On preimages of a class of generalized monotone operators

In this paper we first provide a geometric interpretation of the Minty-Browder monotonicity which allows us to extend this concept to the so called h-monotonicity, still formulated in an analytic way. A topological concept of monotonicity is also known in the literature: it requires the connectedness of all preimages of the operator involved. This fact is important since combined with the local injectivity, it ensures global injectivity. When a linear structure is present on the source space, one can ask for the preimages to even be convex. In an earlier paper, the authors have shown that Minty-Browder monotone operators defined on convex open sets do have convex preimages, obtaining as a by-product global injectivity theorems. In this paper we study the preimages of h-monotone operators, by showing that they are not divisible by closed connected hypersurfaces, and investigate them from the dimensional point of view. As a consequence we deduce that h-monotone local homeomorphisms are actually global homeomorphisms, as the proved properties of their preimages combined with local injectivity still produce global injectivity.