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

Abstract

For a large class of operators of the form[F(\cdot)+A],in whichAis a not necessarily nonsingular realn \times nmatrix, andF(\cdot)is a diagonal strictly monotone-increasing mapping of the set of all realnvectorsE^{n}onto an open subset ofE^{n}, we give necessary and sufficient conditions under whichF(\cdot)+A]possesses a global inverse onE^{n}. Operators of the type[F(\cdot)+A]frequently arise in the analysis of nonlinear networks and are encountered in other areas as well. In particular, forAthe short-circuit conductance matrix of a resistance network, andF(x)the transpose of(f_{1}(x_{1}), f_{2}(x_{2}), \cdots , f_{n}(x_{n}))for allx \in E^{n}in which thef_{j}(\cdot)are the usual exponential diode functions, we give a complete solution to the problem of determining whether or not[F(\cdot)+A]possesses an inverse onE^{n}.