Theorem on an implicit function in quasidifferential calculus
The author proves two generalizations of the implicit function theorem for quasidifferentiable functions. Therefore let \(f:\mathbb{R}^m\times\mathbb{R}^n\to\mathbb{R}^n\) be componentwise quasidifferentiable in the second argument with \(f(x_0,y_0)=0_n\). The first result is a restatement of the implicit function theorem of classical analysis in terms of quasidifferentiable calculus, giving necessary conditions for the existence and uniqueness of a continuous \(y:\mathbb{R}^m\to\mathbb{R}^n\) with \(y(x_0)=y_0\) such that \[ f(x,y(x))=0_n \] for \(x\) near \(x_0\). Given in addition \(g\in\mathbb{R}^m\) and \(q_0\in\mathbb{R}^n\), the second theorem gives necessary conditions for the existence and uniqueness of a continuous \(q:[0,\alpha_0]\to\mathbb{R}^n\) with \(q(0)=q_0\) such that \[ f(x_0+\alpha g,y_0+\alpha q(\alpha))=0_n \] for \(\alpha\in [0,\alpha_0]\), that is, the implicit function exists only in the direction \((g,q_0)\). The differences between both theorems are exhibited in a lot of examples.
scientific article; zbMATH DE number 903431
scientific article