Existence of solutions with turning points for nonlinear singularly perturbed boundary value problems.
Consider the boundary value problem \[ \varepsilon u''=f(x,u,u'), \;0 < \varepsilon \ll 1, \quad g_j (u_{| x=0}, u_{| x=1}, u_{| x=0}', u_{| x=1}' ) =0,\;j=1,2, \tag \(*\) \] on the interval \(0 \leq x \leq 1\). Let \(u_0 \in C^\infty [0,1]\) be a solution to the degenerate equation \(f(x,u,u')=0\) such that there is a point \(x_0 \in (0,1)\) with the property that the function \(a (x):= f_v' (x,u_0 (x), u_0' (x))\) has a simple zero at \(x_0\). The authors derive six conditions guaranteeing that \((*)\) has a unique solution. The proof is based on the existence of a duck trajectory for some auxiliary system of differential equations.
scientific article; zbMATH DE number 1531347