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

In this paper, we obtain the exact rates of decay to the non--hyperbolic equilibrium of the solution of a functional differential equation with maxima and unbounded delay. We study the convergence rates for both locally and globally stable solutions. We also give examples showing how the rate of growth of decay of solutions depends on the rate of growth of the unbounded delay as well as the nonlinearity local to the equilibrium.


Introduction
A large literature has developed in the past decades concerning the rate of decay to equilibrium in delay differential equations with unbounded delay. Some representative papers include Krisztin [21,20], Kato [18], Diblik [11], Cermak [10], and Haddock and Krisztin [14,15]. In the last two papers in particular, the rate of convergence is considered for equations in which the leading order space behaviour at the equilibrium is of smaller than linear order. The results can also be applied to stochastic delay equations in both the nonlinear (Appleby and Rodkina [5]) and linear case (Appleby [1]).
An especially interesting equation which has received much attention is one with proportional delay, called the pantograph equation: fundamental work on the asymptotic behaviour dates back to Kato and McLeod [19], Fox et. al. [12], Ockendon and Tayler [27]. Complex-valued and finite dimensional treatments were considered by Carr and Dyson [8,9], while more recent treatments and generalisations include Iserles [16] and Makay and Terjeki [25].
Another category of functional differential equation are those with maximum functionals on the righthand side. Inspiration for the study of these equations may be traced to work of Halanay [13]. The asymptotic behaviour of solutions of such Halanay type inequalities is considered by e.g., Baker and Tang [6], Mohamad and Gopalsamy [26], Liz and Trofimchuk [23], Ivanov, Liz and Trofimcuk [17] and Liz, Ivanov and Ferreiro [24] for equations with finite memory.
In this paper, we consider non-hyperbolic equations (as in e.g., [15]) with unbounded delay (as in e.g., [19]) as well as equations with max-type functionals (as in e.g., [23]). In particular, we give a complete characterisation of the rate of convergence of the solution x of the delay differential equation x ′ (t) = −ag(x(t)) + bg(x(t − τ (t))), t > 0; x(t) = ψ(t), t ≤ 0 (1.1) and the functional differential equation g(x(s)), t > 0; , x(t) = ψ(t), t ≤ 0 (1.2) to zero as t → ∞. We are interested in equations in which g(0) = 0 but g ′ (0) = 0, so that the equilibrium solution x(t) = 0 for all t ≥ 0 which arises from the initial condition ψ(t) = 0 for all t ≤ 0 is non-hyperbolic. In order to confine attention to a class of equations, we assume that g is regularly varying with index β > 1. Of course we ask that g is increasing and in C 1 on an interval (0, δ). The condition a > b > 0 is natural if we require solutions to be positive and for (at least) solutions with small initial conditions ψ to obey x(t) → 0 as t → ∞.
Granted that a solution obeys x(t) → 0 as t → ∞, we are able to determine the convergence rate of both equation (1.1) and (1.2). This rate can be related to the rate of decay to zero of the solution of the related ordinary differential equation y ′ (t) = −(a − b)g(y(t)), t > 0; y(0) > 0. (1.3) By unifying hypotheses used to prove results under slightly different conditions on the nonlinear function g, our subsidiary results can be consolidated to give the following main theorem.
(a) If τ (t)/t → 0 as t → ∞ and y is the solution of (1.3) then x(t)/y(t) → 1 as t → ∞ (b) If τ (t)/t → q ∈ (0, 1) with a > b(1 − q) −β/(β−1) , and y is the solution of (1.3), then We conjecture that x(t)/y(t) → Λ > 1 as t → ∞. (c) If τ (t)/t → q ∈ (0, 1) with a < b(1 − q) −β/(β−1) , and y is the solution of (1.3), then x(t)/y(t) → ∞ as t → ∞ and moreover 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.
