  In this paper, we develop a new approach to study gain operators built from
the interconnection gains of infinite networks of dynamical systems. Our focus
is on the construction of paths of strict decay which are used for building
Lyapunov functions for the network and thus proving various stability
properties, including input-to-state stability. Our approach is based on the
study of an augmented gain operator whose fixed points are precisely the points
of decay for the original gain operator. We show that plenty of such fixed
points exist under a uniform version of the no-joint-increase condition. Using
these fixed points to construct a path of strict decay, in general, requires
specific dynamical properties of associated monotone operators. For particular
types of gain operators such as max-type operators and subadditive operators,
these properties follow from uniform global asymptotic stability of the system
induced by the original gain operator. This is consistent with former results
in the literature which can readily be recovered from our theory.

Optimization and Control;Dynamical Systems