# Some qualitative properties of linear dynamic equations with multiple delays

- Jan Čermák
^{1}Email author

**2013**:155

https://doi.org/10.1186/1687-1847-2013-155

© Čermák; licensee Springer 2013

**Received: **4 February 2013

**Accepted: **9 May 2013

**Published: **31 May 2013

## Abstract

This paper discusses stability and asymptotic properties of the delay dynamic equation

where ${a}_{j}\in \mathbb{R}$ are scalars, ${\tau}^{j}$ are iterates of a function $\tau :\mathbb{T}\to \mathbb{T}$ and $\mathbb{T}$ is a time scale unbounded above. Under some specific choices of *τ*, this dynamic equation involves several significant particular cases such as linear autonomous differential equations with several delays or linear autonomous higher-order difference equations. For proportional *τ*, we formulate an asymptotic result joint for two different time scales, including a joint form of its proof. For a general *τ*, we investigate stability and asymptotic properties of solutions on the continuous and discrete time scales separately. Besides a related character of the relevant results, we discuss also a possible related character of their proofs.

**MSC:**34N05, 34K25, 39A12, 39A30.

## Keywords

## 1 Introduction

where ${a}_{j}$ are real scalars, ${\tau}^{j}$ are the *j* th iterates of an increasing function $\tau :\mathbb{T}\to \mathbb{T}$ with ${\tau}^{0}\equiv \mathrm{id}$, ${\tau}^{1}\equiv \tau $, $\tau (t)<t$ for $t\in \mathbb{T}$ and $\mathbb{T}$ is a time scale unbounded above. If $\mathbb{T}$ has a finite minimum *m*, we admit $\tau (m)=m$.

Qualitative analysis of delay dynamic equations on time scales has already been the subject of several investigations (see, *e.g.* [1–4] and [5]). In particular, these papers present techniques which enable a joint analysis of delay differential and difference equations. Besides the methods developed directly for delay dynamic equations, there are also some proof procedures utilised originally either for delay differential or difference equations, but they seem to be applicable without any extra difficulties also to a general dynamic case (see, *e.g.* [6] and [7]).

In this paper, we discuss common stability and asymptotic properties of the delay dynamic equation (1.1). Although (1.1) can be taken for a basic type of delay dynamic equations, its qualitative analysis has not been described yet. It might be surprising because delay dynamic equations studied in the above mentioned papers mostly have more complicated forms. A difficulty connected with qualitative analysis of (1.1) follows from the proof methods utilised in its significant particular cases, which are of a quite different nature, and their unification to a general dynamic case can be very complicated. Nevertheless, considering (1.1) with unbounded lags, we describe the cases when a joint investigation of (1.1) is possible.

This paper is organised as follows. In Section 2 we introduce several crucial particular cases of (1.1). Section 3 is devoted to asymptotic estimation of the dynamic pantograph equation. These estimates are generalised and improved in Section 4. Some final remarks conclude the paper.

## 2 Preliminaries

Throughout this paper we adopt the standard time scale notation. In particular, $\rho :\mathbb{T}\to \mathbb{T}$ is the backward jump operator, $\nu :\mathbb{T}\to {\mathbb{R}}_{0}^{+}$ is the backward graininess and the symbol ${y}^{\mathrm{\nabla}}$ means the nabla derivative of $y:\mathbb{T}\to \mathbb{R}$. For precise introductions of these symbols and other related matters, we refer to [8] and [9].

By a solution of (1.1), we mean a function $y:\mathbb{T}\to \mathbb{R}$ which is ld-continuous on $[{\tau}^{k}({t}_{0}),\mathrm{\infty})$, has an ld-continuous nabla derivative on $[{t}_{0},\mathrm{\infty})$ and satisfies (1.1) on $[{t}_{0},\mathrm{\infty})$ for some ${t}_{0}\in \mathbb{T}$. In this definition (as well as throughout the whole paper), we use the classical interval notation without a specification of $\mathbb{T}$ (*e.g.* $[a,b)$ means $\{t\in \mathbb{T}:a\le t<b\}$).

We prefer to study (1.1) as a nabla dynamic equation (instead of its delta analogue) because, as stated above, our aim is to describe qualitative properties of (1.1) common to different time scales. It is well known from numerical analysis of differential equations that there are just discretisations based on backward (nabla) differences which enable to retain the key properties of underlying differential equations.

*i.e.*time scales with a nonzero graininess) it represents a considerable restriction. More precisely, a concrete choice of

*τ*significantly influences the form of $\mathbb{T}$ (and

*vice versa*). Therefore we distinguish the following cases.

- (a)
Equation (1.1) with constant lags.

*h*equal to 1; hence without loosing a generality, we put $h=1$. Then, for $\mathbb{T}=\mathbb{R}$, (1.1) becomes the linear autonomous differential equation with multiple delays

which is a linear autonomous higher-order difference equation (both terms on its left-hand side can be obviously joined to the right-hand side to obtain the standard general form of such an equation, see, *e.g.* [11]).

*y*is tending to zero as $t\to \mathrm{\infty}$. The standard way to investigate this property originates from the zero analysis of characteristic polynomials. If $\mathbb{T}=\mathbb{R}$ then substituting $y(t)=exp(\lambda t)$ into (2.2), we are led to the quasipolynomial

*e.g.*[12]). The matter concerning zeros of $Q(\lambda )$ is a classical polynomial problem. In a theoretical level, it can be solved by use of the Schur-Cohn test or the Jury criterion (see [13]). However, a formulation of explicit necessary and sufficient conditions (in terms of ${a}_{j}$ and

*k*) is still an open problem. From this viewpoint, a unification of both procedures and formulation of explicit necessary and sufficient conditions for the asymptotic stability of the delay dynamic equation

- (b)
Equation (1.1) with proportional lags.