The results show that the rate of convergence is dependent on the delay: while the rate of growth of the delay τ is less than some critical rate, the solution inherits the rate of decay of (1.3) exactly. Once the delay grows more rapidly than a critical rate, the solution no longer inherits the rate of convergence of solutions of (1.3). We are able not only to identify the critical growth rate of the delay at which this happens, but also to determine the exact convergence rate of the solution of e.g., (1.1) whether there is "slowly" growing or "rapidly" growing delay. 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.
2. Notation and Statement of the Problem 2.1. Notation. We recall that a function f is regularly varying at infinity with exponent α ∈ R if lim t→∞ f (λt) f (t) = λ α , for all λ > 0.
For such a function, we write f ∈ RV ∞ (α). A function f is regularly varying at zero with exponent α ∈ R if For such a function, we write f ∈ RV 0 (α). The exploitation of properties of regularly varying functions in studying asymptotic properties of ordinary and functional differential equations is an active field of research. Recent research themes in this direction are recorded in monographs such as [29] and [28] and all properties of regularly varying functions employed in this paper can be found in the classic text [7]. A highly selective list of the properties of regular variation that we have found useful appear in the introduction of [3], a work which concerns ordinary differential equations.

Discussion of Main Results
In this section we state, motivate, and discuss results giving the exact rate of decay to zero of solutions of (2.1) and (2.2). The main tool employed is a type of comparison argument.
In the case when the rate of growth of the delay is "fast", and the equation has long memory, an important auxiliary function σ is introduced which enables the asymptotic behaviour to be determined. Motivation for the method of proof, and the role of the auxiliary function σ is given in the following section, along with easily applicable corollaries of the main results. The ease of applicability of these results relies upon being able to determine the appropriate auxiliary function σ, often as a function asymptotic to τ .
In the case when the rate of growth of the delay is "slow" (or the equation has bounded delay), the function σ is not required. In this case we show that the solution of the delay differential equation converges to zero at exactly the same rate as the ordinary differential equation x ′ (t) = −(a − b)g(x(t)) for t > 0.
The results of the following theorems are given in Section 10. In this section we concentrate on stating the main general results, and discuss the role and necessity of the hypotheses on g, τ and the auxiliary function σ. The implications of the conclusions of the general results are also explored here.
First, we have that solutions of (2.1) and of (2.2) are uniformly bounded. This is used in later theorems to show that solutions of (2.1) and (2.2) tend to zero as t → ∞ and to determine the rate of convergence. Theorem 1. Let τ be a continuous and non-negative function such that −τ = inf t≥0 t − τ (t). Let a > b > 0 and g satisfy (2.4a), (2.4c) and suppose ψ ∈ C([−τ , 0]; (0, ∞)).
We are interested in solutions of (2.1) which tend to zero as t → ∞. In order to guarantee this we assume that An assumption of this type is reasonable; indeed if x(t) → 0 as t → ∞, we require that To see this, suppose to the contrary that lim sup t→∞ t−τ (t) = τ 1 < +∞. Therefore x(s) =: g(x) =: g 2 < +∞.
Therefore x(t) → ∞ as t → ∞, a contradiction, and so (3.3) must hold. We notice that if ψ(t) = ψ(0) > 0 for t ∈ [−τ , 0] and a = b, then the solution of (2.1) is These examples shows that the assumption a > b cannot be relaxed if solutions of both (2.1) and (2.2) are to tend to zero for all initial conditions. 3.1. General results. We start by making some assumptions on g: There is δ 1 > 0 such that g is increasing on (0, δ 1 ); (3.4c) We now state our main result for slowly growing (or bounded) delay.
Theorem 2. Let τ be a continuous and non-negative function such that −τ = inf t≥0 t − τ (t) and which obeys (3.2). Suppose also that there is q ∈ [0, 1) such that Suppose that a > b > 0 in such a manner that Let g satisfy (3.4) and suppose ψ ∈ C([−τ , 0]; (0, ∞)). If the solution of (2.1) viz., obeys x(t) → 0 as t → ∞, and G is defined by (2.5) then Our next general result deals with the case when q ∈ (0, 1) is so large that it does not satisfy (3.9). We modify the hypotheses on g slightly in this case: It turns out that the hypothesis (3.8d) often implies (3.4d).
