Singularly perturbed linear Neumann problem with the characteristic roots on the imaginary axis. A non-resonant case

TL;DR

This paper investigates the existence and asymptotic behavior of solutions to a non-resonant singularly perturbed linear Neumann problem with characteristic roots on the imaginary axis, using integral equation methods.

Contribution

It introduces an analysis of an integral equation approach to study the problem's solutions in the non-resonant case with characteristic roots on the imaginary axis.

Findings

Established existence of solutions for the boundary value problem.

Derived asymptotic behavior of solutions as perturbation parameter approaches zero.

Provided conditions under which solutions are well-behaved.

Abstract

In this note we are dealing with the problem of existence and asymptotic behavior of solutions for the non-resonant singularly perturbed linear Neumann boundary value problem \begin{eqnarray*} \epsilon y"+ky=f(t),\quad k>0,\quad 0<\epsilon<<1,\quad t\in\langle a,b\rangle \end{eqnarray*} \begin{equation*} y'(a)=0,\quad y'(b)=0. \end{equation*} Our approach is based on the analysis of an integral equation equivalent to this problem.