  We provide a novel transcription of monotone operator theory to the
non-Euclidean finite-dimensional spaces $\ell_1$ and $\ell_{\infty}$. We first
establish properties of mappings which are monotone with respect to the
non-Euclidean norms $\ell_1$ or $\ell_{\infty}$. In analogy with their
Euclidean counterparts, mappings which are monotone with respect to a
non-Euclidean norm are amenable to numerous algorithms for computing their
zeros. We demonstrate that several classic iterative methods for computing
zeros of monotone operators are directly applicable in the non-Euclidean
framework. We present a case-study in the equilibrium computation of recurrent
neural networks and demonstrate that casting the computation as a suitable
operator splitting problem improves convergence rates.
