Fixed points of G-monotone mappings in metric and modular spaces

Let C be a bounded, closed and convex subset of a reflexive metric space with a digraph G such that G-intervals along walks are closed and convex. In the main theorem we show that if T C C is a monotone G-nonexpansive mapping and there exists c C such that Tc c, ) ₆, then T has a fixed point provided for each a C, [a, a₆ has the fixed point property for nonexpansive mappings. In particular, it gives an essential generalization of the Dehaish-Khamsi theorem concerning partial orders in complete uniformly convex hyperbolic metric spaces. Some counterparts of this result for modular spaces, and for commutative families of mappings are given too.