Abstract

A nonlocal generalized implicit function theorem is obtained for mappings acting in Hilbert spaces. Its proof is based on the theory of ordinary differential equations. Under natural assumptions, we use this theorem to derive a global generalized implicit function theorem as well as a global implicit function theorem and a global inverse function theorem as particular cases of the former. Using the estimates produced, we prove an assertion on the continuation of an implicit function from a given closed set to the entire parameter space.

中文翻译：

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Hilbert空间中的非局部广义隐函数定理

摘要

对于作用于希尔伯特空间的映射，获得了一个非局部广义隐函数定理。它的证明是基于常微分方程的理论。在自然假设下，我们使用该定理来推导全局广义隐函数定理，全局隐函数定理和全局逆函数定理，作为前者的特殊情况。使用产生的估计，我们证明了隐式函数从给定的闭合集到整个参数空间的连续性的断言。