On the convergence of the Krasnoselskij iteration for strictly pseudocontractive operators

We study the convergence of the nonlinear Krasnoselskij iteration x(k + 1) = (1 − θ)x(k) + θT(x(k)) in real vector spaces of finite dimension equipped with a p-norm, which is relevant for stability analysis and distributed computation in several discrete-time dynamical systems. Specifically, we provide sufficient conditions for the convergence of the Krasnoselskij iteration, derived via implications between the strict pseudocontractivity of the operator T and the nonexpansiveness of (1 − θ)Id + θT. Interestingly, it turns out that strict pseudocontractivity of T is necessary for the Euclidean norm (p = 2) only; not necessary for non-Euclidean norms (p ≠ 2); sufficient for any finite norm p ∈ (1, ∞); not sufficient for the taxi-cab norm (p = 1) and the supremum norm (p = ∞). We numerically verify the above results in the context of recurrent neural networks and multi-agent systems with nonlinear Laplacian dynamics.