# Asymptotic behavior of an odd-order delay differential equation

- Tongxing Li
^{1}and - Yuriy V Rogovchenko
^{2}Email author

**2014**:107

https://doi.org/10.1186/1687-2770-2014-107

© Li and Rogovchenko; licensee Springer. 2014

**Received: **29 January 2014

**Accepted: **22 April 2014

**Published: **9 May 2014

## Abstract

We study asymptotic behavior of solutions to a class of odd-order delay differential equations. Our theorems extend and complement a number of related results reported in the literature. An illustrative example is provided.

**MSC:**34K11.

## Keywords

## 1 Introduction

Professor Ivan Kiguradze is widely recognized as one of the leading contemporary experts in the qualitative theory of ordinary differential equations. His research has been partly summarized in the monograph written jointly with Professor Chanturia [1] where many fundamental results on the asymptotic behavior of solutions to important classes of nonlinear differential equations were collected. In particular, the Kiguradze lemma and Kiguradze classes of solutions are well known to researchers working in the area and are extensively used to advance the knowledge further.

where $t\ge {t}_{0}>0$ and $n\ge 3$ is an odd natural number, $\gamma >0$ is a ratio of odd natural numbers, $r\in {\mathrm{C}}^{1}([{t}_{0},\mathrm{\infty}),\mathbb{R})$, $p,q,g\in \mathrm{C}([{t}_{0},\mathrm{\infty}),\mathbb{R})$, $r(t)>0$, ${r}^{\prime}(t)+p(t)\ge 0$, $p(t)\ge 0$, $q(t)>0$, $g(t)\le t$, and ${lim}_{t\to \mathrm{\infty}}g(t)=\mathrm{\infty}$.

By a solution of (1.1) we mean a function $x\in \mathrm{C}([{T}_{x},\mathrm{\infty}),\mathbb{R})$, ${T}_{x}\ge {t}_{0}$, such that $r{({x}^{(n-1)})}^{\gamma}\in {\mathrm{C}}^{1}([{T}_{x},\mathrm{\infty}),\mathbb{R})$ and $x(t)$ satisfies (1.1) on $[{T}_{x},\mathrm{\infty})$. We consider only those extendable solutions of (1.1) that do not vanish eventually, that is, condition $sup\{|x(t)|:t\ge T\}>0$ holds for all $T\ge {T}_{x}$. We tacitly assume that (1.1) possesses such solutions. As customary, a solution of (1.1) is said to be oscillatory if it has arbitrarily large zeros on the ray $[{T}_{x},\mathrm{\infty})$; otherwise, we call it non-oscillatory.

Analysis of the oscillatory and non-oscillatory behavior of solutions to different classes of differential and functional differential equations has always attracted interest of researchers; see, for instance, [1–19] and the references cited therein. One of the main reasons for this lies in the fact that delay differential equations arise in many applied problems in natural sciences, technology, and automatic control, *cf.*, for instance, Hale [20]. In particular, (1.1) may be viewed as a special case of a more general class of higher-order differential equations with a one-dimensional *p*-Laplacian, which, as mentioned by Agarwal *et al.* [4], have applications in continuum mechanics.

*et al.*[11], Zhang

*et al.*[16], and Zhang

*et al.*[18] studied oscillation of (1.1) assuming that $n\ge 2$ is even, $g(t)\le t$, and

which was studied by Zhang *et al.* [17] who established the following result.

**Theorem 1.1** ([[17], Corollary 2.1])

*Let*

*and assume that*$\delta ({t}_{0})<\mathrm{\infty}$.

*Suppose also that*

*and*,

*for some*${\lambda}_{1}\in (0,1)$,

*Then every solution of* (1.3) *is either oscillatory or converges to zero as* $t\to \mathrm{\infty}$.

