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Localization of Fučík curves for the second order discrete Dirichlet operator
Bulletin des Sciences Mathématiques ( IF 1.3 ) Pub Date : 2021-06-16 , DOI: 10.1016/j.bulsci.2021.103014
Petr Nečesal , Iveta Sobotková

In this paper, we deal with the second order difference equation with asymmetric nonlinearities on the integer lattice and we investigate the distribution of zeros for continuous extensions of positive semi-waves. The distance between two consecutive zeros of two different positive semi-waves depends not only on the parameters of the problem but also on the position of one of these zeros with respect to the integer lattice. We provide an explicit formula for this distance, which allows us to obtain a new simple implicit description of all non-trivial Fučík curves for the discrete Dirichlet operator. Moreover, for fixed parameters of the problem, we show that this distance is bounded and attains its global extrema that are explicitly described in terms of Chebyshev polynomials of the second kind. Finally, for each non-trivial Fučík curve, we provide suitable bounds by two curves with a simple description similar to the description of the first non-trivial Fučík curve.



中文翻译:

二阶离散狄利克雷算子的 Fučík 曲线的定位

在本文中,我们处理整数晶格上具有非对称非线性的二阶差分方程,并研究了正半波连续扩展的零点分布。两个不同的正半波的两个连续零点之间的距离不仅取决于问题的参数,还取决于这些零点之一相对于整数点阵的位置。我们为这个距离提供了一个明确的公式,这使我们能够获得离散 Dirichlet 算子的所有非平凡 Fučík 曲线的新的简单隐式描述。此外,对于问题的固定参数,我们证明了这个距离是有界的,并且达到了用第二类切比雪夫多项式明确描述的全局极值。最后,对于每个非平凡的 Fučík 曲线,

更新日期:2021-06-21
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