Skip to content

Advertisement

  • Research
  • Open Access

Oscillation of forced second-order neutral delay differential equations

Advances in Difference Equations20152015:223

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

  • Received: 9 December 2014
  • Accepted: 26 June 2015
  • Published:

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.

Keywords

  • oscillation
  • second-order
  • forced term
  • neutral differential equation

MSC

  • 34C10
  • 34K11

1 Introduction

In this paper, we are concerned with the oscillation of a forced second-order nonlinear neutral differential equation
$$ \bigl(r(t)\bigl[x(t)+P(t)x\bigl(\tau(t)\bigr)\bigr]' \bigr)'+\sum^{m}_{i=1}Q_{i}(t)f_{i} \bigl(x(t)\bigr) +\sum^{l}_{j=1}R_{j}(t)g_{j} \bigl(x\bigl(\tau(t)\bigr)\bigr)=F(t), $$
(1.1)
where \(t\geq t_{0}>0\), \(m\geq1\), and \(l\geq1\) are integers. We suppose that the following assumptions are satisfied:
(A1): 

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

(A2): 

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

(A3): 

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 \(j=1,2,\ldots,l\);

(A4): 
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
$$ z(t):=x(t)+P(t)x\bigl(\tau(t)\bigr). $$
(1.3)
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 [123] 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
$$\bigl(p(t)x'\bigr)'+q(t)x=f(t). $$
Zhang et al. [22] studied a second-order neutral differential equation
$$ \bigl(r(t)\bigl[x(t)+p(t)x(t-\tau)\bigr]' \bigr)'+Q_{1}(t)f\bigl(x(t)\bigr)+Q_{2}(t)g \bigl(x(t-\tau)\bigr)=H(t), $$
(1.4)
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 (A1)-(A4) 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,
$$\begin{aligned} &J_{k}(u, \rho, \sigma) \\ &\quad=\int_{s_{k}}^{t_{k}} \Biggl\{ \rho \Biggl[u^{2} \Biggl(\sum^{m}_{i=1} \alpha _{i}Q_{i}+r\sigma^{2}-(r \sigma)' \Biggr) -r \biggl(u'+\frac{u\rho'}{2\rho}+u\sigma \biggr)^{2} \Biggr] \Biggr\} (t) \,\mathrm{d}t>0, \end{aligned}$$
(2.1)
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 (A4), 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 (A3), (1.1), (1.2), and (1.3), we have
$$\begin{aligned} \bigl(rz'\bigr)'(t) =&F(t)-\sum ^{m}_{i=1}Q_{i}(t)f_{i} \bigl(x(t)\bigr)-\sum^{l}_{j=1}R_{j}(t)g_{j} \bigl(x\bigl(\tau(t)\bigr)\bigr) \\ \leq&-\sum^{m}_{i=1}\alpha_{i}Q_{i}(t)x(t)- \sum^{l}_{j=1}\beta_{j}R_{j}(t)x \bigl(\tau(t)\bigr) \\ \leq&- \Biggl[\sum^{m}_{i=1} \alpha_{i}Q_{i}(t)x(t)+\sum^{m}_{i=1} \alpha _{i}Q_{i}(t)P(t)x\bigl(\tau(t)\bigr) \Biggr] \\ =&-\sum^{m}_{i=1}\alpha_{i}Q_{i}(t)z(t). \end{aligned}$$
(2.2)
For \(t\geq T\), we define a generalized Riccati substitution by
$$ V(t):=-\rho(t) \biggl[\frac{r(t)z'(t)}{z(t)}+r(t)\sigma(t) \biggr]. $$
(2.3)
Then we have
$$\begin{aligned} V' =&-\rho' \biggl(\frac{rz'}{z}+r \sigma \biggr)-\rho \biggl(\frac {rz'}{z}+r\sigma \biggr)' \\ =&\frac{\rho'}{\rho}V-\rho \biggl(\frac{rz'}{z} \biggr)'-\rho (r\sigma )' \\ =&\frac{\rho'}{\rho}V-\rho\frac{(rz')'}{z}+\rho\frac{r(z')^{2}}{z^{2}}-\rho (r\sigma )'. \end{aligned}$$
