论文标题
非本地差分算子的指数二分法，具有无限范围相互作用
Exponential Dichotomies for Nonlocal Differential Operators with Infinite Range Interactions
论文作者
论文摘要
我们表明，具有无限范围离散和/或连续相互作用的MFDE是基于Faye和Scheel为此类系统开发的Fredholm理论的指数二分法。对于半路，我们完善了Hupkes和Verduyn Lunel的早期方法。对于完整的线路，我们通过概括Mallet-Paret和Verduyn Lunel获得的有限范围结果来构建这些分裂。这些分裂遗漏的有限维空间可以使用Hale Inner产品来表征，但是由此产生的退化问题引发了微妙的问题，这些问题比在有限范围的情况下更难解决。实际上，由于它明确地引用了最小和最大的变化，因此没有通常用于排除归化性的标准“原子性”条件的直接类似物。 我们构建替代标准，以利用有关MFDE结构的更精细信息。当系数相对于适当的移位半群或通常与比较原理相关的标准阳性条件相对于适当的偏移条件，我们的结果是最佳的。我们通过涉及Nagumo方程的明确示例和反例来说明这些结果。
We show that MFDEs with infinite range discrete and/or continuous interactions admit exponential dichotomies, building on the Fredholm theory developed by Faye and Scheel for such systems. For the half line, we refine the earlier approach by Hupkes and Verduyn Lunel. For the full line, we construct these splittings by generalizing the finite-range results obtained by Mallet-Paret and Verduyn Lunel. The finite dimensional space that is `missed' by these splittings can be characterized using the Hale inner product, but the resulting degeneracy issues raise subtle questions that are much harder to resolve than in the finite-range case. Indeed, there is no direct analogue for the standard 'atomicity' condition that is typically used to rule out degeneracies, since it explicitly references the smallest and largest shifts. We construct alternative criteria that exploit finer information on the structure of the MFDE. Our results are optimal when the coefficients are cyclic with respect to appropriate shift semigroups or when the standard positivity conditions typically associated to comparison principles are satisfied. We illustrate these results with explicit examples and counter-examples that involve the Nagumo equation.
