Some results on existence and uniqueness of solutions of nonlinear networks
- 1 September 1971
- journal article
- Published by Institute of Electrical and Electronics Engineers (IEEE) in IEEE Transactions on Circuit Theory
- Vol. 18 (5) , 501-506
- https://doi.org/10.1109/TCT.1971.1083336
Abstract
This paper deals with nonlinear networks which can be characterized by the equationf(x) = y, wheref(\cdot)maps the real Euclideann-spaceR^{n}into itself and is assumed to be continuously differentiablexis a point inR^{n}and represents a set of chosen network variables, andyis an arbitrary point inR^{n}and represents the input to the network. The authors derive sufficient conditions for the existence of a unique solution of the equation for ally \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 classPrecently introduced by Fiedler and Pták, and it includes the familiar uniformly positive-definite matrix as a special case.Keywords
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