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A global implicit function theorem for the $$(n\times m)$$-dimensional case. (Ukrainian. English summary) Zbl 1025.26005
In this paper the authors generalize their results in [Ukr. Math. J. 53, No. 3, 427-437 (2001); translation from Ukr. Mat. Zh. 53, 372-382 (2001; Zbl 0992.26010)] concerning the existence and extendability of solutions of the equation $$g(t,x)=0$$. Now they study the multidimensional case where $$g:\mathbb R^n\times \mathbb R^m \mapsto \mathbb R^m$$. The authors suggest a classification of critical points for $$g(t,x)$$ as well as a definition of the maximal solution $$x(t)$$ to the equation $$g(t,x)=0$$. Their main result is that for any non-critical point $$(t_0,x_0)$$, which is not an isolated point of $$g^{-1}(0)$$, there exists a unique solution $$x(t)$$ defined on a maximal domain $$M_{\max}$$ such that $$x(t_0)=x_0$$ and, in addition, any point $$(t,x(t))$$ is a critical one provided $$t\in \partial M$$. The case where $$g:\mathbb T^n\times \mathbb R^n$$ ($$\mathbb T^m=\mathbb R^m/2\pi \mathbb Z^m$$ denotes $$m$$-dimensional torus) is also discussed.
##### MSC:
 26B10 Implicit function theorems, Jacobians, transformations with several variables 58C15 Implicit function theorems; global Newton methods on manifolds