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 is an element of C such that Tc is an element of [c,-+)G, then T has a fixed point provided for each a is an element of C, [a, a]G 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.