Nonlinear resonance set for nonlinear matrix equations (Q1963938)

Given an \(n\times n\) real matrix \(A\), its Fučik spectrum \(A_{-1}\subset\mathbb{R}^2\) is the set of all \([a,b]^T \in\mathbb{R}^2\) such that the (nonlinear) equation \(Ax=ax^+-bx^-\) has a nontrivial solution. Here \(x^\pm\) has the elements \(x_i^\pm= \max\{\pm x_i,0\}\), where \(x_i\) are the elements of \(x\). The Fučik set is a closed subset of \(\mathbb{R}^2\) (in the Zariski topology). It is relevant in solving nonlinear equations of the form \(Ax=ax^+-bx^-+g(x)\), where \(g(x)=o(\|x\|)\) for \(\|x\|\to\infty\). The authors prove several results describing \(A_{-1}\). The case \(n=2\) is considered in detail.