``Chase on ducks'' in the investigation of singularity perturbed boundary value problems (Q2703893)

Consider the boundary value problem NEWLINE\[NEWLINE\begin{gathered} \varepsilon u''+ a(x,u)u'+ b(x,u)= 0,\quad 0<x< 1,\\ u(0)= u(1)= 0,\quad 0<\varepsilon\ll 1,\end{gathered}\tag{\(*\)}NEWLINE\]NEWLINE with \(a,b\in C^\infty([0, 1],\mathbb{R})\). The authors prove the existence of a solution \(u(x,\varepsilon)\) of \((*)\) satisfying NEWLINE\[NEWLINE\lim_{\varepsilon\to 0} u(x,\varepsilon)= u_0(x)NEWLINE\]NEWLINE uniformly in any interval \([x_1,x_2]\subset (0,1)\), where \(u_0\in C^\infty([0, 1],\mathbb{R})\) is a special solution of the degenerate equation NEWLINE\[NEWLINEa(x,u) u'+ b(x,u)= 0NEWLINE\]NEWLINE having a turning point (and not a discontinuity as usual). For the proof, the authors reduce the problem under consideration to a French duck problem.