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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.
26B10 Implicit function theorems, Jacobians, transformations with several variables
58C15 Implicit function theorems; global Newton methods on manifolds