\(C^1\) solutions for semi-implicit systems of differential equations (Q715719)

The paper studies a semi-implicit differential equation of the form \[ f(x'(t))=g(t,x(t)),\quad x(0)=x_0, \] where \(g:D_g\subset {\mathbb R}\times {\mathbb R}^n\to {\mathbb R}^n\) and \(f:D_f\subset {\mathbb R}^n\to {\mathbb R}^n\) are continuous functions. The existence of continuously differentiable solutions of the considered problem is obtained from a general result concerning the existence of continuously differentiable solutions for nonconvex differential inclusions applied to the differential inclusion \[ x'(t)\in (f^{-1}\circ g)(t,x(t)),\quad x(0)=x_0. \]