Given monotone operators on the positive orthant in n-dimensional
Euclidean space, we explore the relation between inequalities
involving those operators, and induced monotone dynamical
systems. Attractivity of the origin implies stability for these
systems, as well as a certain inequality. Under the right
perspective the converse is also true. In addition we construct an
unbounded path in the set where trajectories of the dynamical system
decay monotonically, i.e., we solve a positive continuous selection
problem.