Neutral Mixed Type Functional Differential Equations

Competing interactions and traveling wave solutions in lattice differential equations

“…This necessitates a Fredholm theory for linearized operators that do not satisfy a strict ellipticity conditions such as (A5), e.g., see [1], together with results on the dimension and structure of the kernel. While the Fredholm theory for problems with infinite range interactions is not well developed, the results in [12] apply to certain infinite range interactions.…”

Traveling Waves and Pattern Formation for Spatially Discrete Bistable Reaction-Diffusion Equations

“…When studying such ill-posed infinite-dimensional problems, exponential dichotomies, Fredholm theory and dimension reduction techniques become the methods of choice. Initiated by the pioneering work of Rustichini [137,138], research in this area features spectral flow results to compute the Fredholm index for operators with finite range shifts [125], infinite-range shifts [68] or neutral terms [119], various state-space decompositions based on exponential dichotomies [82,93,108,128], techniques to construct local [70,106,107] and global [99] center manifolds and extensions of geometric singular perturbation theory [100].…”

“…The challenge is to identify and classify the equilibrium solutions to (117). To this end, we look for stationary solutions of the form v j = v e for even j, v o for odd j, (119) which must hence satisfy the coupled system…”