Discrete nonlinear equations and the Fučı´k Spectrum

We consider matrix-vector equations of the form Ax=f(x) that are motivated by nonlinear oscillating systems such as the Tacoma Narrows Bridge. We identify a particular set, called the Fučı´k Spectrum, which is relevant to questions of solvability, and we develop theorems to describe the spectrum and show how it relates to the solvability of the matrix equation.