*q*-analogue

*e.g.*[14] and [15]), which is, however, not applicable to the corresponding dynamic equation

- (c)
Equation (1.1) with general lags.

*τ*is not specified, then (2.1) does not imply any concrete restriction. Nevertheless, previous special dynamic equations (2.4) and (2.7) can inspire us to meet the condition (2.1) ‘implicitly’ by considering the dynamic equation

instead of (1.1). Indeed, if $\mathbb{T}=\mathbb{Z}$ then (2.8) becomes (2.3), and for $\mathbb{T}=\overline{{q}^{\mathbb{Z}}}$ (2.8) becomes (2.6). In other words, the role of delayed arguments in (2.8) is played by the backward jump operator and its iterates. In Section 4, we give a precise asymptotic description for the solutions of (2.8), including an asymptotic stability condition.

## 3 A dynamic equation with proportional delays

*i.e.*$\tilde{Q}({\lambda}_{\ast})=0$. Then it is easy to check that ${\tilde{Q}}^{\prime}({\lambda}_{\ast})<0$, which along with $\tilde{Q}(0)>0$ implies that such a positive zero must be unique. We denote it by ${\lambda}_{r}$ and add that $0<{\lambda}_{r}<1$ if and only if

Using this notation, we have the following.

**Theorem 3.1**

*Let*

*y*

*be a solution of*(2.7),

*where*${a}_{0}<0$, ${a}_{k}\ne 0$, $q>1$

*and let*$\mathbb{T}={\mathbb{R}}_{0}^{+}$

*or*$\mathbb{T}=\overline{{q}^{\mathbb{Z}}}$.

*Then*

*Proof*All necessary formulae of the basic time scales calculus utilised below (such as product rule, quotient rule or integration by parts) can be found in [8], pp.331-333. Similarly, we use the symbol ${\stackrel{\u02c6}{\mathrm{e}}}_{{a}_{0}}(t,{t}_{0})$ as the nabla exponential function satisfying

(for its precise introduction and properties, we refer to [9], pp.49-55). In particular, the assumption ${a}_{0}<0$ implies $1-{a}_{0}\nu (t)>0$ for all $t\in \mathbb{T}$ (positive regressivity), *i.e.* ${\stackrel{\u02c6}{\mathrm{e}}}_{{a}_{0}}(t,{t}_{0})$ is positive on $\mathbb{T}$.

*y*satisfy (2.7) on $[{t}_{0},\mathrm{\infty})$ for some ${t}_{0}\in \mathbb{T}$, ${t}_{0}>0$. Put $z(t)=y(t)/{t}^{r}$, $t\in \mathbb{T}$, $t\ge {q}^{-k}{t}_{0}$. Substituting into (2.7), we have

*i.e.*

*i.e.*

for both the time scales $\mathbb{T}={\mathbb{R}}_{0}^{+}$ and $\mathbb{T}=\overline{{q}^{\mathbb{Z}}}$, hence the product in (3.6) converges as $i\to \mathrm{\infty}$, *i.e.* *z* is bounded. This proves the asymptotic property (3.2).

*i.e.*the case $r\ge 0$, is a simplified version of the previous one. Since ${({t}^{r})}^{\mathrm{\nabla}}$ is nonnegative in such a case, we can deduce from (3.5) that

and using the same way as above, we arrive at ${S}_{i+1}\le {S}_{i}$. □

**Remark 3.2** The assumption $\mathbb{T}={\mathbb{R}}_{0}^{+}$ or $\mathbb{T}=\overline{{q}^{\mathbb{Z}}}$ was employed within the previous proof only in a verification ${({t}^{r})}^{\mathrm{\nabla}}$ is increasing when $r<0$, and in a check on the asymptotic estimate (3.7). All other utilised proof procedures were independent of a given time scale. To the author’s knowledge, the presented result is new for both the time scales (particularly, for the differential equation (2.5) when $\mathbb{T}={\mathbb{R}}_{0}^{+}$).

## 4 A dynamic equation with general delays

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.

*i.e.*we consider the delay differential equation

*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

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*

*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

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

*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).

Since *ρ* is increasing, (4.5) has a positive and increasing solution *φ*.

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*

*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,

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*

*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 Schur-Cohn 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

*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

- (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).

## 5 Concluding remarks

Comparing results of Theorem 4.1 and Theorem 4.3 (resp. Corollary 4.4), one can observe close similarities between stability and asymptotic properties of the differential equation (4.1) and its dynamic discrete analogue (2.8). As regards their proofs, some common ideas have been involved especially in the proof of Theorem 3.1, but Theorems 4.1 and 4.3 themselves had to be proved separately. In particular, the proof procedure of Theorem 4.3 is not applicable to the continuous case, at least in the presented form. Our conjecture is that the asymptotic estimate (4.12), which is slightly stronger that (4.3), is valid (perhaps in a modified form) also for the corresponding differential equation (4.1). In particular, we believe that the zero analysis of $Q(\lambda )$ instead of $\tilde{Q}(\lambda )$ should be involved here similarly as in the discrete case. However, an issue of the joint proof of such a property on a general time scale (including the continuous one) seems to be a difficult task.

## Declarations

### Acknowledgements

The research was supported by the grant P201/11/0768 of the Czech Science Foundation and by the project FSI-S-11-3 of Brno University of Technology. The author is grateful to the referee for his (or her) suggestions and comments.

## Authors’ Affiliations

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