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(\cdot)+A],
in which
A
is a not necessarily nonsingular real
n \times n
matrix, and
F(\cdot)
is a diagonal strictly monotone-increasing mapping of the set of all real
n
vectors
E^{n}
onto an open subset of
E^{n}
, we give necessary and sufficient conditions under which
F(\cdot)+A]
possesses a global inverse on
E^{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, for
A
the short-circuit conductance matrix of a resistance network, and
F(x)
the transpose of
(f_{1}(x_{1}), f_{2}(x_{2}), \cdots , f_{n}(x_{n}))
for all
x \in E^{n}
in which the
f_{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 on
E^{n}
.