Existence of an operating point for a nonlinear circuit using the degree of mapping

The existence of an operating point within the desired range of the element characteristics for nonlinear uncoupled electric circuits is considered. The case where the desired operating range is bounded is considered and theorems giving explicit bound on the magnitude of the input sources to guarantee the solution falls in the desired range are presented. The case where resistor characteristic is defined on the whole real line is then considered, but only its asymptotic behavior is characterized as monotone. Necessary and sufficient conditions for the existence of a solution for this class of circuits for arbitrary inputs connected at arbitrary points are presented. The principal tool that was used throughout this study is the concept of degree of a mapping.