  In this work, we characterize a class of nonlinear monotone dynamical systems
that have a certain translation invariance property which goes by the name of
plus-homogeneity; usually called "topical" systems. Such systems need not be
asymptotically stable, since they are merely nonexpansive but not contractive.
Thus, we introduce a stricter version of monotonicity, termed "type-K" in honor
of Kamke, and we prove the asymptotic stability of the equilibrium points, as
well as the convergence of all trajectories to such equilibria for type-K
monotone and plus-homogeneous systems: we call them "K-topical". Since topical
maps are the natural nonlinear counterpart of linear maps defined by
row-stochastic matrices, which are a cornerstone in the convergence analysis of
linear multi-agent systems (MASs), we exploit our results for solving the
consensus problem over nonlinear K-topical MASs. We first provide necessary and
sufficient conditions on the local interaction rules of the agents ensuring the
K-topicality of a MAS. Then, we prove that the agents achieve consensus
asymptotically if the graph describing their interactions contains a globally
reachable node. Finally, several examples for continuous-time and discrete-time
systems are discussed to corroborate the enforceability of our results in
different applications.
