We study generalized classes of positive and monotone dynamic
systems in a partially ordered Banach space. Using results from the nonlinear
operators theory, we establish new algebraic conditions for stability of equilibrium states of a class of monotone-type differential and difference systems.
Conditions for the positivity and absolute stability of differential systems with
delay are proposed. Using new technique for constructing the invariant sets of
differential systems, we generalize known positivity conditions for linear and
nonlinear differential systems with respect to typical classes of cones. In addition, we generalize the comparison principle for a finite set of differential
systems and formulate robust stability conditions for some families of differential systems in terms of cone inequalities.