Operators with jumping nonlinearities and combinatorics. (English) Zbl 0671.47056

The solvability of equations with jumping nonlinearities \[ (1)\quad u+\lambda Su^+-\mu Su^-=f \] in finite dimensional spaces is studied. Here S is a linear symmetric operator in \({\mathbb{R}}^ n\), \(\lambda\) and \(\mu\) are real parameters, \(u^+=[u^+_ 1,...,u^+_ n]\), \(u^- [u^-_ 1,...,u^-_ n]\). A relation between this problem and certain combinatorial problems is explained. Denote by \(C^ n\) the cube in \({\mathbb{R}}^ n\) with the vertices all coordinates of which equal either 1 or -1. For any arbitrary hyperplane \(\rho\) in \({\mathbb{R}}^ n\) containing no vertex of \(C^ n\), two nonnegative integers d(\(\rho)\), k(\(\rho)\) depending only on the number and position of the vertices of \(C^ n\) lying in one half-space with respect to \(\rho\) are defined. It is proved that there exist a linear symmetric operator S in \({\mathbb{R}}^ n\) and reals \(\lambda\), \(\mu\) such that \(| \deg (S_{\lambda,\mu},0,B)| =d(\rho)\) and (1) has at least k(\(\rho)\) solutions for every regular \(f\in {\mathbb{R}}^ n\). (Here \(S_{\lambda,\mu}=\lambda Su^+-\mu Su^- ,B\) is an open ball centered at the origin.)
Reviewer: M.Kucera


47J05 Equations involving nonlinear operators (general)
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