Necessary and sufficient conditions for the global invertibility of certain nonlinear operators that arise in the analysis of networks

For a large class of operators of the form in which is a not necessarily nonsingular real matrix, and is a diagonal strictly monotone-increasing mapping of the set of all real vectors onto an open subset of , we give necessary and sufficient conditions under which possesses a global inverse on . Operators of the type frequently arise in the analysis of nonlinear networks and are encountered in other areas as well. In particular, for the short-circuit conductance matrix of a resistance network, and the transpose of for all in which the are the usual exponential diode functions, we give a complete solution to the problem of determining whether or not possesses an inverse on .