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}tin(0,pi) end{equation*} subject to non-local boundary conditions begin{equation*} x(0)=h_1+varepsiloneta_1(x)text{ and } x(pi)=h_2+varepsiloneta_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.
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