We continue to study the properties of covering mappings of metric spaces and present their applications to differential equations. To extend the applications of covering mappings, we introduce the notion of conditionally covering mapping. We prove that the solvability and the estimates for solutions of equations with conditionally covering mappings are preserved under small Lipschitz perturbations. These assertions are used in the solvability analysis of differential equations unsolved for the derivative.

We prove some results concerning solvability, estimates for solutions, and well-posed solvability of equations with conditionally covering mappings. These results are applied to the analysis of differential equations unsolved for the derivative.