This paper answers in the affirmative the long-standing question of nonlinear analysis, concerning the existence of a continuous single-valued local selection of the right inverse to a locally Lipschitzian mapping. Moreover, we develop a much more general result, providing conditions for the existence of a continuous single-valued selection not only locally, but rather on any given ball centered at the point in question. Finally, by driving the radius of this ball to infinity, we obtain the global inverse function theorem, essentially implying the well-known Hadamard’s theorem on a global homeomorphism for smooth mappings and the more general Pourciau’s theorem for locally Lipschitzian mappings.

中文翻译：

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局部 Lipschitzian 方程解的连续选择

本文肯定地回答了非线性分析的长期存在的问题，即关于局部 Lipschitzian 映射的右逆的连续单值局部选择的存在。此外，我们开发了一个更一般的结果，为连续单值选择的存在提供了条件，不仅在局部，而且在以相关点为中心的任何给定球上。最后，通过将这个球的半径驱动到无穷大，我们获得了全局反函数定理，本质上暗示了著名的关于平滑映射的全局同胚的 Hadamard 定理和更一般的适用于局部 Lipschitzian 映射的 Pourciau 定理。