Some results on existence and uniqueness of solutions of nonlinear networks

This paper deals with nonlinear networks which can be characterized by the equation , where maps the real Euclidean -space into itself and is assumed to be continuously differentiable is a point in and represents a set of chosen network variables, and is an arbitrary point in and represents the input to the network. The authors derive sufficient conditions for the existence of a unique solution of the equation for all in terms of the Jacobian matrix . 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 recently introduced by Fiedler and Pták, and it includes the familiar uniformly positive-definite matrix as a special case.