A Constructive Comparison Technique for Determining the Asymptotic Behaviour of Linear Functional Differential Equations with Unbounded Delay

Abstract

In this paper, we obtain exact non-exponential rates of growth and decay of solutions of linear functional differential equations with unbounded delay. As a by-product, exponential asymptotic stability is ruled out for asymptotically stable solutions of these equations. Equations with maximum functionals on the right hand side, as well as perturbed equations, are considered. We also give examples showing how the rate of growth or decay of solutions depends on the rate of growth of the unbounded delay. The results established here are obtained by constructing carefully functions which satisfy upper and lower differential inequalities and one aim of the paper is to elucidate this constructive comparison technique.

Sufficient conditions for polynomial asymptotic behaviour of the stochastic pantograph equation

“…The paper did not find a ready home at the time, but subsequently has steadily attracted citations through its online preprint incarnation. These are included in the bibliography below [4,5,19,23,24,27,33,44,45,48,49,50,51,52,55,56]. Since a number of other works quote [6], we feel it best that the paper be subject to formal review, and as the first author's introduction to the subject came through a paper in the EJQTDE [38], we felt it fitting, after a long (but not unbounded) delay, to submit a revised version of it here.…”

Exact and memory-dependent decay rates to the non-hyperbolic equilibrium of differential equations with unbounded delay and maximum functional

“…However, examples of equations whose asymptotic behaviour is determined by finding an appropriate σ are given at the end of the section. More about the role of the function σ, and its connections with the solution of a class of functional equations (called Schröder equations [22]) is written in [2].…”

“…In order to do this, we must determine how the auxiliary function σ can be chosen for a given problem. The role of the function σ is explained further in [2]. In the problems here to which Theorem 9 or 24 could be applied, we do not have that τ is asymptotic to σ, as scrutiny of the calculations involved in Examples 15 and 14 reveal.…”

“…The result in (d) generalises to the nonlinear setting results in Krisztin,Čermak etc., adapting the approach in [2] used to obtain sharp asymptotic estimates for linear equations. Since the equation does not have infinite memory (so we cannot have τ (t)/t → q > 1 as t → ∞), the results (a)-(d) can reasonably be said to provide a quite complete picture of the relationship between the rate of convergence, the strength of the nonlinearity g, and the rate of growth of the unbounded delay τ for the class of nonlinearities considered.…”

“…As a byproduct, the results also cover the case of bounded delay. We use comparison-type arguments inspired especially by Appleby and Patterson [3,4] which deal with non-hyperbolic ordinary and stochastic differential equations, and Appleby and Buckwar [2] which deals with equations of the form (1.1) and (1.2) with g(x) = x. For such a function, we write f ∈ RV ∞ (α).…”