Weakly nonlocal boundary value problems with application to geology

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.