Abstract
In this paper, we investigate the existence of common fixed points of monotone Lipschitzian semigroup in hyperbolic metric spaces under the natural condition that the images under the action of the semigroup at certain point are comparable to the point. In particular, we prove that if one map in the semigroup is a monotone contraction mapping, then such common fixed point exists. We introduce a notion of the uniform convexity in every direction (UUCED) of a hyperbolic space. In the case of monotone nonexpansive semigroup we prove the existence of common fixed points if the hyperbolic metric space is uniformly convex in every direction (UUCED).