Some combinatorial results about the operators with jumping nonlinearities. (English) Zbl 0652.47040

This paper contains some applications of the results of the author [Nonlinear Analysis Theory, Methods, Appl. 12, 341-364 (1988)]. Assume S:\({\mathbb{R}}\) \(n\to {\mathbb{R}}^ n \)is a linear operator and for any \(\lambda\),\(\mu\in {\mathbb{R}}\) define \[ S_{\lambda,\mu}u=u+\lambda Su\quad +-\mu Su\quad - \] with u \(+=\max (u,0)\), u \(-=\max (-u,0)\). Then, \(S_{\lambda,\mu}\) is called operator with jumping nonlinearity. In the quoted paper, the author has proved some theorems on the relation between the absolute value of the Brouwer degree \(\deg (S_{\lambda,\mu}B,0)\) and the number of solutions of the equation \(S_{\lambda,\mu}u=f\), \(f\in {\mathbb{R}}^ n.\) In the present paper, the author, by means of a combinatorial method, constructs various examples of operators to which those results can be applied.
Reviewer: G.Caristi


47J05 Equations involving nonlinear operators (general)
55M25 Degree, winding number
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