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# Notes on oscillation of linear delay differential equations

*Advances in Difference Equations*
**volume 2018**, Article number: 162 (2018)

## Abstract

This paper deals with the oscillation criteria for the linear delay differential equations. We present new sufficient conditions for the oscillation of all solutions of such equations. The results improve and complement some earlier ones in the literature.

## 1 Introduction

In this article we consider the linear delay differential equation

where the functions \(p, \tau \in C([t_{0},\infty),(0,\infty)), \tau (t)\) is nondecreasing, \(\tau (t)< t\) for \(t\ge t_{0}\) and \(\lim_{t\to \infty }\tau (t)=\infty \).

Our aim is to establish new sufficient conditions for the oscillation of all solutions of Eq. (1). This problem has been recently investigated by many authors. See, for example, [1–14] and the references cited therein.

A continuously differentiable function defined on \([\tau (T _{0}),\infty)\) for some \(T_{0}\ge t_{0}\) and satisfying Eq. (1) for \(t\ge T_{0}\) is called a solution of Eq. (1). Such a solution is called oscillatory if it has arbitrarily large zeros. Otherwise, it is called nonoscillatory.

We will denote by *k* the lower limit of the average

as \(t\to \infty \), i.e.,

and constant *L* is defined by

We refer the readers to the papers [6, 13] for the historical and chronological review of the results. We mention some results in the literature (cf. [6, 13]) for the purpose of this article. In 1972 Ladas, Lakshmikantham, and Papadakis [11] proved that if the following holds

then every solution of Eq. (1) oscillates.

In 1979 Ladas [10] and in 1982 Koplatadze and Chanturija [7] showed that the same conclusion holds if

We point out that if the inequality

holds eventually, then, according to a result in [7], Eq. (1) has a nonoscillatory solution.

It is obvious that there is a gap between conditions (2) and (3) when the limit

does not exist. How to fill this gap is an interesting problem which has been recently investigated by several authors, e.g., [1–6, 8, 9, 12–15].

In the case \(0< k\le 1/e\), all conditions in the papers [4–6, 8, 9, 12–14] are dependent on \(0< L<1\). The aim of this article is to establish such conditions for oscillation of solutions of Eq. (1) which are independent of *L*.

We assume for the analysis of asymptotic behavior of the function

that Eq. (1) has a solution \(x(t)\) which is positive for all large *t*.

In the second section we will use the next lemma by Jaroš and Stavroulakis [5].

### Lemma 1.1

([5])

*Suppose that*
\(k>0\)
*and Eq*. (1) *has an eventually positive solution*
\(x(t)\). *Then*
\(k\le 1/e\)
*and*

*where*
\(\lambda_{1}\)
*is the smaller and*
\(\lambda_{2}\)
*is the greater root of the equation*
\(\lambda =e^{k\lambda }\).

## 2 Oscillatory properties

In this section we will study the oscillatory properties of Eq. (1).

### Lemma 2.1

*Let*
\(x(t)\)
*be an eventually positive solution of Eq*. (1) *and*
\(0< k\le \frac{1}{e}\). *Suppose that*

*where*
\(\beta \in [\lambda_{1}, \lambda_{2}]\), *and*
\(\lambda_{1}\)
*is the smaller and*
\(\lambda_{2}\)
*is the greater root of the equation*
\(\lambda =e^{k\lambda }\). *Then*

### Proof

Let \(t>t_{0}\) be large enough so that \(\tau (t)>t_{0}\). Integrating (1) from \(\tau (t)\) to *t*, we obtain

Let \(0<\lambda <\lambda_{1}\). Then the function

is decreasing for appropriate \(t_{1}\ge t_{0}\) (cf. [6, 13]). Indeed by Lemma 1.1

for \(t\ge t_{2}\), where \(t_{2}\ge t_{1}\) is sufficiently large, and consequently,

which implies \(\varphi '(t)<0\) for \(t\ge t_{2}\). Substituting (6) into (5), we derive for \(t\ge t_{2}\) that

From (4) it follows that there exists a constant *c* such that \(c>1-\frac{1-\varepsilon }{\beta }\), where \(0<\varepsilon <\beta [c-(1-\frac{1}{ \beta })]\leq 1\), and

Then, for *λ* sufficiently close to \(\lambda_{1}\), we get

where \(t_{3}\ge t_{2}\) is sufficiently large. If it is not true, then for all \(0<\lambda <\lambda_{1}\) we have

By letting \(\lambda \rightarrow \lambda_{1}\), the last inequality leads to

This inequality contradicts (8). Thus we obtain from (7)

Therefore, we have

The proof is complete. □

### Theorem 2.1

*Let*
\(0< k\le \frac{1}{e}\). *Suppose that*

*Then all solutions of Eq*. (1) *oscillate*.

### Proof

Assume that Eq. (1) eventually has a positive solution \(x(t)\). It follows from Lemma 2.1 that

This contradicts the result of Lemma 1.1 and completes the proof of the theorem. □

### Theorem 2.2

*Let*
\(0< k\le \frac{1}{e}\). *Suppose that there exists*
\(\gamma \in ( \lambda_{1},\lambda_{2})\)
*such that*

*Then all solutions of Eq*. (1) *oscillate*.

### Proof

Assume that Eq. (1) has an eventually positive solution \(x(t)\). By Lemma 2.1, condition (9) implies that

With regard to condition (10) and repeating the procedure as in the proof of Lemma 2.1, we get

This contradicts the result of Lemma 1.1. The proof is complete. □

In the next example we observe the case \(k=1/e\). Then \(\lambda_{1}= \lambda_{2}=e\).

### Example

Consider the linear delay differential equation

where \(p>0\), \(a>0\).

Then

We get

Then we have

We conclude that

and by Theorem 2.1 all solutions of Eq. (11) oscillate.

We point out that the results in [6, 13] are dependent on the constant \(0< L<1\), while our results do not depend on the constant *L*.

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## Acknowledgements

The authors gratefully acknowledge the Scientific Grant Agency VEGA of the Ministry of Education of Slovak Republic and the Slovak Academy of Sciences for supporting this work under the Grant No. 1/0812/17. The authors would like to thank the anonymous referees for their valuable comments.

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Dorociaková, B., Chupáč, R. & Olach, R. Notes on oscillation of linear delay differential equations.
*Adv Differ Equ* **2018**, 162 (2018). https://doi.org/10.1186/s13662-018-1608-9

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DOI: https://doi.org/10.1186/s13662-018-1608-9