标题： Fučík谱对于离散系统：曲线及其切线 显示英文标题

标题： Fučík spectrum for discrete systems: curves and their tangent lines

摘要： 在本文中，我们研究了方阵$A$的 Fučík 谱，并提供了从点$(\lambda,\lambda)$出发的 Fučík 曲线存在的必要且充分条件，其中$\lambda$是$A$的实特征值。我们扩展了 Maroncelli（2024）的最新结果，并去除了他对$A$对称性和$\lambda$简单性的假设。我们证明了 Fučík 曲线的数量可以显著超过$\lambda$的重数，并确定了它们可以出发的所有可能方向。 我们还处理了当$\lambda$的代数重数大于几何重数的情况，并表明在这种情况下，Fučík 曲线可能会失去光滑性，并给出它们“单侧切线”的斜率。 最后，我们提供了两种可能的推广：对角线以外的情况以及具有格子结构的希尔伯特空间上一般弗雷德霍姆算子的 Fučík 谱。 显示英文摘要

摘要： In this paper, we study the Fu\v{c}\'{\i}k spectrum of a square matrix $A$ and provide necessary and sufficient conditions for the existence of Fu\v{c}\'{\i}k curves emanating from the point $(\lambda,\lambda)$ with $\lambda$ being a real eigenvalue of $A$. We extend recent results by Maroncelli (2024) and remove his assumptions on symmetry of $A$ and simplicity of $\lambda$. We show that the number of Fu\v{c}\'{\i}k curves can significantly exceed the multiplicity of $\lambda$ and determine all the possible directions they can emanate in. We also treat the situation when the algebraic multiplicity of $\lambda$ is greater than the geometric one and show that in such a case the Fu\v{c}\'{\i}k 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 Fu\v{c}\'{\i}k spectrum of a general Fredholm operator on the Hilbert space with a lattice structure.