In this section, we discuss the problem when a type of lags is not specified, i.e. we consider the delay dynamic equation (1.1). Our aim is to extend the asymptotic result formulated in Theorem 3.1.
First let \mathbb{T}=\mathbb{R}, i.e. we consider the delay differential equation
{y}^{\prime}(t)=\sum _{j=0}^{k}{a}_{j}y({\tau}^{j}(t)),\phantom{\rule{1em}{0ex}}t\in I,
(4.1)
where I is a real interval unbounded above. A very useful method converting differential equations with general lags into equations with prescribed lags is based on the utilisation of suitable functional equations (see [17] and [18]). If the prescribed lags are constant, then the corresponding functional equation is that of Abel (for an elegant application of this approach in the oscillation theory of delay differential equations, we refer, e.g. to [19]). In our case, the prescribed lags are proportional, hence along with (4.1), we consider also the linearisation equation
\phi (\tau (t))={q}^{1}\phi (t),\phantom{\rule{1em}{0ex}}t\in I,
(4.2)
which is called the Schröder equation. Let I=[m,\mathrm{\infty}), \tau \in {C}^{1}(I), \tau (m)=m, \tau (t)<t for all t>m, {\tau}^{\prime} is positive and nonincreasing on I and q=1/{\tau}^{\prime}(m)>1. Then there exists a solution \phi \in {C}^{1}(I) of (4.2), which is positive on (m,\mathrm{\infty}) and has a positive and nonincreasing derivative on I. For this and other relevant results on the Schröder equation, we refer to [20].
In the next assertion, we assume that τ and φ have the properties stated above and {\lambda}_{r} has the same meaning as in Theorem 3.1 (see the discussion preceding this theorem).
Theorem 4.1 Let y be a solution of (4.1), where {a}_{0}<0 and {a}_{k}\ne 0. Then
y(t)=O\left({(\phi (t))}^{r}\right)\phantom{\rule{1em}{0ex}}\mathit{\text{as}}\phantom{\rule{0.1em}{0ex}}t\to \mathrm{\infty},\phantom{\rule{1em}{0ex}}r={log}_{q}{\lambda}_{r}.
(4.3)
Proof We give only its brief outline. It is enough to replace the function {t}^{r} by {(\phi (t))}^{r} and follow the proof of Theorem 3.1. Indeed, if we put z(t)=y(t)/{(\phi (t))}^{r}, {t}_{1}={\tau}^{k}({t}_{0}) and {t}_{i}={\tau}^{i}({t}_{0}), where i=1,2,\dots , then the proof procedures presented above can be repeated step by step. In particular, (3.3) is replaced by
{a}_{0}{(\phi (t))}^{r}+\sum _{j=1}^{k}{a}_{j}{\left(\phi ({\tau}^{j}(t))\right)}^{r}=0,
which is satisfied just due to (4.2). Thus we arrive at
{S}_{i+1}\le {S}_{1}\prod _{\ell =1}^{i}(1\frac{{({(\phi (t))}^{r})}^{\prime}{}_{t={t}_{\ell}}}{{a}_{0}{(\phi ({t}_{\ell +1}))}^{r}})\le {S}_{1}\prod _{\ell =1}^{i}(1\frac{r}{{a}_{0}{q}^{r}}\frac{{\phi}^{\prime}({t}_{\ell})}{\phi ({t}_{\ell})})
instead of (3.6). Since {\phi}^{\prime} is bounded,
\frac{{\phi}^{\prime}({t}_{\ell})}{\phi ({t}_{\ell})}=O\left({q}^{\ell}\right)\phantom{\rule{1em}{0ex}}\text{as}\ell \to \mathrm{\infty}.
The remaining parts of the proof are identical with those stated above. □
Remark 4.2 If \tau (t)={q}^{1}t, then (4.2) is satisfied by the identity function. Consequently, Theorem 4.1 is a direct generalisation of Theorem 3.1 (when \mathbb{T}={\mathbb{R}}_{0}^{+}). Moreover, the above stated assumptions on τ particularly imply that
{\tau}^{\prime}(t)\le {q}^{1}<1\phantom{\rule{1em}{0ex}}\text{for all}t\in I,
(4.4)
i.e. the lags t{\tau}^{j}(t), j=1,2,\dots ,k are unbounded as t\to \mathrm{\infty}.
A reformulation of Theorem 4.1 and its proof for discrete time scales is not straightforward (e.g. we have used here the chain rule which is not valid on a general time scale). Therefore, we do not follow this way and give a simple alternative proof of a related assertion for discrete time scales. This assertion does not only extend, but even improves the asymptotic property (4.3).
Considering discrete time scales only, we investigate the delay dynamic equation (2.8) instead of (1.1). Therefore, instead of (4.2) we consider the Schröder equation
\phi (\rho (t))={q}^{1}\phi (t),\phantom{\rule{1em}{0ex}}t\in \mathbb{T}.
