Singular Perturbations of a General Boundary Value Problem

This paper treats the boundary problem \[\begin{gathered} y' = A(t)y + B(t)z, \hfill \\ \varepsilon z' = C(t)y + D(t)z, \hfill \\ M(\varepsilon )\left( {\begin{array}{*{20}c} {y(0,\varepsilon )} \\ {z(0,\varepsilon )} \\ \end{array} } \right) + N(\varepsilon )\left( {\begin{array}{*{20}c} {y(1,\varepsilon )} \\ {z(1,\varepsilon )} \\ \end{array} } \right) = \left( {\begin{array}{*{20}c} {c_1 (\varepsilon )} \\ {c_2 (\varepsilon )} \\ \end{array} } \right). \hfill \\ \end{gathered} \] The main difference of our approach and that of earlier writers is that we are able to reduce the system to a purely diagonalized form under even less stringent assumptions.