An inequality in the theory of networks with monotone elements

We consider a certain inequality that arises in the study of iterative methods for solving equations in a Hilbert space, and give equivalent characterizations of the inequality. We then show that the inequality is satisfied by the members of a large class of networks of monotone (possibly dynamic) two-terminal elements. This establishes the applicability of a simple algorithm that, for a large class of monotone resistive networks, will converge to a solution of the network equations whenever a solution exists, and that will generate an unbounded sequence of iterates if no solution exists