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.
Fredholm Differential Operators with Unbounded Coefficients
证明了一个算子系数无界的一阶线性微分算子G在实直线上的函数空间上是Fredholm当且仅当相应的强连续发展族在两个半轴上都是指数二分的，且一对二分投影的值域是Fredholm，且G的Fredholm指标等于这对二分投影的Fredholm指标.算子G是与演化族相关联的演化半群的生成元。当演化族是适定微分方程u '（t）=A（t）u（t）的传播子时，一般地，算子A（t）是无界的，算子G是算子-d/dt+A（t）的闭包。因此，本文提供了K. Palmer，and by A.我和本-阿特兹戈伯格
