Asymptotic expansion of the solution of a singularly perturbed boundary value problem with interior layer (Q2640027)

The existence of a singular solution \(y=y(t,\epsilon)\) with the property \[ \lim_{\epsilon \to 0}y(t,\epsilon)=u_{1,0}(t)\quad if\quad 0<t<t^*,\quad =u_{2,0}(t)\quad if\quad t^*<t<1, \] \(t^*\in (0,1)\), is proved for a singularly perturbed system of the form \(\epsilon^ 2y''=h(t,y),\) \(0<t<1\), \(0<\epsilon \ll 1\); \(y(0)=A\), \(y(1)=B\). Here \(u_{1,0}(t)\), \(u_{2,0}(t)\) denote sufficiently smooth solutions of the equation \(h(t,y)=0\).