Some results on existence and uniqueness of solutions of nonlinear networks

This paper deals with nonlinear networks which can be characterized by the equation
f(x) = y
, where
f(\cdot)
maps the real Euclidean
n
-space
R^{n}
into itself and is assumed to be continuously differentiable
x
is a point in
R^{n}
and represents a set of chosen network variables, and
y
is an arbitrary point in
R^{n}
and represents the input to the network. The authors derive sufficient conditions for the existence of a unique solution of the equation for all
y \in R^{n}
in terms of the Jacobian matrix
\partial f/ \partial x
. It is shown that if a set of cofactors of the Jacobian matrix satisfies a "ratio condition," the network has a unique solution. The class of matrices under consideration is a generalization of the class
P
recently introduced by Fiedler and Pták, and it includes the familiar uniformly positive-definite matrix as a special case.