(4.5)
Also, as a direct analogue to (4.4), we assume that
{\rho}^{\mathrm{\nabla}}(t)\le {q}^{1}<1\phantom{\rule{1em}{0ex}}\text{for all}t\in \mathbb{T}.
(4.6)
Since ρ is increasing, (4.5) has a positive and increasing solution φ.
The reason why the asymptotic formula (4.3) (as well as (3.2)) does not seem to be quite optimal, consists in the fact that (4.3) involves the zero analysis of the polynomial \tilde{Q}(\lambda ). However, the polynomial Q(\lambda ), whose coefficients are in a direct correspondence with coefficients of studied equations, may be more natural than \tilde{Q}(\lambda ). Therefore, our next aim is to clarify this correspondence. Before doing this, we state some preliminaries on zeros of Q(\lambda ) and their relationship to the corresponding difference equation
\sum _{j=0}^{k}{a}_{j}\omega (sj)=0.
(4.7)
For any positive real θ, we introduce the set \mathrm{\Lambda}(\theta ) of all complex zeros of Q(\lambda ) with the modulus θ. If \mathrm{\Lambda}(\theta ) is nonempty for a given θ, then by a characteristic solution of (4.7) corresponding to \mathrm{\Lambda}(\theta ), we understand a finite sum of solutions of (4.7) corresponding to all values \lambda \in \mathrm{\Lambda}(\theta ). Of course, the form of such solutions depends on multiplicity of λ and it is described in details e.g. in [11].
Using this we have the following theorem.
Theorem 4.3 Let y be a solution of (2.8), where {a}_{0},{a}_{k}\ne 0 and let \mathbb{T} be a discrete time scale such that {a}_{0}\nu (t)\ne 1 for all t\in \mathbb{T} and (4.6) holds for some q>1. Then there exists \theta >0 such that \mathrm{\Lambda}(\theta ) is nonempty and
y(t)=\omega ({log}_{q}\phi (t))+O\left({(\phi (t))}^{\gamma}\right)\phantom{\rule{1em}{0ex}}\mathit{\text{as}}\phantom{\rule{0.1em}{0ex}}t\to \mathrm{\infty},\phantom{\rule{1em}{0ex}}\gamma ={log}_{q}(\theta \epsilon ),
(4.8)
where ω is a nontrivial characteristic solution of (4.7) corresponding to \mathrm{\Lambda}(\theta ), φ is a positive and increasing solution of (4.5) and 0<\epsilon <\theta is a suitable real scalar.
Proof Let y satisfy (2.8) on [{t}_{0},\mathrm{\infty}) for some {t}_{0}\in \mathbb{T}. By the definition,
\frac{y(t)y(\rho (t))}{\nu (t)}=\sum _{j=0}^{k}{a}_{j}y({\rho}^{j}(t)),\phantom{\rule{1em}{0ex}}t\in \mathbb{T},\phantom{\rule{1em}{0ex}}t\ge {t}_{0}.
(4.9)
Since
{\rho}^{\mathrm{\nabla}}(t)=\frac{\rho (t){\rho}^{2}(t)}{t\rho (t)}=\frac{\nu (\rho (t))}{\nu (t)},
it holds \nu (\rho (t))\le {q}^{1}\nu (t) in view of (4.6). Comparing with (4.5), one gets
\phi (t)=O(\nu (t))\phantom{\rule{1em}{0ex}}\text{as}t\to \mathrm{\infty}.
(4.10)
Now put s={log}_{q}\phi (t) and z(s)=y(t), t\in \mathbb{T}, t\ge {t}_{0}. Then y(\rho (t))=z(s1) and, more generally, y({\rho}^{j}(t))=z(sj) by use of (4.5). Thus (4.9) becomes
\frac{z(s)z(s1)}{\nu ({\phi}^{1}({q}^{s}))}=\sum _{j=0}^{k}{a}_{j}z(sj),\phantom{\rule{1em}{0ex}}s\in \mathbb{Z},s\ge {s}_{0},
{s}_{0}={log}_{q}\phi ({t}_{0}). Rewrite it as
[{a}_{0}b(s)]z(s)+[{a}_{1}+b(s)]z(s1)+\sum _{j=2}^{k}{a}_{j}z(sj)=0,
where
b(s)=\frac{1}{\nu ({\phi}^{1}({q}^{s}))}=O\left({q}^{s}\right)\phantom{\rule{1em}{0ex}}\text{as}s\to \mathrm{\infty}
due to (4.10). Equivalently,
{a}_{0}z(s)+\sum _{j=1}^{k}[{a}_{j}+{c}_{j}(s)]z(sj)=0,
(4.11)
where {c}_{j}(s) are appropriate functions satisfying {c}_{j}(s)=O({q}^{s}) as s\to \mathrm{\infty}, j=1,2,\dots ,k. Consequently, (4.11) is the higherorder difference equation of Poincaré type. Its limiting equation is just (4.7) with the characteristic polynomial Q(\lambda ). Moreover, the limits {a}_{j} are approached at an exponential rate. Then, by Theorem 2.3 of [21], there exists \theta >0 such that \mathrm{\Lambda}(\theta ) is nonempty, and for some 0<\epsilon <\theta, it holds
z(s)=\omega (s)+O\left({(\theta \epsilon )}^{s}\right)\phantom{\rule{1em}{0ex}}\text{as}s\to \mathrm{\infty},
where ω is a nontrivial characteristic solution of (4.7) corresponding to \mathrm{\Lambda}(\theta ). Now, using the backward substitution, we obtain (4.8). □
Let κ be a zero of Q(\lambda ) and {m}_{\kappa} be its multiplicity. We call this zero maximal if κ is (among all other zeros of Q(\lambda )) maximal in the modulus and if multiplicities of other possible zeros of Q(\lambda ) having the maximal modulus do not exceed {m}_{\kappa}. Of course, Q(\lambda ) may have several maximal zeros (with the same modulus and multiplicity).
