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On the domain of the implicit function and applications. (English) Zbl 1108.26015

The implicit function theorem in multivariable calculus asserts that if \(F_i(x,y),\; i=1,\dots,n;\; x\in \mathbb R^m, y\in \mathbb R^n\), are continuously differentiable in a neighborhood of the point \((x_0,y_0)\), and \(F_i(x_0,y_0)=0\) for each \(i\), and the Jacobian \(D_yF(x_0,y_0)\) is invertible, then there are continuous functions \(g_1, \dots, g_n\) and a positive radius \(r\) such that \(g_i(x_0)=y_0\) and \(F_i(x,g_1(x),\ldots,g_n(x))=0\) for \(i=1,\ldots,n\) and \(| x-x_0| <r\). The theorem has been extended to Lipschitz functions by F. H. Clarke [Pac. J. Math. 64, 97–102 (1976; Zbl 0331.26013); “Optimization and nonsmooth analysis” (1983; Zbl 0582.49001)].
The author derives a positive lower bound for this radius \(r\) for the case of Lipschitz functions. It is further shown that an implicit function exists under certain conditions in the case of a set of inequalities, and again an estimate for the radius of the domain is obtained. An application to the local Lipschitz behavior of solution functions is discussed.

MSC:

26B10 Implicit function theorems, Jacobians, transformations with several variables
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