*n*odd. Furthermore, in this case the methods in [11, 18] which employ Riccati substitutions cannot be applied to the analysis of (1.1). Therefore, the objective of this paper is to extend the techniques exploited in [17] to the study of (1.1) in the case when the integral in (1.2) is finite, that is, for all$\phantom{\rule{0.25em}{0ex}}{T}_{1}\ge T\ge {t}_{0}$,

As usual, all functional inequalities considered in this paper are supposed to hold for all *t* large enough. Without loss of generality, we may deal only with positive solutions of (1.1), because under our assumption that *γ* is a ratio of odd natural numbers, if $x(t)$ is a solution of (1.1), so is $-x(t)$.

## 2 Main results

We need the following auxiliary lemmas.

**Lemma 2.1**

*Assume that*(1.2)

*is satisfied and let*$x(t)$

*be an eventually positive solution of*(1.1).

*Then there exists a sufficiently large*${t}_{1}\ge {t}_{0}$

*such that*,

*for all*$t\ge {t}_{1}$,

*Proof*Let $x(t)$ be an eventually positive solution of (1.1). Then there exists a ${T}_{0}\ge {t}_{0}$ such that $x(t)>0$ and $x(g(t))>0$ for all $t\ge {T}_{0}$. By virtue of (1.1),

is decreasing for $t\ge {T}_{0}$. Therefore, ${x}^{(n-1)}(t)$ does not change sign eventually, that is, there exists a ${t}_{1}\ge {T}_{0}$ such that either ${x}^{(n-1)}(t)>0$ or ${x}^{(n-1)}(t)<0$ for all $t\ge {t}_{1}$.

*t*, ${T}_{1}\ge T$, we conclude that

which implies that ${x}^{(n)}(t)<0$. This completes the proof. □

**Lemma 2.2** (Agarwal *et al.* [3])

*Assume that*$u\in {\mathrm{C}}^{n}([{t}_{0},\mathrm{\infty}),{\mathbb{R}}^{+})$, ${u}^{(n)}(t)$

*is non*-

*positive for all large*

*t*

*and not identically zero on*$[{t}_{0},\mathrm{\infty})$.

*If*${lim}_{t\to \mathrm{\infty}}u(t)\ne 0$,

*then for every*$\lambda \in (0,1)$,

*there exists a*${t}_{\lambda}\in [{t}_{0},\mathrm{\infty})$

*such that*

*holds on* $[{t}_{\lambda},\mathrm{\infty})$.

**Lemma 2.3** (Agarwal *et al.* [4])

*The equation*

*where*$\gamma >0$

*is a quotient of odd natural numbers*, $r\in {\mathrm{C}}^{1}([{t}_{0},\mathrm{\infty}),(0,\mathrm{\infty}))$,

*and*$a\in \mathrm{C}([{t}_{0},\mathrm{\infty}),\mathbb{R})$

*is non*-

*oscillatory if and only if there exist a number*$T\ge {t}_{0}$

*and a function*$v\in {\mathrm{C}}^{1}([T,\mathrm{\infty}),\mathbb{R})$

*such that*,

*for all*$t\ge T$,

**Theorem 2.4**

*Assume that*

*Then every solution*$x(t)$

*of*(1.1)

*is either oscillatory or satisfies*

*provided that either*

- (i)
(1.2)

*holds or* - (ii)(1.4)
*is satisfied and*,*for some*${\lambda}_{1}\in (0,1)$,$\underset{t\to \mathrm{\infty}}{lim\hspace{0.17em}sup}{\int}_{{t}_{0}}^{t}[q(s){(\frac{{\lambda}_{1}}{(n-2)!}{g}^{n-2}(s)\delta (s))}^{\gamma}E({t}_{0},s)-\phi (s)]\phantom{\rule{0.2em}{0ex}}\mathrm{d}s=\mathrm{\infty}.$(2.6)

*Proof*Assume that (1.1) has a non-oscillatory solution $x(t)$ which is eventually positive and such that

*t*. Let

However, it follows from the result due to Werbowski [[15], Corollary 1] that the latter inequality does not have positive solutions under the assumption (2.4), which is a contradiction. The proof of part (i) is complete.

