C1C1 solutions for fully nonlinear systems of differential equations of first order

This note is concerned with the existence of continuously differentiable solutions for the nonlinear system of differential equationsf(x′(t))=g(t,x(t)),f(x′(t))=g(t,x(t)),x(0)=x0,x(0)=x0, where Ω   is an open set containing (0,x0)(0,x0), g:Ω⊆R×Rn→Rng:Ω⊆R×Rn→Rn is continuous and f:Rn→Rnf:Rn→Rn satisfies Im(g)⊆Im(f)Im(g)⊆Im(f). The set of points x such that f is not locally Lipschitz in an open neighborhood of x   is denoted by ΛfΛf. We prove the existence of at least one C1C1 solution x:[0,T]→Rnx:[0,T]→Rn to the system if f is continuous, coercive and if each y in the setf(Λf∪{x∉Λf:∂f(x)is not of maximal rank}) has exactly one preimage in RnRn.