  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.
