Existence Theorems and a Solution Algorithm for Piecewise-Linear Resistor Networks

This paper deals with nonlinear networks which can be characterized by the equation ${\bf f}({\bf x}) = {\bf y}$, where ${\bf f}$ is a continuous piecewise-linear mapping from $R^n $ into itself. The main theorem asserts that the existence of solutions ${\bf x} \in R^n $ of ${\bf f}({\bf x}) = {\bf y}$ for an arbitrary given ${\bf y} \in R^n $ is guaranteed by fairly general conditions based on the theory of the degree of mapping. Then it is shown that an iterative algorithm (generalized Katzenelson algorithm) leads to a solution in a finite number of iteration steps. Finally, a comprehensive study of physical nonlinear elements demonstrates that the theory can be applied to most of the currently used nonlinear networks.