Existence of solutions with turning points for nonlinear singularly perturbed boundary-value problems

We consider the following singularly perturbed boundary-value problem:\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\varepsilon u'' = f\left( {x,u,u'} \right), 0< \varepsilon \ll 1, g_j \left( {u\left| {_{x = 0} ,} \right.u\left| {_{x = 1} ,u'\left| {_{x = 0} ,} \right.} \right.u'\left| {_{x = 1} } \right.} \right) = 0, j = 1,2,$$ \end{document} on the interval 0 ≤x ≤ 1. We study the existence and uniqueness of its solutionu(x, ε) having the following properties:u(x, ε) →u0(x) asε → 0 uniformly inx ε [0, 1], whereu0(x) εC∞ [0, 1] is a solution of the degenerate equationf(x, u, u′)=0; there exists a pointx0 ε (0, 1) such thata(x0)=0,a′(x0) > 0,a(x) < 0 for 0 ≤x <x0, anda(x) > 0 forx0 <x ≤ 1, wherea(x)=f′v(x,u0(x),u′0(x)).