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 Ptak, and it includes the familiar uniformly positive-definite matrix as a special case.