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 () 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 R^n in terms of the Jacobian matrix f/ 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.