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# Oscillation of forced second-order neutral delay differential equations

*Advances in Difference Equations*
**volumeÂ 2015**, ArticleÂ number:Â 223 (2015)

## Abstract

The objective of this paper is to study oscillation of a forced second-order neutral differential equation. By using the generalized Riccati substitution and integral technique, a new sufficient condition is obtained which insures that all solutions to the studied equation are oscillatory. An illustrative example is included.

## 1 Introduction

In this paper, we are concerned with the oscillation of a forced second-order nonlinear neutral differential equation

where \(t\geq t_{0}>0\), \(m\geq1\), and \(l\geq1\) are integers. We suppose that the following assumptions are satisfied:

- (A
_{1}): -
\(r\in\mathrm{C}^{1}([t_{0}, \infty),(0, \infty))\), \(P, Q_{i}, R_{j}\in\mathrm{C}([t_{0}, \infty),[0, \infty))\), \(f_{i}, g_{j}\in \mathrm{C}(\mathbb{R},\mathbb{R})\), \(yf_{i}(y)>0\), and \(yg_{j}(y)>0\) for \(y\neq0\), \(i=1,2,\ldots,m\), and \(j=1,2,\ldots,l\);

- (A
_{2}): -
\(\tau\in\mathrm{C}([t_{0}, \infty),\mathbb{R})\), \(\tau (t)\leq t\), and \(\lim_{t\rightarrow\infty}\tau(t)=\infty\);

- (A
_{3}): -
there exist constants \(\alpha_{i}>0\) and \(\beta_{j}>0\) such that \({f_{i}(y)}/{y} \geq\alpha_{i}\) and \({g_{j}(y)}/{y} \geq\beta_{j}\) for \(y\neq0\), \(i=1,2,\ldots,m\), and \(j=1,2,\ldots,l\);

- (A
_{4}): -
for any \(T\geq t_{0}\), there exist \(T\leq s_{1}< t_{1}\leq s_{2}< t_{2}\) such that

$$F(t)\left \{ \textstyle\begin{array}{@{}l@{\quad}l} \leq0, & t\in[s_{1}, t_{1}],\\ \geq0, & t\in[s_{2}, t_{2}], \end{array}\displaystyle \right . $$and

$$ \sum^{l}_{j=1} \beta_{j}R_{j}(t)\geq\sum^{m}_{i=1} \alpha_{i}Q_{i}(t)P(t),\quad t\in[s_{1}, t_{1}]\cup[s_{2}, t_{2}]. $$(1.2)

Throughout the paper, we define

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

As is well known, the study of qualitative theory of differential equations is of importance both in theory and applications. For instance, the problems of oscillatory behavior of neutral differential equations have a number of practical applications in the study of distributed networks containing lossless transmission lines which arise in high-speed computers where the lossless transmission lines are used to interconnect switching circuits. For some related contributions on oscillation of various classes of neutral differential equations, we refer the reader to [1â€“23] and the references cited therein.

In what follows, we provide some background details that motivated our study. El-Sayed [4] and Wong [19] investigated the second-order forced linear differential equation

Zhang *et al.* [22] studied a second-order neutral differential equation

where \(Q_{1}\) and \(Q_{2}\) are nonnegative functions. Equation (1.4) is a special case of (1.1). In the sequel, using a generalized Riccati substitution which differs from those exploited in [4, 19, 22], a new oscillation criterion for (1.1) is presented. Furthermore, an illustrative example is provided.

## 2 Main results

### Theorem 2.1

*Assume that conditions* (A_{1})-(A_{4}) *are satisfied and let*
\(B_{k}=\lbrace u\in\mathrm{C}^{1}[s_{k}, t_{k}]: u(t)\not\equiv0, u(s_{k})=u(t_{k})=0\rbrace\), \(k=1\), 2. *If there exist functions*
\(u\in B_{k}\), \(\rho\in\mathrm{C}^{1}([t_{0}, \infty), (0, \infty))\), *and*
\(\sigma\in\mathrm{C}^{1}([t_{0}, \infty), \mathbb{R})\)
*such that*, *for*
\(k=1\), 2,

*then every solution of* (1.1) *is oscillatory*.

### Proof

Suppose that *x* is a nonoscillatory solution of (1.1) which is eventually positive. Then *z* defined by (1.3) is also eventually positive. Using (A_{4}), for any \(T\geq t_{0}\), there exist \(t_{1}>s_{1}\geq T\) such that \(F(t)\leq0\) for \(t\in[s_{1}, t_{1}]\). From (A_{3}), (1.1), (1.2), and (1.3), we have

For \(t\geq T\), we define a generalized Riccati substitution by

Then we have

By virtue of (2.3), we obtain

For \(t\in[s_{1}, t_{1}]\), substituting (2.2) and (2.5) into (2.4), we conclude that

Let \(u\in B_{1}\) be given as in the hypothesis. Multiplying (2.6) by \(u^{2}\) and integrating the resulting inequality from \(s_{1}\) to \(t_{1}\), we have

Integrating (2.7) by parts and using the fact that \(u(s_{1})=u(t_{1})=0\), we deduce that

That is,

Hence,

which is equivalent to

where \(J_{1}(u, \rho, \sigma)\) is as in (2.1). Since \(J_{1}(u, \rho, \sigma)>0\), inequality (2.8) yields

which is a contradiction. This contradiction proves that *x* is not eventually positive.

When *x* is eventually negative, we use \(u\in B_{2}\) and \(F(t)\geq0\) on \([s_{2}, t_{2}]\) to arrive at a similar contradiction. The proof is complete.â€ƒâ–¡

### Example 2.1

For \(t\geq1\), consider the forced second-order neutral delay differential equation

Let \(r(t)=1\), \(P(t)=1/2\), \(\tau(t)=t/2\), \(m=l=1\), \(Q_{1}(t)=8\), \(R_{1}(t)=4t^{2}\), \(f_{1}(y)=g_{1}(y)=y\), \(\alpha_{1}=\beta_{1}=1\), \(u=\sin t\), \(\rho(t)=1\), and \(\sigma(t)=0\). Set \(s_{1}=(2n+1)\pi\), \(t_{1}=(2n+2)\pi\), \(s_{2}=(2n+3)\pi\), and \(t_{2}=(2n+4)\pi\). Then

Similarly, \(J_{2}(u, \rho, \sigma)=7\pi/2\). Hence, by Theorem 2.1, every solution of (2.9) is oscillatory.

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## Acknowledgements

The authors are grateful to the editors and two anonymous referees for a very thorough reading of the manuscript and for kindly prompting improvements in presentation. This research is supported by NNSF of P.R. China (Grant No.Â 61403061), NSF of Shandong Province (Grant No. ZR2012FL06), and the AMEP of Linyi University, P.R. China.

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

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Jiang, Y., Fu, Y., Wang, H. *et al.* Oscillation of forced second-order neutral delay differential equations.
*Adv Differ Equ* **2015**, 223 (2015). https://doi.org/10.1186/s13662-015-0556-x

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DOI: https://doi.org/10.1186/s13662-015-0556-x