Nonlinear resonance set for nonlinear matrix equations Original Research Article

We study nonlinear matrix equations of the form Ax=f(x), where f is mildly nonlinear. We identify a particular set, called the Fuimageik spectrum or resonance 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 the constant coefficients in f may cause a radical change in the character of the solutions of Ax=f(x).