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# Oscillation of second-order nonlinear delay differential equations with nonpositive neutral coefficients

*Advances in Difference Equations*
**volume 2015**, Article number: 35 (2015)

## Abstract

We present several oscillation criteria for a second-order nonlinear delay differential equation with a nonpositive neutral coefficient. Two examples are given to illustrate the main results.

## Introduction

In this work, we study the oscillation of a nonlinear second-order neutral delay differential equation

where \(z(t)=x(t)-p(t)x(\tau(t))\) and \(\alpha>0\) is the ratio of two odd integers. Throughout, we assume that the following hypotheses are satisfied:

- (H
_{1}): -
\(r, p, q\in\mathrm{C}([t_{0}, \infty),\mathbb{R})\), \(r(t)>0\), \(0\leq p(t)\leq p_{0}<1\), \(q(t)\geq0\), and

*q*is not identically zero for large*t*; - (H
_{2}): -
\(\tau\in\mathrm{C}([t_{0}, \infty),\mathbb{R})\), \(\tau (t)\leq t\), and \(\lim_{t\rightarrow\infty}\tau(t)=\infty\);

- (H
_{3}): -
\(\sigma\in\mathrm{C}^{1}([t_{0}, \infty),\mathbb{R})\), \(\sigma'(t)>0\), \(\sigma(t)\leq t\), and \(\lim_{t\rightarrow\infty}\sigma(t)=\infty\);

- (H
_{4}): -
\(f\in\mathrm{C}(\mathbb{R},\mathbb{R})\), \(uf(u)>0\) for all \(u\neq0\), and there exists a positive constant

*k*such that$$\frac{f(u)}{u^{\alpha}}\geq k\quad \text{for all } u\neq0. $$

By a solution to (1.1), we mean a function \(x\in\mathrm{C}([T_{x},\infty),\mathbb{R})\), \(T_{x}\geq t_{0}\) which has the property \(r(z')^{\alpha}\in\mathrm{C}^{1}([T_{x},\infty),\mathbb{R})\) and satisfies (1.1) on the interval \([T_{x},\infty)\). We consider only those solutions of (1.1) which satisfy condition \(\sup\{|x(t)|:t\geq T\}>0\) for all \(T\geq T_{x}\) and assume that (1.1) possesses such solutions. As usual, a solution of (1.1) is called oscillatory if it has arbitrarily large zeros on \([T_{x},\infty)\); otherwise, it is said to be nonoscillatory. Equation (1.1) is termed oscillatory if all its solutions oscillate.

In recent years, there has been increasing interest in studying oscillation of solutions to different classes of differential equations due to the fact that they have numerous applications in natural sciences and engineering; see, *e.g.*, Hale [1] and Wong [2]. In particular, many papers deal with oscillatory behavior of second-order and third-order delay differential equations; see, for instance, [2–15] and the references cited therein.

In what follows, we provide some background details regarding the study of oscillation of second-order differential equations which motivated our study. Oscillation criteria for (1.1) and its particular cases have been reported in [2–5, 9–12, 14, 15]. A commonly used assumption is

although several authors studied the oscillation of (1.1) in the case where

In particular, Wong [2] and Yang *et al.* [15] obtained several oscillation theorems for (1.1) under the assumptions that

and

see also the paper by Qin *et al.* [14] where inaccuracies in [15] were pointed out.

Baculíková and Džurina [6] investigated the asymptotic properties of the couple of third-order neutral differential equations

assuming that

On the basis of the ideas exploited by [6], we derive some new oscillation results for (1.1). In the sequel, all functional inequalities are assumed to hold for all *t* large enough. Without loss of generality, we can deal only with positive solutions of (1.1).

## Lemmas

In this section, we give two lemmas that will be useful for establishing oscillation criteria for (1.1).

### Lemma 2.1

*Let conditions* (H_{1})-(H_{4}) *and* (1.4) *be satisfied and assume that*
*x*
*is a positive solution of* (1.1). *Then*
*z*
*satisfies the following two possible cases*:

- (C
_{1}): -
\(z(t)>0\), \(z'(t)>0\), \((r(t)(z'(t))^{\alpha})'\leq0\);

- (C
_{2}): -
\(z(t)<0\), \(z'(t)>0\), \((r(t)(z'(t))^{\alpha})'\leq0\),

*for*\(t\geq t_{1}\),

*where*\(t_{1}\geq t_{0}\)

*is sufficiently large*.

