Some results on existence and uniqueness of solutions of nonlinear networks

Abstract

This paper deals with nonlinear networks which can be characterized by the equation <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">f(x) = y</tex> , where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">f(\cdot)</tex> maps the real Euclidean <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> -space <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R^{n}</tex> into itself and is assumed to be continuously differentiable <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x</tex> is a point in <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R^{n}</tex> and represents a set of chosen network variables, and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">y</tex> is an arbitrary point in <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R^{n}</tex> and represents the input to the network. The authors derive sufficient conditions for the existence of a unique solution of the equation for all <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">y \in R^{n}</tex> in terms of the Jacobian matrix <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\partial f/ \partial x</tex> . 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 <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">P</tex> recently introduced by Fiedler and Pták, and it includes the familiar uniformly positive-definite matrix as a special case.