Theorem 4. Let τ be a continuous and non-negative function such that −τ = inf t≥0 t − τ (t) and which obeys (3.2). Suppose also σ is a non-negative, continuous function on [−τ , ∞), Let a > b > 0 and g satisfy (3.8) and suppose ψ ∈ C([−τ , 0]; (0, ∞)). If the solution x of (2.1) obeys x(t) → 0 as t → ∞ , then (3.15) 4. Slowly Growing and Proportional Delay for Equations with Regularly Varying Coefficient 4.1. Slowly growing delay. Apart from the positivity of g, which guarantees positive solutions, we require conditions on g local to the equilibrium 0 in order to determine the rate of convergence of solutions in the case when the delay grows sublinearly. Theorem 2 can be applied to equations with coefficients in RV 0 (β); in the first instance we consider the case where τ (t)/t → 0 as t → ∞. The hypotheses on g become: There is δ 1 > 0 such that g is increasing on (0, δ 1 ); (4.1c) There is β > 1 such that g ∈ RV 0 (β). (4.1d) We now state our main result for slowly growing (or bounded) delay.
The result shows that when the delay τ grows sublinearly (or is bounded), converging solutions of (2.1) have the same asymptotic behaviour as the non-delay differential equation (4.5) because y(t) → 0 as t → ∞ and the hypothesis g ∈ RV 0 (β) implies Therefore, if y is the solution of (4.5) we have lim t→∞ x(t) y(t) = 1.

4.2.
Proportional delay and asymptotic behaviour equivalent to nondelay case. If the delay grows proportionately to t in the sense that (3.5) holds for some q ∈ (0, 1) the rate of decay of (2.1) is not the same as (4.5). We can prove the following result.
However if q is sufficiently small, it can be shown that the main asymptotic behaviour of the differential equation (4.5) is preserved, in the sense that x(t) is bounded above and below by G −1 (t) times a constant as t → ∞. Theorem 7. Let τ be a continuous and non-negative function such that −τ = inf t≥0 t − τ (t) and which obeys (3.2). Suppose also that τ obeys (3.5) for some q ∈ (0, 1), that a > b > 0 and moreover that a and b obey . (4.8) Suppose g satisfies (4.1) and suppose ψ ∈ C([−τ , 0]; (0, ∞)). If the solution x of (2.1) obeys x(t) → 0 as t → ∞, and G is defined by (2.5) then there is Λ 0 > 0 such that Recalling that the solution y of the non-delay differential equation (4.5) obeys (4.6), Theorem 7 shows that as claimed. We conjecture when a, b, q and β obey (4.7), and τ obeys (3.5) that we can strengthen the conclusion of Theorem 7 to obtain the limit (4.10) where Λ is defined by (4.8).
In fact, by the methods of Theorem 6 it can be shown that if there is a λ such that then we must have λ = Λ.
On the other hand, if τ obeys (3.5) and a, b, q and β obey (with a > b > 0) the method of proof of Theorem 6 shows that there is no λ ∈ (0, ∞) such that x obeys (4.11). In the next section we investigate the case covered by (4.12) as well as the case when the delay grows so quickly that τ (t)/t → 1 as t → ∞.

4.3.
Proportional delay and asymptotic behaviour not equivalent to nondelay case. Our next results demonstrates that once τ grows faster that qt (where q ∈ (0, 1) is so large that it obeys (4.12)), the asymptotic behaviour of (2.1) is no longer asymptotic to or bounded by the solution y of the ordinary differential equation (4.5). The exact rate of convergence can be determined in the case when τ obeys (3.5) when q ∈ (0, 1) obeys (4.12). Of course, the nonlinearity g and the constants a and b still play an important role in determining the asymptotic behaviour.