Corollary 4.4 Let y be a solution of (2.8), where {a}_{0},{a}_{k}\ne 0 and let \mathbb{T} be a discrete time scale such that {a}_{0}\nu (t)\ne 1 for all t\in \mathbb{T} and (4.6) holds for some q>1. Further, let κ be a maximal zero of Q(\lambda ) and {m}_{\kappa} be its multiplicity. Then
y(t)=O\left({(\phi (t))}^{\alpha}{({log}_{q}\phi (t))}^{{m}_{\kappa}1}\right)\phantom{\rule{1em}{0ex}}\mathit{\text{as}}\phantom{\rule{0.1em}{0ex}}t\to \mathrm{\infty},\alpha ={log}_{q}\kappa ,
(4.12)
where φ is a positive and increasing solution of (4.5).
Remark 4.5 As an immediate consequence, we get that under the assumptions of Theorem 4.3, (2.8) is asymptotically stable (i.e. its any solution y tends to zero as t\to \mathrm{\infty}) if all the zeros of Q(\lambda ) are located inside the unite circle. In this connection, we have already mentioned the SchurCohn criterion, which can be applied to any polynomial Q(\lambda ) with concrete (fixed) coefficients and order. However, this criterion does not enable us to formulate explicit stability conditions in terms of (general) coefficients {a}_{j} and k. Such explicit conditions are known only in a very few particular cases (see, e.g. [22, 23] and [24]).
Now we illustrate conclusions of Theorem 4.1 and Theorem 4.3 by a simple example involving a type of delay not considered yet (to avoid a discussion on zeros of Q(\lambda ), we put k=1).
Example 4.6 Let y be a solution of the dynamic equation
{y}^{\mathrm{\nabla}}(t)=ay(t)+by\left({t}^{1/2}\right),\phantom{\rule{1em}{0ex}}t\in \mathbb{T},
(4.13)
where a, b are nonzero real scalars. Because of a type of τ, we assume that \mathbb{T} has a minimum m=1. Then q=1/{\tau}^{\prime}(m)=2 and the corresponding Schröder equation (4.2) becomes
\phi \left({t}^{1/2}\right)=\frac{1}{2}\phi (t),
(4.14)
which admits the solution \phi (t)={log}_{2}t having the required properties. We distinguish two cases:

(i)
If \mathbb{T}=\mathbb{R} (more precisely \mathbb{T}=[1,\mathrm{\infty})) and a<0, then by Theorem 4.1
y(t)=O\left({\left(\frac{b}{a}\right)}^{{log}_{2}{log}_{2}t}\right)\phantom{\rule{1em}{0ex}}\text{as}t\to \mathrm{\infty}.
(4.15)

(ii)
If \mathbb{T}=\{{2}^{{2}^{n}}:n\in \mathbb{Z}\}\cup \{1\}, then (4.5) becomes (4.14) and by Theorem 4.3
y(t)=c{(\frac{b}{a})}^{{log}_{2}{log}_{2}t}+O\left({\left(\right\frac{b}{a}\epsilon )}^{{log}_{2}{log}_{2}t}\right)\phantom{\rule{1em}{0ex}}\text{as}t\to \mathrm{\infty},
where c\in \mathbb{R} and 0<\epsilon <b/a are suitable scalars.
Remark 4.7 Equation (4.13) with \mathbb{T}=\mathbb{R} has been studied in [25] as the differential equation with advanced power argument, i.e. when \tau (t)={t}^{\gamma}, \gamma >1 (for extensions to the case of a general advanced argument τ, see also [26]). It is interesting to observe that asymptotic formulae derived in these papers are very close to the property (4.15).