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.

翻译：本文研究欧拉-伯努利四阶边值问题（BVP）$w^{(4)}=f(x,w)$，$x\in \intcc{a,b}$，其中端点处$w$和$w''$的值给定，且右端项$f$的行为受生物力学、机电学及考虑接触力的结构工程应用所驱动。特别地，我们考虑$f$关于第二个变量单调递减且有上界的情形。首先，我们证明该边值问题解的存在性与唯一性。随后研究其数值解，通过有限差分法进行空间离散。与原始连续空间问题类似，离散问题始终存在唯一解。当$f$为分段线性函数时，离散问题可转化为绝对值方程。我们证明该绝对值方程的解可通过不动点迭代获得，且绝对值方程的解收敛于连续边值问题的解。最后通过数值算例展示不动点迭代的性能。