For these results, we place slightly different hypotheses on g local to zero than the conditions (4.1) imposed in Theorem 5 or 7; now we require g to not only be increasing, but to have a positive derivative close to zero, and we ask that g ′ , rather than g, be regularly varying at 0. The hypotheses are the following.

Slowly Growing Delay for Equations with Regularly Varying Coefficient
We now attempt to determine the asymptotic behaviour of solutions when the delay grows according to τ (t)/t → 1 as t → ∞. It transpires that the following theorem enables us to achieve this, provided a related limiting functional equation involving τ can be solved which involves an auxiliary function σ. The result follows by an application of Theorem 4.
Theorem 9. Let τ be a continuous and non-negative function such that −τ = inf t≥0 t − τ (t) and which obeys (3.2). Suppose also that σ and τ obey (3.11)-(3.14). Let a > b > 0 and g satisfy (4.13) and suppose ψ ∈ C([−τ , 0]; (0, ∞)). If the solution of (2.1) viz., x ′ (t) = −ag(x(t)) + bg(x(t − τ (t)), t ≥ 0 is a direct consequence of (2.4d) and (5.1). (2.4d) implies The hypotheses on the auxiliary function σ under which Theorem 9 holds will be explored and motivated in greater depth in the next section. Note however, that the conditions on the size of σ and τ are asymptotic: the short run behaviour of τ and σ is seen not to be important in being able to determine the rate of convergence.
Neither are differentiability or monotonicity conditions required on τ . This feature of Theorem 9, allow analysis to be extended to delay-differential equations with relatively badly behaved τ . All that turns out to be important is the asymptotic rate of growth of τ . The presence of unbounded delay has just been mentioned, but it is not explicitly present in the statement of Theorem 9. However, the conditions (3.12) and (3.13) on σ, together with (3.2), force lim t→∞ τ (t) = +∞. (5.3) Therefore, by also assuming (3.2) in Theorem 9, the delay will be unbounded even though this is not explicitly stated. We have already noted that (3.2) is a reasonable assumption if we want x(t) → 0 as t → ∞. To show that (5.3) must hold, first note that as (3.13) holds, there exists T 1 > 0 such that for all t > T 1 we have Since σ(t) → ∞ as t → ∞, for every M > 0 there exists T 2 (M ) > 0 such that σ(t) > M for all t > T 2 (M ). Also as t − τ (t) → ∞ as t → ∞, there exists . Then for t > T (M ) we have Hence τ (t) > M/2 for t > T (M ). Since M > 0 is arbitrary we have (5.3). Therefore, by the hypotheses in Theorem 9, the delay is unbounded even though this is not explicitly stated. We have already noted that (3.2) is a reasonable assumption if we want x(t) → 0 as t → ∞.
We next show that Theorem 9 covers precisely the rapidly growing delay which is not covered by Theorems 5, 7 and 8 which cover the case when τ (t)/t → q ∈ [0, 1) as t → ∞. The question now is: how does the condition (3.14) relate to the case not already covered by the results to date, namely the case when τ (t)/t → 1 as t → ∞. Roughly speaking, we will now show that if the delay grows like t, then solutions grow at the rate determined by (5.2). To do this, we state an auxiliary result which shows how the linear or sublinear growth of σ implies linear of sublinear growth in τ .

5.1.
