Uniqueness of solution for linear transmission lines with nonlinear terminal resistors

Linear transmission lines with nonlinear terminal resistors are considered. Each resistor is assumed to be voltage controlled or current controlled and weakly active. These distributed circuits are shown to have a unique solution, which passes at a given instant through a given value, if the characteristics of the terminal resistors satisfy certain inequalities. In particular, if the resistors are locally passive, then the solution is unique. Transmission lines with terminal resistors that are locally active may admit several solutions that have the same initial value. In these cases, the uniqueness of solution is obtained adding "small" capacitances in parallel to the voltage controlled resistors and "small" inductances in series to the current controlled resistors.