(2.4)
By virtue of (2.3), we obtain
$$ \biggl(\frac{z'}{z} \biggr)^{2}= \biggl( \frac{V}{-\rho r}-\sigma \biggr)^{2} = \biggl(\frac{V}{\rho r} \biggr)^{2}+\sigma^{2}+2\frac{V\sigma}{\rho r}. $$
(2.5)
For \(t\in[s_{1}, t_{1}]\), substituting (2.2) and (2.5) into (2.4), we conclude that
$$\begin{aligned} V' =&\frac{\rho'}{\rho}V-\rho\frac{(rz')'}{z}+\rho r \biggl(\frac {V^{2}}{\rho^{2}r^{2}}+\sigma^{2} +2\frac{V\sigma}{\rho r} \biggr)-\rho (r\sigma )' \\ =&-\rho\frac{(rz')'}{z}+\rho \bigl[r\sigma^{2}-(r \sigma)' \bigr]+ \biggl(\frac{\rho'}{\rho} +2\sigma \biggr)V+ \frac{V^{2}}{\rho r} \\ \geq&\rho \Biggl[\sum^{m}_{i=1} \alpha_{i}Q_{i}+r\sigma^{2}-(r \sigma)' \Biggr]+ \biggl(\frac{\rho'}{\rho} +2\sigma \biggr)V+ \frac{V^{2}}{\rho r}. \end{aligned}$$
(2.6)
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
$$\begin{aligned} \int_{s_{1}}^{t_{1}}u^{2}V' \,\mathrm{d}t\geq{}&\int_{s_{1}}^{t_{1}}u^{2}\rho \Biggl[\sum^{m}_{i=1}\alpha_{i}Q_{i}+r \sigma^{2}-(r\sigma)' \Biggr]\,\mathrm{d}t +\int _{s_{1}}^{t_{1}} \biggl(\frac{\rho'}{\rho}+2\sigma \biggr)Vu^{2}\,\mathrm{d}t \\ &{}+\int_{s_{1}}^{t_{1}} \frac{V^{2}}{\rho r}u^{2}\,\mathrm{d}t. \end{aligned}$$
(2.7)
Integrating (2.7) by parts and using the fact that \(u(s_{1})=u(t_{1})=0\), we deduce that
$$\begin{aligned} -\int_{s_{1}}^{t_{1}}2uu'V\,\mathrm{d}t\geq{}& \int_{s_{1}}^{t_{1}}u^{2}\rho \Biggl[\sum ^{m}_{i=1}\alpha_{i}Q_{i}+r \sigma^{2}-(r\sigma)' \Biggr]\,\mathrm{d}t \\ &{}+\int _{s_{1}}^{t_{1}} \biggl(\frac{\rho'}{\rho}+2\sigma \biggr)Vu^{2}\,\mathrm{d}t +\int_{s_{1}}^{t_{1}} \frac{V^{2}}{\rho r}u^{2}\,\mathrm{d}t. \end{aligned}$$
That is,
$$\int_{s_{1}}^{t_{1}} \biggl[\frac{u^{2}V^{2}}{\rho r}+2uV \biggl(u'+u \biggl(\frac{\rho'}{2\rho}+\sigma \biggr) \biggr) \biggr] \,\mathrm{d}t +\int_{s_{1}}^{t_{1}}u^{2}\rho \Biggl[ \sum^{m}_{i=1}\alpha_{i}Q_{i}+r \sigma^{2}-(r\sigma )' \Biggr]\,\mathrm{d}t \leq0. $$
Hence,
$$\begin{aligned} &\int_{s_{1}}^{t_{1}} \biggl[\frac{uV}{\sqrt{\rho r}} +\sqrt{ \rho r} \biggl(u'+\frac{u\rho'}{2\rho}+u\sigma \biggr) \biggr]^{2}\,\mathrm{d}t \\ &\quad{} +\int_{s_{1}}^{t_{1}} \Biggl[u^{2}\rho \Biggl(\sum^{m}_{i=1}\alpha_{i}Q_{i}+r \sigma ^{2}-(r\sigma)' \Biggr) -\rho r \biggl(u'+\frac{u\rho'}{2\rho}+u\sigma \biggr)^{2} \Biggr]\,\mathrm{d}t\leq0, \end{aligned}$$
which is equivalent to
$$ \int_{s_{1}}^{t_{1}} \biggl[ \frac{uV}{\sqrt{\rho r}} +\sqrt{\rho r} \biggl(u'+\frac{u\rho'}{2\rho}+u\sigma \biggr) \biggr]^{2}\,\mathrm{d}t+J_{1}(u, \rho, \sigma)\leq0, $$
(2.8)
where \(J_{1}(u, \rho, \sigma)\) is as in (2.1). Since \(J_{1}(u, \rho, \sigma)>0\), inequality (2.8) yields
$$\int_{s_{1}}^{t_{1}} \biggl[\frac{uV}{\sqrt{\rho r}} +\sqrt{ \rho r} \biggl(u'+\frac{u\rho'}{2\rho}+u\sigma \biggr) \biggr]^{2}\,\mathrm{d}t\leq-J_{1}(u, \rho, \sigma)< 0, $$
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. □