Concrete examples of σ obeying (3.11)-(3.14). We now state general results which enable to explicitly construct σ obeying (3.11)-(3.14) while at the same time only making assumptions concerning the asymptotic behaviour of τ . Proposition 1. Let τ be a continuous and non-negative function such that −τ = inf t≥0 t − τ (t) and for which There exists β ∈ (0, 1) such that lim Then there is a function σ which obeys (3.11), (3.12) and (3.13) such that Proposition 2. Let τ be a continuous and non-negative function such that −τ = inf t≥0 t − τ (t). Suppose that ϕ ∈ C[0, ∞); R) is such that ϕ is increasing on [0, ∞) and lim t→∞ ϕ(t) = ∞; Then there is a function σ which obeys (3.11), (3.12) and (3.13) such that Factors influencing the rate of decay of x. We note that the relationship between the rate of growth of the unbounded delay τ and the rate of decay of the solution x of (2.1) to 0 as t → ∞ (which depends on σ) is embodied in the condition (3.13). The limit (3.13) relates the asymptotic behaviour of σ to that of τ . We see that the faster that σ(t) → ∞ as t → ∞, the faster that 1/σ(t) → 0, so in order for (3.13) to hold, τ (t) must tend to infinity faster as t → ∞ to compensate for the rapid decay of 1/σ(t). Also, the faster that σ tends to infinity, the slower that t 0 1/σ(s) ds tends so infinity as t → ∞, and so by (5.1), the slower that x(t) → 0 as t → ∞. Therefore, we see that the faster that τ (t) → ∞ as t → ∞, the slower that x(t) → ∞ as t → ∞. This makes intuitive sense, as the longer the "memory" of the equation, the slower the convergence of asymptotically stable solutions to the equilibrium should be.
The limit (5.1) reveals that the rate of decay of x(t) → 0 increases as a increases and decreases as b increases, as should be expected; the greater the negative instantaneous feedback and the less the positive delayed feedback of the delayed term, the more rapidly solutions of (2.1) should converge to zero.
The limit (5.1) also reveals that the stronger the nonlinearity g local to zero, the faster the rate of convergence of x(t) → 0 as t → ∞. Consider the solutions x 1 and x 2 of (2.1) in the case when g = g 1 and g = g 2 respectively. By (5.1), we have Therefore This limit is interesting in itself as it shows the impact of different nonlinearities on the rate of convergence of solutions, even when the auxiliary function σ is not known.
To give a concrete example where we get different convergence rates arising from different nonlinearities, suppose that g 1 ∈ RV 0 (β 1 ) and so g 2 (x)/g 1 (x) → 0 as x → 0 + and therefore g 1 dominates g 2 local to zero. We should therefore expect that x 1 tends to zero more rapidly than x 2 . By (5.13) and (5.12) we have Hence as t → ∞, we have . Thus x 1 converges to zero more quickly to zero than x 2 as t → ∞, as we anticipated.

Motivation For
Results. In rough terms, Theorems 9 and 24 are proven by constructing one-parameter families of functions x L,ǫ and x U,ǫ such that where the monotonicity of g ensures that the functions x L,ǫ and x U,ǫ are welldefined. These functions are constructed so that they are upper and lower solutions of the solution x of e.g., (2.1). This is achieved because x 1 (ǫ), x 2 (ǫ), C 1 (ǫ) and C 2 (ǫ) can be chosen so that there are T 1 (ǫ), T 2 (ǫ) > 0 such that We choose x 1 (ǫ) so small and x 2 (ǫ) so large so that . The values of x 1 (ǫ) and x 2 (ǫ) play no role in the differential inequality. The parameters C 1 (ǫ) and C 2 (ǫ) are chosen so that the differential inequalities are satisfied on [T 1 (ǫ), ∞) and [T 2 (ǫ), ∞) respectively. The values of T 1 (ǫ) and T 2 (ǫ) are chosen so as to use asymptotic information about σ and τ that is present in (3.12) and in (3.13) especially: this information is mainly used to satisfy the differential inequalities. The comparison principle now implies that x L,ǫ (t) < x(t) < x U,ǫ (t) for all t ≥ 0, and hence that g(x L,ǫ (t)) < g(x(t)) < g(x U,ǫ (t)) for all t ≥ 0. The upper and lower estimates on g(x(t)) are known explicitly by the construction (5.14). Finally, we send the parameter ǫ → 0 + : the exact asymptotic limit (5.1) is obtained because C 1 (ǫ) and C 2 (ǫ) have been designed so that both tend to the same limit as ǫ → 