  In many cases, groundwater flow in an unconfined aquifer can be simplified to
a one-dimensional Sturm-Liouville model of the form: \begin{equation*}
x''(t)+\lambda x(t)=h(t)+\varepsilon f(x(t)),\hspace{.1in}t\in(0,\pi)
\end{equation*} subject to non-local boundary conditions \begin{equation*}
x(0)=h_1+\varepsilon\eta_1(x)\text{ and } x(\pi)=h_2+\varepsilon\eta_2(x).
\end{equation*} In this paper, we study the existence of solutions to the above
Sturm-Liouville problem under the assumption that $\varepsilon$ is a small
parameter. Our method will be analytical, utilizing the implicit function
theorem and its generalizations.
