On the Cauchy problem for implicit differential equations of higher orders

The article is devoted to the study of implicit differential equations based on statements about covering mappings of products of metric spaces. First, we consider the system of equations
$$\begin{equation*} \Phi_i(x_i,x_1,x_2,\ldots,x_n)=y_i, \ \ \ i=\overline{1,n}, \end{equation*}$$
where $\Phi_i: X_i \times X_1 \times \ldots \times X_n \to Y_i,$ $y_i \in Y_i,$ $X_i$ and $Y_i$ are metric spaces, $i=\overline{1,n}.$ It is assumed that the mapping $\Phi_i$ is covering in the first argument and Lipschitz in each of the other arguments starting from the second one. Conditions for the solvability of this system and estimates for the distance from an arbitrary given element $x_0 \in X$ to the set of solutions are obtained. Next, we obtain an assertion about the action of the Nemytskii operator in spaces of summable functions and establish the relationship between the covering properties of the Nemytskii operator and the covering of the function that generates it. The listed results are applied to the study of a system of implicit differential equations, for which a statement about the local solvability of the Cauchy problem with constraints on the derivative of a solution is proved. Such problems arise, in particular, in models of controlled systems. In the final part of the article, a differential equation of the $n$-th order not resolved with respect to the highest derivative is studied by similar methods. Conditions for the existence of a solution to the Cauchy problem are obtained.