动力系统方法（DSM）用于全局同胚的合理性证明
Justification of the Dynamical Systems Method (DSM) for global homeomorphisms
AI总结：
针对Hilbert空间中的非线性算子方程F(u)=f，在F为全局同胚且Fréchet导数逆有界（非一致有界）条件下，证明牛顿法连续类比(*)强收敛到解，并推广到单调算子情形。
摘要：
动力系统方法（DSM）被证明适用于求解算子方程 $F(u)=f$，其中 $F$ 是 Hilbert 空间 $H$ 中的非线性算子。假设 $F$ 是 $H$ 到 $H$ 的全局同胚，且 $F\in C^1_{loc}$，即它具有关于 $u$ 连续的 Fréchet 导数 $F'(u)$，算子 $[F'(u)]^{-1}$ 对所有 $u\in H$ 存在且有界，$||[F'(u)]^{-1}||\leq m(u)$，其中 $m(u)>0$ 是依赖于 $u$ 的常数，且不一定关于 $u$ 一致有界。在这些假设下，证明了牛顿法的连续类比 $\dot{u}=-[F'(u)]^{-1}(F(u)-f), \quad u(0)=u_0, \quad (*)$ 对任意 $f\in H$ 和任意 $u_0\in H$ 强收敛到方程 $F(u)=f$ 的解。此前，在不假设 $F'(u)$ 是 Lipschitz 连续的情况下，Cauchy 问题 (*) 的解的全局（甚至局部）存在性尚未建立。还考虑了 $F$ 不是全局同胚而是 $H$ 中的单调算子的情况。
英文摘要：
The Dynamical Systems Method (DSM) is justified for solving operator equations $F(u)=f$, where $F$ is a nonlinear operator in a Hilbert space $H$. It is assumed that $F$ is a global homeomorphism of $H$ onto $H$, that $F\in C^1_{loc}$, that is, it has a continuous with respect to $u$ Fréchet derivative $F'(u)$, that the operator $[F'(u)]^{-1}$ exists for all $u\in H$ and is bounded, $||[F'(u)]^{-1}||\leq m(u)$, where $m(u)>0$ is a constant, depending on $u$, and not necessarily uniformly bounded with respect to $u$. It is proved under these assumptions that the continuous analog of the Newton's method $\dot{u}=-[F'(u)]^{-1}(F(u)-f), \quad u(0)=u_0, \quad (*)$ converges strongly to the solution of the equation $F(u)=f$ for any $f\in H$ and any $u_0\in H$. The global (and even local) existence of the solution to the Cauchy problem (*) was not established earlier without assuming that $F'(u)$ is Lipschitz-continuous. The case when $F$ is not a global homeomorphism but a monotone operator in $H$ is also considered.
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