A one-dimensional prescribed curvature equation modeling the corneal shape

We prove existence, uniqueness and stability of solutions of the prescribed curvature problem \begin{equation*} \begin{cases} \bigl({u'}/{\sqrt{1 + u'^2}}\bigr)' = au -{b}/{\sqrt{1 + u'^2}} \quad \text{in }[0,1]\\ u'(0)=u(1)=0, \end{cases} \end{equation*} for any given $a>0$ and $b>0$. We also develop a linear monotone iterative scheme for approximating the solution. This equation has been proposed as a model of the corneal shape in the recent paper \cite{OkPl}, where a simplified version obtained by partial linearization has been investigated.