  Incremental stability properties are considered for certain systems of
forced, nonlinear differential equations with a particular positivity
structure. An incremental stability estimate is derived for pairs of
input/state/output trajectories of the Lur'e systems under consideration, from
which a number of consequences are obtained, including the incremental
exponential input-to-state stability property and certain input-output
stability concepts with linear gain. Incremental stability estimates provide a
basis for an investigation into the response to convergent and (almost)
periodic forcing terms, and is treated presently. Our results show that an
incremental version of the real Aizerman conjecture is true for positive Lur'e
systems when an incremental gain condition is imposed on the nonlinear term, as
we describe. Our argumentation is underpinned by linear dissipativity theory --
a property of positive linear control systems.
