as homoclinic solutions. Using an abstract and flexible continuation theorem, we find even global branches of such homoclinic solutions for parametrized nonautonomous ordinary differential equations. Our approach is based on degree‐theoretical arguments. In particular, Landesman–Lazer conditions are proposed to obtain the existence of homoclinic solutions by means of a nonzero degree.

中文翻译：

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Fredholm映射II的延续原理：应用于同宿解决方案

在研究时变扰动消失的情况下自治常微分方程的行为时 $Ť\to \pm \infty$，它们的平衡一般作为同宿解决方案在本地持续。使用抽象且灵活的连续性定理，我们甚至找到了针对参数化非自治常微分方程的此类同宿解的全局分支。我们的方法基于度理论论证。特别是，提出了Landesman–Lazer条件，以通过非零度来获得同宿解。