Non-Resonant Non-Hyperbolic Singularly Perturbed Neumann Problem

In this brief note, we study the problem of asymptotic behavior of the solutions for non-resonant, singularly perturbed linear Neumann boundary value problems $\epsilon y^{″}+ky=f\left(t\right)$, $y^{\prime }\left(a\right)=0$, $y^{\prime }\left(b\right)=0$, $k>0,$ with an indication of possible extension to more complex cases. Our approach is based on the analysis of an integral equation associated with this problem.