0 + . Roughly speaking for each estimate, we need two adjustable constants x i and C i to satisfy two inequalities: one for the differential inequality on (T i (ǫ), ∞) and one for the "initial condition" on The free parameter ǫ is used at the end of the proof to match exactly the upper and lower estimates. In fact, to give sufficient flexibility in the construction of the upper and lower estimates, we sometimes have additional free parameters C 1 and C 2 which can be sent to C 1 (ǫ) and C 2 (ǫ) in advance of taking the limit as ǫ → 0 + . These are the broad guidelines followed in constructing the upper and lower estimates, and do not cover all the subtleties encountered: sometimes the objectives are in conflict and the construction can become quite delicate and require some iteration. As a general rule, it is more difficult to construct a very good upper estimate as there is some interaction between all three terms in the differential inequality for x U . For the lower estimate, the presence of the derivative term can generally be ignored by using the fact that the estimates constructed are decreasing functions; therefore the relative size of the two terms on the righthand side of the differential inequality for x L is all that matters. The monotonicity of the estimates also allows the analysis to be extended easily to the equation (2.2) with a maximum functional, and simplifies the choice of estimates that must be taken in order to satisfy constraints on the "initial conditions".
We motivate now the functional forms of x L and x U and the hypotheses required in Theorems 9 and 24. If a function y is written in the form as g is in C 1 , it is easily seen that with a similar equality holding when y is on the lefthand side. Therefore analysis of the righthand side of (5.16) is the same whether we consider equation (2.1) or (2.2). A reasonable objective now is to ensure that the term in curly braces in (5.16) is negligible (at least as t → ∞) so that y can be close to a solution of (2.1). In order that y(t) tends to zero, we need t 0 1/σ(s) ds → ∞ as t → ∞, while a condition of the form σ(t) → ∞ as t → ∞ will preclude exponential decay. Moreover, as mentioned in the discussion after Theorem 9, the assumption that σ(t) → ∞ as t → ∞ is consistent with τ (t) → ∞ as t → ∞. This explains the rationale behind the conditions (3.12). Moreover, if y(t) → 0 as t → ∞, then g ′ (y(t)) tends to a nontrivial limit by (2.4d), so as we suppose that σ(t) → ∞ as t → ∞, the first term in the curly brackets in (5.16) tends to zero as t → ∞. Therefore, in order for y to be in some sense "close to" a solution of (2.1), we need a condition of the form which makes sense of the hypothesis (3.13) on σ, if we choose C = log(a/b). Now, if we take logarithms across (5.15) and use C = log(a/b) and (3.13) we have Taking limits as t → ∞ gives Since y should be close to the solution x of (2.1), this motivates the claimed result (5.1), and therefore the construction of x L and x U in (5.14).
Of course, this argument is a long way from being a rigorous proof; it however motivates the choice of conjecture, and an identity of the form (5.16) in fact plays an important role in the proof of Theorems 9 and 24.
In some sense, our calculation leaves the functional form of σ undetermined: it is left as an open question whether a function σ exists which obeys the conditions (3.12) and (3.13) required in order to approximately fit y as a solution. This leaves the question of how to find such a function open. 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]. 6. Summary of Main Results and Examples in Regularly Varying Case 6.1. Unifying the main results. Since the condition that σ(t)/t → 0 as t → ∞ implies that τ (t)/t → 0 as t → ∞, σ(t)/t tends to a finite limit as t → ∞ implies τ (t)/t → q as t → ∞ for some q ∈ (0, 1), and (3.14) implies that τ (t)/t → 1 as t → ∞, we can unify Theorems 5, 7, 8 and and 9 by means of the parameter λ in (5.4).

6.2.
Examples. In the following section, we show the versatility of the results in the last section, by considering a selection of examples with with different rates of growth in the delay τ . We state the results for each example in turn.
