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In this paper, we investigate the Euler-Bernoulli fourth-order boundary valueproblem (BVP) $w^{(4)}=f(x,w)$, $x\in \intcc{a,b}$, with specified values of$w$ and $w''$ at the end points, where the behaviour of the right-hand side $f$is motivated by biomechanical, electromechanical, and structural applicationsincorporating contact forces. In particular, we consider the case when $f$ isbounded above and monotonically decreasing with respect to its second argument.First, we prove the existence and uniqueness of solutions to the BVP. We thenstudy numerical solutions to the BVP, where we resort to spatial discretizationby means of finite difference. Similar to the original continuous-spaceproblem, the discrete problem always possesses a unique solution. In the caseof a piecewise linear instance of $f$, the discrete problem is an example ofthe absolute value equation. We show that solutions to this absolute valueequation can be obtained by means of fixed-point iterations, and that solutionsto the absolute value equation converge to solutions of the continuous BVP. Wealso illustrate the performance of the fixed-point iterations through anumerical example.

Euler-Bernoulli beams with contact forces: existence, uniqueness, and numerical solutions