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

This paper deals with nonlinear networks which can be characterized by the equation f( x) = y, where f is a continuous piecewise-linear mapping from Rⁿ into itself. The main theorem asserts that the existence of solutions x ∈ Rⁿ of f( x) = y for an arbitrary given y ∈ Rⁿ 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.