  In this study we establish connections between asymptotic functions and
properties of solutions to important problems in wireless networks. We start by
introducing a class of self-mappings (called asymptotic mappings) constructed
with asymptotic functions, and we show that spectral properties of these
mappings explain the behavior of solutions to some maxmin utility optimization
problems. For example, in a common family of max-min utility power control
problems, we prove that the optimal utility as a function of the power
available to transmitters is approximately linear in the low power regime.
However, as we move away from this regime, there exists a transition point,
easily computed from the spectral radius of an asymptotic mapping, from which
gains in utility become increasingly marginal. From these results we derive
analogous properties of the transmit energy efficiency. In this study we also
generalize and unify existing approaches for feasibility analysis in wireless
networks. Feasibility problems often reduce to determining the existence of the
fixed point of a standard interference mapping, and we show that the spectral
radius of an asymptotic mapping provides a necessary and sufficient condition
for the existence of such a fixed point. We further present a result that
determines whether the fixed point satisfies a constraint given in terms of a
monotone norm.
