The Fučík spectrum of the discrete Dirichlet operator

Abstract

In this paper, we deal with the discrete Dirichlet operator of the second order and we investigate its Fučík spectrum, which consists of a finite number of algebraic curves. For each non-trivial Fučík curve, we are able to detect a finite number of its points, which are given explicitely. We provide the exact implicit description of all non-trivial Fučík curves in terms of Chebyshev polynomials of the second kind. Moreover, for each non-trivial Fučík curve, we give several different implicit descriptions, which differ in the level of depth of used nested functions. Our approach is based on the Möbius transformation and on the appropriate continuous extension of solutions of the discrete problem. Let us note that all presented descriptions of Fučík curves have the form of necessary and sufficient conditions. Finally, our approach can be also directly used in the case of difference operators of the second order with other local boundary conditions.