On Nonseparated Three‐Point Boundary Value Problems for Linear Functional Differential Equations

Abstract

For a system of linear functional differential equations, we consider a three-point problem with nonseparated boundary conditions determined by singular matrices. We show that, to investigate such a problem, it is often useful to reduce it to a parametric family of two-point boundary value problems for a suitably perturbed differential system. The auxiliary parametrised two-point problems are then studied by a method based upon a special kind of successive approximations constructed explicitly, whereas the values of the parameters that correspond to solutions of the original problem are found from certain numerical determining equations. We prove the uniform convergence of the approximations and establish some properties of the limit and determining functions.

Periodic successive approximations and interval halving

“…This complication can be overcome by using the functions for a fixed m, which will lead one to the so-called approximate determining equations. More precisely, similarly to [15], one can show that, under certain natural assumptions, the exact determining system (9.3) can be replaced by its approximate analogue m . ; / D 0; H m .Á; / D 0;…”

A new approach to non-local boundary value problems for ordinary differential systems

“…The last theorem suggests an approach to the study of the non-local problem (1) and (2) by looking for its solution among those of the family of equations (13), which are, as Theorem 7 shows, motivated by auxiliary problems with separated twopoint conditions (7). The study of the problem then consists of two parts, namely, the analytic part, when the integral Eq.…”

“…This closely correlates with the idea of the LyapunovSchmidt reduction (see, e.g., [12]). The solvability of the determining system (41) (47), which can be carried out by analogy to [3,4,13], is not treated here.…”

“…A rigorous proof of the existence of the solution in question would involve an analysis of the approximate determining Eqs. (46) and (47) in the spirit of [4,13].…”

Further results on the investigation of solutions of integral boundary value problems

“…The solvability analysis based on properties of the equations (5.30) can be carried out by analogy to [6] on the base of topological degree methods [2], but it is not treated here.…”