Solutions φ(x) of the functional equation φ(φ(x)) = f(x) are called iterative roots of the given function f(x). They are of interest in dynamical systems, chaos and complexity theory and also in the modeling of certain industrial and financial processes. The problem of computing this “square root” of a function or operator remains a hard task. While the theory of functional equations provides some insight for real and complex valued functions, iterative roots of nonlinear mappings from Rn to Rn are less studied from a theoretical and computational point of view. Here we prove existence of iterative roots of a certain class of monotone mappings in Rn spaces and demonstrate how a method based on neural networks can find solutions to some examples that arise from simple physical dynamical systems.