On the Continuous Dependence of the Solution of a Boundary Value Problem on Boundary Conditions: Elements of P-Regularity Theory

Abstract

The existence of a continuous dependence of the solution to a boundary value problem on a parameter is studied. In the presence of the p-regularity property, it is proved that there exists a solution depending continuously on a small parameter. The main result of this 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 an operator and left and right inverse operators are introduced. 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.