\(C^1\) solutions for fully nonlinear systems of differential equations of first order (Q1028281)

Consider the system of differential equations \[ f(x'(t))= g(t,x(t)),\quad x(0) = x_{0}. \tag{1} \] Theorem. Let \(\Omega\) be an open set containing \((0,x_{0}), g:\Omega\subset\mathbb{R}\times\mathbb{R}^{n}\to\mathbb{R}^{n}\) is a continuous function. Suppose \(f:\mathbb{R}^{n}\to\mathbb{R}^{n}\) is continuous and \( \Lambda_{f}= \{x\in\mathbb{R}^{n}: f\) is not locally Lipschitz in an open neighborhood of \(x\)\}. If \(f\) satisfies the following conditions: {\parindent7mm \begin{itemize}\item[i)] \(\mathrm{Im}(g)\subseteq \mathrm{Im}(f)\), \item[ii)] \(f\) is coercive, \item[iii)] \(f(\Lambda_{f}\cup\{x\notin\Lambda_{f}:\partial f(x)\text{ is not of maximal rank}\})\subseteq\{y\in \mathrm{Im}(f):\mathrm{card}\, f^{-1}(y) =1\},\) then there exists at least one \(C^{1}\) solution \(x:[0,T]\to\mathbb{R}^{n}\) to the Cauchy problem (1). \end{itemize}}