On singular singularly-perturbed vector boundary value problems (Q1366914)

The author considers the differential equations \[ x'=U(t,x,y,\varepsilon z,\varepsilon),\quad y'=V(t,x,y,z,\varepsilon),\quad \varepsilon^2z'=W(t,x,y,\varepsilon^2z,\varepsilon), \] together with nonlinear boundary conditions \[ H_1(x(0),y(0),\varepsilon z(0),\varepsilon)=0,\quad H_2(x(0),y(0),z(0),x(1),y(1),z(1),\varepsilon)=0, \] where \(\varepsilon >0\), \(x\in\mathbb{R}^m\), \(y\) and \(z\in\mathbb{R}^n\). A first problem concerns the construction of a formal asymptotic expansion of the form \[ \begin{pmatrix} x_N\\ y_N\\ z_N\end{pmatrix} (t,\varepsilon)=\begin{pmatrix} X_N(t,\varepsilon)+\varepsilon^2\overline X_N(\tau,\varepsilon)+\varepsilon^2\widehat X_N(\sigma,\varepsilon) \\ Y_N(t,\varepsilon)+\varepsilon\overline Y_N(\tau,\varepsilon)+\varepsilon\widehat Y_N(\sigma,\varepsilon) \\ Z_N(t,\varepsilon)+\overline Z_N(\tau,\varepsilon)+\widehat Z_N(\sigma,\varepsilon)\end{pmatrix}, \] where \(\tau=t/\varepsilon\) and \(\sigma=(1-t)/\varepsilon\) are boundary layer variables and the functions \(X_N\), \(Y_N\), \(Z_N\), \(\overline X_N\), \(\overline Y_N\), \(\overline Z_N\), \(\widehat X_N\), \(\widehat Y_N\) and \(\widehat Z_N\) are polynomials of degree \(N\) in \(\varepsilon\). The second part of the paper concerns the validity of such an expansion. It is proved that, under appropriate assumptions, there exists a unique solution \((x,y,z)(t,\varepsilon)\) of the boundary value problem such that \[ (x,y,z)(t,\varepsilon)=(x_N,y_N,z_N)(t,\varepsilon)+O(\varepsilon^{N+1}). \] The proof is based on a study of the corresponding linear problem and Banach's fixed point theorem. An example is given.