Antiperiodic boundary value problem for an implicit ordinary differential equation

The paper is devoted to the investigation of the antiperiodic boundary value problem for an implicit nonlinear ordinary differential equation
$$f(t,x,\dot x)=0, \quad x(0)+x(\tau)=0.$$
We assume that the mapping $f:\mathbb{R}\times \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}^k$ defining the equation under consideration is smooth and satisfies the condition of uniform nondegeneracy of the first derivative
$$ \inf \bigl\{ {\rm cov} f'_v (t,x,v):\, (t,x,v)\in \mathbb{R}\times \mathbb{R}^n \times \mathbb{R}^n \bigr\}>0. $$
Here ${\rm cov} A$ is the Banach constant of the linear operator $A.$ The assumption of uniform non-degeneracy holds, in particular, for the mapping $f$ defining an explicit ordinary differential equation. For implicit equations, sufficient conditions for the existence of a solution to an antiperiodic boundary value problem are obtained, and estimates for solutions are found. Corollaries for normal ordinary differential equations are formulated. To prove the main result, the original implicit equation is reduced to an explicit differential equation by applying a nonlocal implicit function theorem. Then we prove an auxiliary assertion on the solvability of the equation $x+\psi(x)=0,$ which is an analog of Brouwer's fixed point theorem. It is shown that the mapping $\psi,$ that assigns the value of the solution of the Cauchy problem at the point $\tau$ to an arbitrary initial point $x_0,$ is well defined and satisfies the assumptions of the auxiliary statement. This reasoning completes the proof of the existence of a solution to the boundary value problem.