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[F(·)+A],in whichAis a not necessarily nonsingular realn × nmatrix, andF(·)is a diagonal strictly monotone-increasing mapping of the set of all realnvectorsEⁿonto an open subset ofEⁿ, we give necessary and sufficient conditions under whichF(·)+A]possesses a global inverse onEⁿ. Operators of the type[F(·)+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(f₁(x₁), f₂(x₂), ⋯ , fₙ(xₙ))for allx ∈ Eⁿin which thefⱼ(·)are the usual exponential diode functions, we give a complete solution to the problem of determining whether or not[F(·)+A]possesses an inverse onEⁿ.