In this paper we introduce a new algorithm for solving perturbed nonlinear functional equations which admit a right-invertible linearization, but with an inverse that loses derivatives and may blow up when the perturbation parameter $\varepsilon$ goes to zero. These equations are of the form $F_\varepsilon(u)=v$ with $F_\varepsilon(0)=0$, $v$ small and given, $u$ small and unknown. The main difference from the by now classical Nash–Moser algorithm is that, instead of using a regularized Newton scheme, we solve a sequence of Galerkin problems thanks to a topological argument. As a consequence, in our estimates there are no quadratic terms. For problems without perturbation parameter, our results require weaker regularity assumptions on $F$ and $v$ than earlier ones, such as those of Hörmander [17]. For singularly perturbed functionals $F_\varepsilon$, we allow $v$ to be larger than in previous works. To illustrate this, we apply our method to a nonlinear Schrödinger Cauchy problem with concentrated initial data studied by Texier–Zumbrun [26], and we show that our result improves significantly on theirs.

中文翻译：

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具有奇异微扰和导数损失的映射的一个投影定理

在本文中，我们介绍了一种用于求解受扰非线性函数方程的新算法，该方程允许右可逆线性化，但逆函数会丢失导数，并且当扰动参数 $\varepsilon$ 变为零时可能会爆炸。这些方程的形式为 $F_\varepsilon(u)=v$，其中 $F_\varepsilon(0)=0$，$v$ 小且给定，$u$ 小且未知。与现在经典的 Nash-Moser 算法的主要区别在于，我们不使用正则化的牛顿方案，而是通过拓扑参数解决一系列 Galerkin 问题。因此，在我们的估计中没有二次项. 对于没有扰动参数的问题，我们的结果需要对 $F$ 和 $v$ 的正则性假设比早期的假设要弱，例如 Hörmander [17] 的假设。对于奇异扰动的泛函 $F_\varepsilon$，我们允许 $v$ 比以前的工作大。为了说明这一点，我们将我们的方法应用于 Texier-Zumbrun [26] 研究的具有集中初始数据的非线性薛定谔柯西问题，并且我们表明我们的结果显着改善了他们的结果。