On global implicit function theorem
Let \(E\) be a Banach space and \(F : E\times \mathbb{R}^{m}\to \mathbb{R}^{m}\) be differentiable and \(\det \frac{\partial F}{\partial y}(x,y)\neq 0\) for all \((x,y)\) in a suitable subset of \(E\times \mathbb{R}^{m}\). Let \((x_0, y_0)\in E\times \mathbb{R}^{m}\) such that \(F(x_0, y_0)=0\). The authors prove the existence of unique continuous implicit function \(\phi: E\to \mathbb{R}^{m}\) such that \(\phi(x_0)=y_0\), \(F(x, \phi(x))=0\) for all \(x\in E\), \(\phi\) is differentiable on \(E\setminus \pi_{1}(K)\) and \(\phi^{\prime}(x)=f(x, \phi(x))\) for all \(x\in E\setminus \pi_{1}(K)\), \[ f(x,y)=-\frac{\partial F}{\partial y}(x, y)^{-1}\frac{\partial F}{\partial x}(x,y) \] for all \((x,y)\in (E\times \mathbb{R}^{m})\setminus K\), where \(K\) is a suitable subset of \(E\times \mathbb{R}^{m}\) and \(\pi_{1}\) is the projection of \(K\). Analogous results are also included under weaker conditions.
global existence of the solutions of differential equations
scientific article; zbMATH DE number 6864242
scientific article