Positivity, robust stability and comparison of dynamic systems

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.