The integral representation of solutions to a boundary value problem on the half-line for a system of linear ODEs with singularity of first kind

TL;DR

This paper develops an integral representation for solutions to a boundary value problem involving a linear ODE system with a first-order pole at zero, focusing on the resonant case and conditions for solution existence.

Contribution

It introduces a generalized Green function for the problem and establishes conditions for reducing the non-homogeneous problem to a Noetherian form with nonzero index.

Findings

Constructed a generalized Green function for the system.

Identified conditions for the reduction to a Noetherian problem.

Analyzed the resonant case with nontrivial homogeneous solutions.

Abstract

We consider a problem of finding vanishing at infinity $C^1([0,\oo))$-solutions to non-homogeneous system of linear ODEs which has the pole of first order at $x=0$. The resonant case where the corresponding homogeneous problem has nontrivial solutions is of main interest. Under the conditions that the homogeneous system is exponentially dichotomic on $[1,\oo)$ and the residue of system's operator at $x=0$ does not have eigenvalues with real part 1, we construct the so called generalized Green function. We also establish conditions under which the main non-homogeneous problem can be reduced to the Noetherian one with nonzero index.