Exponential dichotomies for linear non-autonomous functional differential equations of mixed type

Abstract

ABSTRACT. Functional differential equations with forward and backward delays arise naturally, for instance, in the study of travelling waves in lattice equations and as semi-discretizations of partial differential equations (PDEs) on unbounded domains. Linear functional differential equations of mixed type are typically ill-posed, i.e., there exists, in general, no solution to a given initial condition. We prove that Fredholm properties of these equations imply the existence of exponential dichotomies. Exponential dichotomies can be used in discretized PDEs and in lattice differential equations to construct multi-pulses, to perform Evans-function type calculations, and to justify numerical computations using artificial boundary conditions.

Lin's method and homoclinic bifurcations for functional differential equations of mixed type

“…It is known that exponential dichotomies form a very powerful tool when dealing with ill-posed problems, since they split the state space into separate parts that do admit a semiflow. The existence of such exponential splittings for parameter-independent homogeneous linear MFDEs, was established independently and simultaneously by Verduyn Lunel and Mallet-Paret [15] on the one hand and Härterich and coworkers [6] on the other, using very different methods.…”

The dynamical behaviour near solitary waves in Hamiltonian lattice differential equations

“…In order to solve the associated initial value problem, we have to specify a function on the interval [−M, M] and typically one considers initial data in the space C 0 := C 0 ([−M, M], R N ). However, the initial value problem is ill-posed [18]! Advance delay equations have been studied almost exclusively over the past ten years [13,14,15,18,27,29,30] among with the pioneering work [32].…”

“…However, the initial value problem is ill-posed [18]! Advance delay equations have been studied almost exclusively over the past ten years [13,14,15,18,27,29,30] among with the pioneering work [32]. Let us now assume the existence of a solitary wave solution…”

“…In this section we collect some known facts about linear functional differential equations of mixed type which we will use in the sequel, see also [29,18]. We investigate the linear equationẏ…”

“…Furthermore, the algebraic multiplicity of λ * as an eigenvalue of A + (which is the dimension of its generalized eigenspace) coincides with the order of λ * as a zero of det (·); we refer to [13,18] for proofs of these statements.…”