Invariance and Monotonicity of Nonlinear Iterated Systems.

In this paper, the positive invariance of sets defined by nonlinear inequalities with respect to nonlinear maps from a partially ordered set into itself is investigated. The results of this paper extend former results relative to the positive invariance of polyhedral sets with respect to linear systems defined in ordered semi-rings as well as former results relative to nonlinear dynamical systems defined in the real n-space. The v-monotonicity of nonlinear maps is also been investigated. It is shown that, opposite to the case of linear maps, the v-monotonicity of a nonlinear map is not a necessary condition for a set defined by nonlinear inequalities to be positively invariant. [ABSTRACT FROM AUTHOR]