### Proof

Suppose that there exists a \(t_{1}\geq t_{0}\) such that \(x(t)>0\), \(x(\tau(t))> 0\), and \(x(\sigma(t))> 0\) for \(t\geq t_{1}\). It follows from (1.1) that

Hence, \(r(z')^{\alpha}\) is nonincreasing and of one sign. That is, there exists a \(t_{2}\geq t_{1}\) such that \(z'(t)>0\) or \(z'(t)<0\) for \(t\geq t_{2}\).

If \(z'(t)>0\) for \(t\geq t_{2}\), then we have (C_{1}) or (C_{2}). We prove now that \(z'(t)<0\) cannot occur. If \(z'(t)<0\) for \(t\geq t_{2}\), then

for \(t\geq t_{2}\), where \(c=-r(t_{2})(z'(t_{2}))^{\alpha}>0\). Thus, we conclude that

By virtue of condition (1.4), \(\lim_{t\rightarrow\infty}{z(t)}=-\infty\). We consider now the following two cases separately.

Case 1. If *x* is unbounded, then there exists a sequence \(\{t_{k}\}\) such that \(\lim_{k\rightarrow\infty}{t_{k}}=\infty\) and \(\lim_{k\rightarrow\infty}{x(t_{k})}=\infty\), where \(x(t_{k})=\max\{x(s); t_{0}\leq s \leq t_{k}\}\). Since \(\lim_{t\rightarrow\infty}\tau(t)=\infty\), \(\tau(t_{k})>t_{0}\) for all sufficiently large *k*. By \(\tau(t)\leq t\),

Therefore, for all large *k*,

which contradicts the fact that \(\lim_{t\rightarrow\infty}{z(t)}=-\infty\).

Case 2. If *x* is bounded, then *z* is also bounded, which contradicts \(\lim_{t\rightarrow\infty}{z(t)}=-\infty\). Hence, *z* satisfies one of the cases (C_{1}) and (C_{2}). This completes the proof. □

### Lemma 2.2

*Assume that*
*x*
*is a positive solution of* (1.1) *and*
*z*
*satisfies case* (C_{2}). *Then*

### Proof

By \(z<0\) and \(z'>0\), we deduce that

where *l* is a finite constant. That is, *z* is bounded. As in the proof of Case 1 in Lemma 2.1, *x* is also bounded. Using the fact that *x* is bounded, we obtain

We claim that \(a=0\). If \(a>0\), then there exists a sequence \(\{t_{m}\}\) such that \(\lim_{m\rightarrow\infty}{t_{m}}=\infty\) and \(\lim_{m\rightarrow\infty}{x(t_{m})}=a\). Let \(\varepsilon=a(1-p_{0})/2p_{0}\). Then for all large *m*, \(x(\tau(t_{m}))< a+\varepsilon\), and so

which is a contradiction. Thus, \(a=0\) and \(\lim_{t\rightarrow\infty}{x(t)}=0\). The proof is complete. □

## Oscillation results

In what follows, we denote

where \(t_{1}\geq t_{0}\) is sufficiently large.

### Theorem 3.1

*Let conditions* (H_{1})-(H_{4}) *and* (1.4) *be satisfied*. *If there exists a positive function*
\(\rho\in\mathrm{C}^{1}([t_{0}, \infty),\mathbb{R})\)
*such that*, *for all sufficiently large*
\(t_{1}\geq t_{0}\),

*where*
\(\rho'_{+}(t)=\max\{0,\rho'(t)\}\), *then every solution*
*x*
*of* (1.1) *is either oscillatory or satisfies*
\(\lim_{t\rightarrow\infty}x(t)=0\).

### Proof

Without loss of generality, we may assume that there exists a \(t_{1}\geq t_{0}\) such that \(x(t)>0\), \(x(\tau(t))>0\), and \(x(\sigma(t))>0\) for \(t\geq t_{1}\). Then we have (2.1). From Lemma 2.1, *z* satisfies one of the cases (C_{1}) and (C_{2}). We consider each of two cases separately.

Suppose first that case (C_{1}) holds. By the definition of *z*,

Using (2.1) and condition \(\sigma(t)\leq t\), we conclude that

It follows now from (2.1) that

We define a function *ω* by

Then \(\omega(t)>0\) for \(t\geq t_{1}\) and

Using (2.1) and (3.2)-(3.5), we get

Integrating (3.6) from \(t_{2}\) (\(t_{2}>t_{1}\)) to *t*, we obtain

which contradicts (3.1).

If *z* satisfies (C_{2}), then \(\lim_{t\rightarrow\infty}{x(t)}=0\) due to Lemma 2.2. The proof is complete. □

Let \(\rho(t)=1\). We can obtain the following criterion for (1.1) using Theorem 3.1.

