Internal layers for a singularly perturbed second-order quasilinear differential equation with discontinuous right-hand side (Q1709667)

Consider the boundary value problem \[ \begin{gathered} \varepsilon y''= A(x,y)y'+ f(y,x),\quad 0<x<1,\\ y(0)= y^0,\quad y(1)= y^1,\end{gathered}\tag{\(*\)} \] where \(\varepsilon\) is a small positive parameter. It is assumed that \(A\) and \(f\) have at \(x_0\in (0,1)\) a discontinuity. The authors derive conditions under which the boundary value problem \((*)\) has a smooth solution with a boundary layer near \(x_0\). They present an asymptotic expansion of the solution in \(\varepsilon\).