Fuč'ık spectrum for discrete systems: curves and their tangent lines

2024/12/16 by Gabriela Holubová, Holubová, Gabriela, Petr Nečesal +1
Engineering · Computer Science · #Advanced Numerical Analysis Techniques #Digital Image Processing Techniques #Computational Geometry and Mesh Generation

paper · pdf · doi:10.48550/arxiv.2412.11709

Abstract

In this paper, we study the Fuč'ık spectrum of a square matrix A and provide necessary and sufficient conditions for the existence of Fuč'ık curves emanating from the point (λ,λ) with λ being a real eigenvalue of A. We extend recent results by Maroncelli (2024) and remove his assumptions on symmetry of A and simplicity of λ. We show that the number of Fuč'ık curves can significantly exceed the multiplicity of λ and determine all the possible directions they can emanate in. We also treat the situation when the algebraic multiplicity of λ is greater than the geometric one and show that in such a case the Fuč'ık curves can loose their smoothness and provide the slopes of their "one-sided tangent lines". Finally, we offer two possible generalizations: the situation off the diagonal and Fuč'ık spectrum of a general Fredholm operator on the Hilbert space with a lattice structure.

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