The fixed point iteration of positive concave mappings converges geometrically if a fixed point exists

We prove that the fixed point iteration of arbitrary positive concave mappings with nonempty fixed point set converges geometrically for any starting point. We also show that positivity is crucial for this result to hold, and the concept of (nonlinear) spectral radius of asymptotic mappings provides us with information about the convergence factor. As a practical implication of the results shown here, we rigorously explain why some power control and load estimation algorithms in wireless networks, which are particular instances of the fixed point iteration, have empirically shown good convergence speed, even though the algorithms are derived by considering a more general class of mappings (namely, standard interference mappings) for which its usage with the fixed point iteration can result in sublinear convergence.