On the continuous dependence of the solution of a boundary value problem on boundary conditions. Elements of p-regularity theory

The problem of the existence of a continuous dependence of the solution of a boundary value problem on a parameter is considered. In this paper, it is proved that, in the presence of the p-regularity property, there exists a solution that continuously depends on a small parameter. The main result of the paper is based on theorems representing different versions of the implicit function theorem. In the case of degenerate mappings, the theorems are used to analyze a boundary value problem with a small parameter. In the case of absolute degeneration, a -p-factor operator is found. The concept of the p-kernel of the operator is introduced, as well as the left and right inverse operators. Theorems are formulated that are special versions of the generalized Lyusternik theorem and the implicit function theorem in the degenerate case. An implicit function theorem is formulated and proved in the case of a nontrivial kernel.