Nonlinear resonance set for nonlinear matrix equations

We study nonlinear matrix equations of the form Ax = f(x), where f is mildl y nonlinear. We identify a particular set, called the Fucik spectrum or res onance set, which is relevant to questions of solvability of the equation. We develop theorems to describe the spectrum and show how it relates to the solvability of the matrix equation. We see that a small perturbation in th e constant coefficients in f may cause a radical change in the character of the solutions of Ax = f(x). (C) 1999 Elsevier Science Inc. All rights rese rved.