$p$-Regularity theory and the existence of a solution to a boundary value problem continuously dependent on boundary conditions

For a given boundary value problem, the existence of a solution depending continuously on the boundary conditions is analyzed. Previously, such a fact has been known only for the Cauchy problem, which is a classical result in the theory of differential equations. We prove a similar result for boundary value problems in the case when they are $p$-regular. In the general case, this result does not hold. Several implicit function theorems are proved in the degenerate case, which is a development of $p$-regularity theory concerning the existence of a solution to nonlinear differential equations. The results are illustrated by an example of a classical boundary value problem, namely, a degenerate Van der Pol equation is considered, for which the existence of a solution depending continuously on the boundary conditions of the perturbed problem is proved.