  In this paper, we investigate the Euler-Bernoulli fourth-order boundary value
problem (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 applications
incorporating contact forces. In particular, we consider the case when $f$ is
bounded above and monotonically decreasing with respect to its second argument.
First, we prove the existence and uniqueness of solutions to the BVP. We then
study numerical solutions to the BVP, where we resort to spatial discretization
by means of finite difference. Similar to the original continuous-space
problem, the discrete problem always possesses a unique solution. In the case
of a piecewise linear instance of $f$, the discrete problem is an example of
the absolute value equation. We show that solutions to this absolute value
equation can be obtained by means of fixed-point iterations, and that solutions
to the absolute value equation converge to solutions of the continuous BVP. We
also illustrate the performance of the fixed-point iterations through a
numerical example.
