Some qualitative properties of linear dynamic equations with multiple delays
© Čermák; licensee Springer 2013
Received: 4 February 2013
Accepted: 9 May 2013
Published: 31 May 2013
This paper discusses stability and asymptotic properties of the delay dynamic equation
where are scalars, are iterates of a function and 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.
where are real scalars, are the j th iterates of an increasing function with , , for and is a time scale unbounded above. If has a finite minimum m, we admit .
Qualitative analysis of delay dynamic equations on time scales has already been the subject of several investigations (see, e.g. [1–4] and ). 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.  and ).
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.
Throughout this paper we adopt the standard time scale notation. In particular, is the backward jump operator, is the backward graininess and the symbol means the nabla derivative of . For precise introductions of these symbols and other related matters, we refer to  and .
By a solution of (1.1), we mean a function which is ld-continuous on , has an ld-continuous nabla derivative on and satisfies (1.1) on for some . In this definition (as well as throughout the whole paper), we use the classical interval notation without a specification of (e.g. means ).
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.
Equation (1.1) with constant lags.
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. ).
Equation (1.1) with proportional lags.
Equation (1.1) with general lags.
instead of (1.1). Indeed, if then (2.8) becomes (2.3), and for (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
Using this notation, we have the following.
(for its precise introduction and properties, we refer to , pp.49-55). In particular, the assumption implies for all (positive regressivity), i.e. is positive on .
for both the time scales and , hence the product in (3.6) converges as , i.e. z is bounded. This proves the asymptotic property (3.2).
and using the same way as above, we arrive at . □
Remark 3.2 The assumption or was employed within the previous proof only in a verification is increasing when , 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 ).
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.
which is called the Schröder equation. Let , , , for all , is positive and nonincreasing on I and . Then there exists a solution of (4.2), which is positive on and has a positive and nonincreasing derivative on I. For this and other relevant results on the Schröder equation, we refer to .
In the next assertion, we assume that τ and φ have the properties stated above and has the same meaning as in Theorem 3.1 (see the discussion preceding this theorem).
The remaining parts of the proof are identical with those stated above. □
i.e. the lags , are unbounded as .
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 of all complex zeros of with the modulus θ. If is nonempty for a given θ, then by a characteristic solution of (4.7) corresponding to , we understand a finite sum of solutions of (4.7) corresponding to all values . Of course, the form of such solutions depends on multiplicity of λ and it is described in details e.g. in .
Using this we have the following theorem.
where ω is a nontrivial characteristic solution of (4.7) corresponding to , φ is a positive and increasing solution of (4.5) and is a suitable real scalar.
where ω is a nontrivial characteristic solution of (4.7) corresponding to . Now, using the backward substitution, we obtain (4.8). □
Let κ be a zero of and be its multiplicity. We call this zero maximal if κ is (among all other zeros of ) maximal in the modulus and if multiplicities of other possible zeros of having the maximal modulus do not exceed . Of course, may have several maximal zeros (with the same modulus and multiplicity).
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 ) if all the zeros of are located inside the unite circle. In this connection, we have already mentioned the Schur-Cohn criterion, which can be applied to any polynomial with concrete (fixed) coefficients and order. However, this criterion does not enable us to formulate explicit stability conditions in terms of (general) coefficients and k. Such explicit conditions are known only in a very few particular cases (see, e.g. [22, 23] and ).
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 , we put ).
- (i)If (more precisely ) and , then by Theorem 4.1(4.15)
- (ii)If , then (4.5) becomes (4.14) and by Theorem 4.3
where and are suitable scalars.
Remark 4.7 Equation (4.13) with has been studied in  as the differential equation with advanced power argument, i.e. when , (for extensions to the case of a general advanced argument τ, see also ). 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 instead of 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.
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.
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