ON THE THEORY OF AN INTEGRAL-EQUATION

A strongly nonlinear integral equation of the Hammerstein-type. which arises from an important engineering application, is considered. Becau se of the presence of the function sign x(t) in the integrand, the sol ution of the equation is reduced to finding the zerocrossings of the u nknown function x(t) and to analysis of the ''structural stability'' o f its waveform which is required to be of alternative polarity and to have only isolated zeroS. A realistic condition for the linear operato r included in the equation is found which gives a precise, constructiv ely built, solution. The ''norm'' for x(t) is introduced according to an energetic meaning which is associated with the application, and the case of the precise solution is associated with some extreme features of the power consumption of the relevant system. Logarithmic sensitiv ity of the norm to the changes in the parameter included in the equati on is explained to be an important characteristic of the system- The p hysical discussion is concentrated in the appendixes, and a generaliza tion of the equation, concerned with the applications, is given there. The work is specifically intended to draw the attention of mathematic ians and applied scientists to the equation and a method for solving i t, which do not appear in the classical theory. As a collateral but im portant result, a new criterion for the analysis of oscillatory system s is found from the theory of the equation. (C) 1994 Academic Press, I nc.