Stability of generalized monotone maps with respect to their characterizations

We show that known types of generalized monotone maps are not stable with respect to their characterizations (i.e., the characterizations are not maintained during an arbitrary map of this type is disturbed by an element with sufficiently small norm) then introduce $s$-quasimonotone maps, which are stable with respect to their characterization. For gradient maps, $s$-quasimonotonicity is related to $s$-quasiconvexity of the underlying function. A necessary and sufficient condition for a univariate polynomial to be $s$-quasimonotone is given. Furthermore, some stability properties of $s$-quasiconvex functions are presented.