For a class of monotone operators T on the positive orthant of n-dimensional Euclidean
space we introduce the concept of decay sets. These consist of points x satisfying
T(x)<x. Considering the induced dynamical system x(k 1)=T(x(k)), we establish results
relating stability properties of the origin, order conditions on T, and topological
properties of decay sets. In particular, we construct paths in the decay sets and derive a
quasi-invertibility result of the operator (Id-T).
These results are applied to derive generalized small-gain type conditions for the input-
to-state stability(ISS) of large-scale interconnections of (individually input-to-state
stable) control systems: The interconnection topology together with the ISS gains yields a
monotone operator with an inherent graph structure. We provide trajectory estimate
based small-gain theorems and also construct ISS Lyapunov functions for the composite
system. It is also shown how an algorithm due to Eaves can be used to numerically verify
the small-gain condition.