Investigation of a Class of Nonlinear Integral Equations and Applications to Nonlinear Network Theory

Abstract

Nonlinear oscillations in some networks can be described by the operator equation Au + Fu = J, where A is an unbounded linear operator on a Hilbert or Banach space, F is a nonlinear operator, and B = A + F is monotone. This is true for the general passive one-loop network, consisting of an e.m.f. E(t), and arbitrary linear passive stable one-port L, and a nonlinear one-port N with a monotone voltage-current characteristic i = Fu, where i is the current through N and u is the voltage on N. The theory presented below makes it possible to study nonlinear oscillations qualitatively, including questions of existence, uniqueness, stability in the large, convergence, and calculation of stationary regimes by means of an iterative process. Our assumptions concerning the network are more general than those usually adopted in the literature. No assumptions concerning the “smallness” of the nonlinearity or filter property of the linear one-port are made. Our results for nonlinear networks of the class defined above are final in the sense that if we omit the assumption concerning passivity of the network the results will not hold.