### Corollary 3.1

*Let conditions* (H_{1})-(H_{4}) *and* (1.4) *be satisfied*. *If*

*then the conclusion of Theorem *
3.1
*remains intact*.

### Theorem 3.2

*Let*
\(\alpha\geq1\)
*hold and conditions* (H_{1})-(H_{4}) *and* (1.4) *be satisfied*. *If there exists a positive function*
\(\rho\in\mathrm{C}^{1}([t_{0}, \infty),\mathbb{R})\)
*such that*, *for all sufficiently large*
\(t_{1}\geq t_{0}\),

*where*
\(\rho'_{+}(t)=\max\{0,\rho'(t)\}\), *then the conclusion of Theorem *
3.1
*remains intact*.

### Proof

As above, suppose that *x* is a positive solution of (1.1). By virtue of Lemma 2.1, *z* satisfies one of (C_{1}) and (C_{2}). We discuss each of the two cases separately.

Assume first that *z* has property (C_{1}). We obtain (3.3) and (3.4). Define now *ω* as in the proof of Theorem 3.1. Then \(\omega>0\) and

On the other hand, by (3.3) and (3.4),

Substituting (3.9) into (3.8), we obtain

Integrating (3.10) from \(t_{2}\) (\(t_{2}>t_{1}\)) to *t*, we have

which contradicts (3.7).

If *z* satisfies (C_{2}), then \(\lim_{t\rightarrow\infty}{x(t)}=0\) when using Lemma 2.2. This completes the proof. □

## Examples and discussion

### Example 4.1

For \(t\geq1\), consider a second-order neutral differential equation

It follows from Corollary 3.1 that every solution *x* of (4.1) is either oscillatory or satisfies property \(\lim_{t\rightarrow\infty}x(t)=0\). For instance, \(x(t)=\sin4t\) is an oscillatory solution of this equation.

### Example 4.2

For \(t\geq1\), consider a second-order nonlinear neutral differential equation

where \(z(t)=x(t)-x(t/3)/2\) and \(\gamma>0\) is a constant. Let \(\alpha=3\), \(r(t)=t^{2}\), \(q(t)=t^{-2}\), \(\sigma(t)=t/2\), \(k=\gamma\), and \(\rho(t)=t\), and note that \(\xi(t)\geq3k_{0}t\) for every \(k_{0}\in(0,1)\). By virtue of Theorem 3.2, every solution *x* of (4.2) is either oscillatory or satisfies \(\lim_{t\rightarrow\infty}x(t)=0\) if \(\gamma>1/(54k_{0}^{2})\) for some \(k_{0}\in(0,1)\). However, it follows from Theorem 3.1 that every solution *x* of (4.2) is either oscillatory or satisfies \(\lim_{t\rightarrow\infty}x(t)=0\) if \(\gamma>2/(27k_{0}^{3})\) for some \(k_{0}\in(0,1)\). Note that

for every \(k_{0}\in(0,1)\). Therefore, Theorem 3.2 improves Theorem 3.1 in some cases. But Theorem 3.1 can be applied to (1.1) in the case when \(0<\alpha<1\). Observe that results reported in [2, 14, 15] cannot be applied to (4.2) since (1.3) fails to hold for this equation.

### Remark 4.1

We establish two classes of oscillation criteria for (1.1) without requiring the restrictive conditions (1.3). Note that these results are based on the assumption (1.2) and, as fairly noticed by one of the referees, these results cannot be applied to the case where \(p(t)=1\).

### Remark 4.2

Note that Theorems 3.1 and 3.2 and Corollary 3.1 guarantee that every solution *x* of (1.1) is either oscillatory or satisfies \(\lim_{t\rightarrow\infty}x(t)=0\) and, unfortunately, these results cannot distinguish solutions with different behaviors. Since the sign of *z* is not known, it is not easy to establish sufficient conditions which ensure that all solutions of (1.1) are just oscillatory and do not satisfy \(\lim_{t\rightarrow\infty}x(t)=0\).

### Remark 4.3

On the basis of Remarks 4.1 and 4.2, two interesting problems for future research can be formulated as follows:

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## Acknowledgements

The authors are grateful to the editors and three referees for a very thorough reading of the manuscript and for pointing out several inaccuracies. This research is supported by NNSF of P.R. China (Grant No. 61403061), NSF of Shandong Province (Grant No. ZR2012FL06), and the AMEP of Linyi University, P.R. China.

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## Additional information

### Competing interests

The authors declare that they have no competing interests.

### Authors’ contributions

All four authors contributed equally to this work. They all read and approved the final version of the manuscript.

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### MSC

- 34K11

### Keywords

- oscillation
- second-order neutral equation
- nonlinear delay differential equation