Invertibility of Lipschitz Continuous Mappings and Its Application to Electrical Network Equations

Conditions for unique solvability of nonlinear simultaneous equations satisfying Lipschitz conditions and an application to nonlinear network equations are proposed. It is shown that the global invertibility of a Lipschitz continuous mapping f of $\mathbb{R}^n $ into itself and the Lipschitz continuity of $f^{ - 1} $ are verified by investigating the positivity of some principal minors of the Jacobian matrix of f in spite of the existence of nondifferentiable points of f. This is a generalization of previous works, especially Fujisawa and Kuh’s theorem, obtained for continuously differentiable or piecewise-linear mappings. This generalization is given by employing the Lebesgue integration of the Jacobian matrix over an open interval of $\mathbb{R}^n $. This result is applied to network equations, both resistive and dynamical; especially to the latter, the Lipschitz continuity of such inverse mappings is of great importance to guarantee the uniqueness of the solutions. In addition, network examples for which the above result is useful are given, and it is demonstrated that simple conditions for the unique solvability are obtained in terms of differential coefficients of network element characteristics.