A new method for the study of boundary value problems for linear and quasi-linear systems of ODE

TL;DR

This paper introduces a novel integral equation-based method for analyzing boundary value problems in nonlinear ODE systems, simplifying the process by avoiding Green's functions and facilitating solvability conditions.

Contribution

The paper presents a new approach that reduces complex boundary value problems to integral equations, making analysis more straightforward and applicable to many-point boundary conditions.

Findings

Simplifies boundary value problem analysis without Green's functions.

Provides an algorithm for establishing unique solvability conditions.

Applicable to nonlinear and quasi-linear ODE systems.

Abstract

The method is proposed for the study of many-point boundary value problems for systems of nonlinear ODE, by reducing them to special equivalent integral equations, and allows us [in contrast with the known method [1]] to consider boundary and initial value problems. Here we avoid the mechanism of Green's function whose construction is quite nontrivial, especially in case of many-point boundary value problems. The proposed algorithm [2, 3] makes it easier to write out the condition of unique solvability of such problems. Key words: Boundary-value problems, Singular perturbation theory, Spectrum, Asymptotic behavior.