  In this paper, we discuss the fixed point property for an infinite family of
order-preserving mappings which satisfy the Lipschitzian condition on
comparable pairs. The underlying framework of our main results is a metric
space of any global upper curvature bound $\kappa \in \mathbb{R}$, i.e., a
$\rm{CAT}(\kappa)$ space. In particular, we prove the existence of a fixed
point for a nonexpasive semigroup on comparable pairs. Then, we propose and
analyze two algorithms to approximate such a fixed point.