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 relationship between σ and τ is nontrivial and must be determined for each problem by analysis of (3.13), thereby justifying general theorems such as Theorem 9 and 24. The introduction of the function σ also enables us to remove monotonicity and differentiability conditions on τ often required in the study of differential equations with delay. Moreover, the asymptotic form of the condition (3.13) shows that it is the behaviour of τ (t) as t → ∞ that determines the asymptotic behaviour of solutions of (2.1) and (2.2); the behaviour of τ (t) on any compact interval [0, T ] is not material.
In each case the common hypotheses are that the solution x of equation (2.1) is studied, with τ a continuous and non-negative function, where −τ = inf t≥0 t − τ (t) is finite and t − τ (t) → ∞ as t → ∞. We have a > b > 0, the initial function ψ ∈ C([−τ , 0]; (0, ∞)). We suppose that x(t) → 0 as t → ∞. All results stated here for solutions of (2.1) apply equally to the max-type equation (2.2).
We consider for concreteness two functions for g: let Such a function obeys all the conditions in (6.1). We can also consider the non-polynomial function g 2 such that g 2 : [0, δ) → [0, ∞) obeys g 2 (x) = x β log(1/x) for 0 < x ≤ δ < 1 and g 2 (0) = 0. Such a function g 2 also obeys all hypotheses in (6.1). Therefore we can determine the asymptotic behaviour of x using Theorem 9 or Theorem 5 if g = g 1 or g = g 2 .
In order to apply Theorem 5 it is first necessary to determine the asymptotic behaviour of To do this we compute G 1 and G 2 . First we have that .
For G 2 we see that From this it can be shown that We should note that it is not necessary that g assume exactly the form of g 1 or g 2 above in order for us to determine asymptotic results. Suppose merely that where i = 1, 2. Then g ∈ RV 0 (β), and moreover we can show that Therefore, if g is a function which is positive on (0, ∞), is increasing and continuously differentiable on an interval (0, δ 1 ), and obeys g(0) = 0, we can still apply Theorem 5.
As we move down through this list of examples, the rate of growth of the delay becomes faster; the rate of growth of the auxiliary function σ will also be shown to be faster; and the rate of decay of the solutions becomes slower. Example 13 deals with an equation with (approximately) proportional delay. In this case our results are consistent with more precise results which have been determined for the so-called pantograph equation when β = 1.

Equations With Non-Regularly Varying Coefficients
We can apply theorem 2 and 3 to equations where the function g is not in RV 0 (β) for some β > 1. The results are as follows. We first consider hypotheses on g which enable us to consider equations with slowly growing delay τ : There is δ 1 > 0 such that g is increasing on (0, δ 1 ); (7.1c) The hypothesis (7.1d) represents a strengthening of hypothesis (3.4d) in the case γ = 1. We may now apply Theorem 2 directly to the equation (2.1).
Theorem 16. Let τ be a continuous and non-negative function such that −τ = inf t≥0 t − τ (t) and which obeys (3.2). Suppose also that there is q ∈ [0, 1) such that τ obeys (3.5). Suppose that a > b > 0 in such a manner that Let g satisfy (7.1) and suppose ψ ∈ C([−τ , 0]; (0, ∞)). If the solution x of (2.1) obeys x(t) → 0 as t → ∞, and G is defined by (2.5) then Therefore if y is the solution of the corresponding non-delay equation (4.5), we have that Our next result deals with the case when q ∈ (0, 1) is so large that it does not satisfy (3.9). The hypotheses on g in this case are a special case of the hypotheses (3.8) It turns out that the hypothesis (7.4d) often implies (3.4d). The following is therefore a simple application of Theorem 3.
Theorem 17. Let τ be a continuous and non-negative function such that −τ = inf t≥0 t − τ (t) and which obeys (3.2). Suppose also that τ obeys (3.5) for some q ∈ (0, 1), that a > b > 0 and moreover that Finally, we can apply Theorem 4 to get the following result in the case when the delay grows rapidly.
