Existence and Approximations for Order-Preserving Nonexpansive Semigroups over $$\mathrm{CAT}(\kappa )$$ Spaces

In this paper, we discuss the fixed point property for an infinite family of order-preserving mappings which satisfy the Lipschitz 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 \(\mathrm CAT(\kappa )\) space. In particular, we prove the existence of a fixed point for a nonexpansive semigroup on comparable pairs. Then, we propose and analyze two algorithms to approximate such a fixed point.