Uniqueness of solution for nonlinear resistive circuits containing CCCS's or VCVS's whose controlling coefficients are finite

Necessary and sufficient conditions are given for the uniqueness of solution of nonlinear resistive circuits made of strictly monotoneincreasing nonlinear resistors, dc sources, and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</tex> linear current-controlled current sources (CCCS's) or linear voltage-controlled voltage sources (VCVS's) whose controlling coefficients <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">{\alpha}_{\mu}</tex> are bounded by <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0&lt;{\alpha}_{\mu}&lt;{\alpha}_{\mu}max'{\mu} = 1,2,{\cdots}, k</tex> . These conditions are cast in explicit topological terms and are therefore easy to check.