Theorem 18. Let τ be a continuous and non-negative function such that −τ = inf t≥0 t − τ (t) and which obeys (3.2). Suppose also that σ and τ obey (3.11)-(3.14). Let a > b > 0 and g satisfy (7.4) and suppose ψ ∈ C([−τ , 0]; (0, ∞)). If the solution x of (2.1) obeys x(t) → 0 as t → ∞ , then Examples. The difficult conditions to verify are (7.1d) (for Theorem 16) and (7.4d) (for Theorem 17). We now consider two examples of g that are so flat at 0 that the above results apply. However, this necessitates finding the asymptotic behaviour of rather complicated functions such as g ′ • g −1 and g • G −1 . We record our findings in the following lemmata.
Lemma 2. Suppose that α > 0 and that g obeys Lemma 3. Suppose that α > 0 and that g obeys (7.11) With these asymptotic results to hand, we can obtain precise rates of convergence to 0 of the solution of (2.1) for the functions g above.

Equations with Maximum Functionals
We may also consider the asymptotic behaviour of solutions of the equation (2.2). The proofs of the asymptotic results for this equation are the same as those for (2.1) except at one stage of the proof. , This is because the functions used as upper and lower solutions in Theorem 5 and 9 are also employed for (2.2), in the sense that the functional dependence of the comparison functions on the solution of the underlying functional differential equation and data are the same in both proofs (they are not the same functions because the solutions of (2.1) and (2.2) are not the same, in general). These comparison functions are monotone decreasing and small on their domain of definition, and g is assumed to be increasing in some neighbourhood to the right of zero, so we may write where x U is the upper comparison function, and T > 0 is sufficiently large. The same identity holds for lower comparison functions. Therefore the upper comparison function which satisfies the differential inequality with an analogous pair of inequalities holding for the lower comparison functions.
The comparison principle now shows that these upper and lower comparison functions bound the solution above and below. Therefore we have a direct analogue of Theorems 9. If an analogue of Theorem 5 can be shown, then the analogue of Theorem 10 follows directly. The relevant results are now stated.

Proof of Theorem 2.
We first need to prove that ds ≥ −at, t ≥ 0.
9.4. Proof of Theorem 4. We first need to prove a preliminary lemma.
Therefore we have .
Therefore for t ≥ T 5 (ǫ) we have from which (4.3) follows, by first taking limits as t → ∞, and then letting ǫ → 0 + .

Proof of Results for Proportional Delay
11.1. Proof of Theorem 6. We employ the following result in the proof of Theorem 6.

Proof of Theorem 7
We are going to prove this result in two parts: first, we will show that there exists Λ 1 > 0 such that where G 0 is defined by (9.3). From this the result holds. Then we will use this estimate to show that where Λ is given by (4.8).
12.1. Proof of (12.2). We show when a, b, q and β obey (4.7), and τ obeys (3.5) that we can to obtain the limit where Λ is defined by (4.8). To do this we first need two preliminary lemmata.
We are now in a position to prove (12.2). To do this, it is important first to show that Λ 1 ∈ (0, ∞) defined by obeys Λ 1 ≥ Λ. The argument used to prove this can then be adapted easily to prove (12.2). We formulate the desired result in the following proposition.
We will show that the following statements are true: The consequence of these statements is that (12.6) holds. To see this, note that by using statement (a) successively, we have that Since λ n ′ +1 > Λ 1 by applying statement (b) we have (12.6). The limit (12.6) now leads to a contradiction, because if lim t→∞ x(t)/G −1 (t) is finite and positive, it must be Λ. Therefore Λ 1 = Λ. But Λ 1 < Λ by hypothesis, so this contradiction forces Λ 1 ≥ Λ. It therefore remains to prove the inferences (a), (b).
Taking logarithms and arguing in a similar manner we obtain (7.14).