Boundary value problem for a singularly perturbed nonlinear system (Q1369500)

This paper considers the problem \[ \varepsilon y''=f(t,y,y',\varepsilon), \;y(0)=a(\varepsilon),\quad y(1)=b(\varepsilon), \tag{1} \] where \(y\in\mathbb{R}^n\), together with the reduced problem \[ f(t,y,y',0)=0, \;y_i(1)=b_i(0), i=1,\ldots,k, \;y_i(0)=a_i(0), i=k+1,\ldots,n. \] Under usual assumptions and provided \(\bar y_i(0)-a_i(0)\), \(i=1,\ldots,k\) and \(\bar y_i(1)-b_i(0)\), \(i=k+1,\ldots,n\) are small enough, the author proves that given a solution \(\bar y(t)\) of the reduced problem, problem (1) has a solution \(y(t,\varepsilon)\) such that \[ \begin{aligned} y(t,\varepsilon)&=\bar y(t)+O(\varepsilon+\exp(-\mu t/\varepsilon) +\exp(\mu (t-1)/\varepsilon)),\\ y'(t,\varepsilon)&= \bar y'(t)+O(\varepsilon+ \varepsilon^{-1}\exp(-\mu t/\varepsilon)+ \varepsilon^{-1}\exp(\mu (t-1)/\varepsilon)),\end{aligned} \] The method of proof uses essentially a diagonalization of the linear part of the equations.