The Fučík spectrum for discrete systems and some nonlinear existence theorems (Q6141067)

This paper deals with studying the existence solutions to nonlinear Fučík problems of the form \[ Ax = \alpha {x^ + } - \beta {x^ - } + g(x), \] where \(A\) is a symmetric \(n\times n\) matrix, \(\alpha, \beta\) are real numbers and \(g : \mathbb{R}^n \to \mathbb{R}^n\) is continuous. Ordered pairs of real numbers \((\alpha, \beta)\) that yield nontrivial solutions to the equation \(Ax = \alpha {x^ + } - \beta {x^ - } \) are called Fučík pairs, and the associated nontrivial solutions \(x\) are referred to as Fučík eigenvectors. The collection of all Fučík pairs is referred to as the Fučík spectrum and denoted by \(\mathcal{F}_A\). In [\textit{C. Margulies} and \textit{W. Margulies}, Linear Algebra Appl. 293, No. 1--3, 187--197 (1999; Zbl 0944.15014)], it was demonstrated that the Fučík spectrum consists of subsets of collection of algebraic curves. Then, an existence theorem was established for solutions to above mentioned form of Fučík problem with assumptions regarding the positioning of \((\alpha, \beta) \in \mathcal{F}_A^c\). However, no specific properties of the Fučík curves was determined when the dimension of \(A\) exceeds 2, and existence results were not discussed in cases of resonance, wherein \((\alpha, \beta) \in \mathcal{F}_A\). In this paper, the author aims to explore various qualitative properties of the Fučík spectrum, \(\mathcal{F}_A\), under more relaxed conditions on the \(n \times n\) symmetric matrix \(A\). Also, the existence of solutions to the considered Fučík problem is investigated under different conditions on the nonlinearity \(g\). Specifically, the author focuses on the case where \((\alpha, \beta) \in \mathcal{F}_A\), which is termed the resonance case. It is mentioned that the interest in resonance is both theoretical and application oriented. Overall, the current paper presents several ideas for characterizing some of the properties of the Fučík curves and it is concluded with some open questions regarding the Fučík spectrum of the general discrete problem \(Ax = \alpha {x^ + } - \beta {x^ - } \).