Under certain conditions on $g(x,u)$ we establish the existence and asymptotic behavior for small $\varepsilon > 0$ of multiple asymptotic solutions of the nonlinear boundary value problem \[ \begin{gathered} \varepsilon u'' + u' - g(x,u) = 0,\quad 0 0, \hfill \\ u'(1) + bu(1) = B > 0,\quad b > 0. \hfill \\ \end{gathered} \] Formal techniques of singular perturbation theory clearly reveal the mechanism which controls the appearance of multiple solutions. Their existence is then established rigorously by iteration schemes and the so-called “shooting method” for ordinary differential equations.