*t*to

*T*, we obtain

*t*. Therefore, (2.11) yields

*t*, we have

*φ*, we derive from (2.11) that

which contradicts (2.6). This completes the proof for the part (ii). □

**Remark 2.5** For a result similar to the one established in part (i) in Theorem 2.4, see also Zhang *et al.* [[16], Theorem 5.3].

**Remark 2.6** For $p(t)=0$, Theorem 2.4 includes Theorem 1.1.

In the remainder of this section, we use different approaches to arrive at the conclusion of Theorem 2.4. First, we employ the integral averaging technique to replace assumption (2.6) with a Philos-type condition.

*H*has a non-positive continuous partial derivative $\partial H/\partial s$ with respect to the second variable satisfying the condition

for some function $\xi \in {L}_{\mathrm{loc}}(\mathbb{D},\mathbb{R})$.

**Theorem 2.7**

*Let*$\delta (t)$

*be as in Theorem*2.4

*and suppose that*(1.4)

*and*(2.4)

*hold*.

*Assume that there exists a function*$H\in {\mathcal{P}}_{\gamma}$

*such that*

*for all* ${t}_{1}\ge {t}_{0}$ *and for some* ${\lambda}_{1}\in (0,1)$. *Then the conclusion of Theorem* 2.4 *remains intact*.

*Proof*Assuming that $x(t)$ is an eventually positive solution of (1.1) that satisfies (2.7) and proceeding as in the proof of Theorem 2.4, we arrive at the inequality (2.12) which holds for all $\lambda \in (0,1)$. Multiplying (2.12) by $H(t,s)$ and integrating the resulting inequality from ${t}_{1}$ to

*t*, we obtain

which contradicts assumption (2.13). This completes the proof. □

Finally, we formulate also a comparison result for (1.1) that leads to the conclusion of Theorem 2.4.

**Theorem 2.8**

*Let*$\delta (t)$

*be as above*,

*and assume that*(1.4)

*and*(2.4)

*hold*.

*If a second*-

*order half*-

*linear ordinary differential equation*

*is oscillatory for some* ${\lambda}_{1}\in (0,1)$, *then the conclusion of Theorem* 2.4 *remains intact*.

*Proof* Assuming again that $x(t)$ is an eventually positive solution of (1.1) that satisfies (2.7) and proceeding as in the proof of Theorem 2.4, we obtain (2.12) which holds for all $\lambda \in (0,1)$. By virtue of Lemma 2.3, we conclude that (2.14) is non-oscillatory, which is a contradiction. The proof is complete. □

## 3 Example

The following example illustrates possible applications of theoretical results obtained in the previous section.

**Example 3.1**For $t\ge 1$, consider the third-order differential equation

for some ${\lambda}_{1}\in (0,1)$. Hence, by Theorem 2.4, every solution of (3.1) is either oscillatory or satisfies (2.5). As a matter of fact, $x(t)={\mathrm{e}}^{-t}$ is a solution of this equation satisfying condition (2.5).

**Remark 3.2** Note that Theorems 2.4, 2.7, and 2.8 ensure that every solution $x(t)$ of (1.1) is either oscillatory or satisfies (2.5) and, unfortunately, these results cannot distinguish solutions with different behaviors. Since the sign of the derivative ${x}^{\prime}(t)$ is not known, it is difficult to establish sufficient conditions which guarantee that all solutions of (1.1) are just oscillatory and do not satisfy (2.5). Neither is it possible to use the technique exploited in this paper for proving that all solutions of (1.1) satisfy (2.5). Therefore, these two interesting problems remain for future research.

## Declarations

### Acknowledgements

The authors express their sincere gratitude to both anonymous referees for the careful reading of the original manuscript and useful comments that helped to improve the presentation of the results and accentuate important details.

## Authors’ Affiliations

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