Example 2.1

For \(t\geq1\), consider the forced second-order neutral delay differential equation
$$ \biggl(x(t)+\frac{1}{2}x \biggl(\frac{t}{2} \biggr) \biggr)''+ 8x (t )+4t^{2}x \biggl( \frac{t}{2} \biggr)=\sin t. $$
(2.9)
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
$$\begin{aligned} J_{1}(u, \rho, \sigma) =&\int_{s_{1}}^{t_{1}} \Biggl\{ \rho \Biggl[u^{2} \Biggl(\sum^{m}_{i=1} \alpha _{i}Q_{i}+r\sigma^{2}-(r \sigma)' \Biggr) -r \biggl(u'+\frac{u\rho'}{2\rho}+u\sigma \biggr)^{2} \Biggr] \Biggr\} (t) \,\mathrm{d}t \\ =&\int_{(2n+1)\pi}^{(2n+2)\pi} \bigl(8\sin^{2}t- \cos^{2}t \bigr)\,\mathrm{d}t =\frac{7}{2}\pi. \end{aligned}$$
Similarly, \(J_{2}(u, \rho, \sigma)=7\pi/2\). Hence, by Theorem 2.1, every solution of (2.9) is oscillatory.

Declarations

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. 61403061), NSF of Shandong Province (Grant No. ZR2012FL06), and the AMEP of Linyi University, P.R. China.

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
Qingdao Technological University, Feixian, Shandong, 273400, P.R. China
(2)
College of Economics, Ocean University of China, Qingdao, Shandong, 266100, P.R. China
(3)
LinDa Institute of Shandong Provincial Key Laboratory of Network Based Intelligent Computing, Linyi University, Linyi, Shandong, 276005, P.R. China
(4)
School of Informatics, Linyi University, Linyi, Shandong, 276005, P.R. China

References

  1. Baculíková, B, Džurina, J: Oscillation theorems for second-order nonlinear neutral differential equations. Comput. Math. Appl. 62, 4472-4478 (2011) MathSciNetView ArticleMATHGoogle Scholar
  2. Baculíková, B, Džurina, J: Oscillation theorems for higher order neutral differential equations. Appl. Math. Comput. 219, 3769-3778 (2012) MathSciNetView ArticleGoogle Scholar
  3. Baculíková, B, Džurina, J, Li, T: Oscillation results for even-order quasilinear neutral functional differential equations. Electron. J. Differ. Equ. 2011, 143 (2011) Google Scholar
  4. El-Sayed, MA: An oscillation criterion for a forced second order linear differential equation. Proc. Am. Math. Soc. 118, 813-817 (1993) MathSciNetGoogle Scholar
  5. Erbe, L, Kong, Q, Zhang, BG: Oscillation Theory for Functional Differential Equations. Dekker, New York (1995) Google Scholar
  6. Grace, SR, Lalli, BS: Oscillation of nonlinear second order neutral delay differential equations. Rad. Mat. 3, 77-84 (1987) MathSciNetMATHGoogle Scholar
  7. Győri, I, Ladas, G: Oscillation Theory of Delay Differential Equations with Applications. Clarendon Press, Oxford (1991) MATHGoogle Scholar
  8. Hale, JK: Theory of Functional Differential Equations. Springer, New York (1977) View ArticleMATHGoogle Scholar
  9. Hasanbulli, M, Rogovchenko, YuV: Oscillation criteria for second order nonlinear neutral differential equations. Appl. Math. Comput. 215, 4392-4399 (2010) MathSciNetView ArticleMATHGoogle Scholar
  10. Ladde, GS, Lakshmikantham, V, Zhang, BG: Oscillation Theory of Differential Equations with Deviating Arguments. Dekker, New York (1987) MATHGoogle Scholar
  11. Li, T, Agarwal, RP, Bohner, M: Some oscillation results for second-order neutral differential equations. J. Indian Math. Soc. 79, 97-106 (2012) MathSciNetMATHGoogle Scholar
  12. Li, T, Agarwal, RP, Bohner, M: Some oscillation results for second-order neutral dynamic equations. Hacet. J. Math. Stat. 41, 715-721 (2012) MathSciNetGoogle Scholar
  13. Li, T, Han, Z, Zhao, P, Sun, S: Oscillation of even-order neutral delay differential equations. Adv. Differ. Equ. 2010, 184180 (2010) MathSciNetView ArticleMATHGoogle Scholar
  14. Li, T, Rogovchenko, YuV: Asymptotic behavior of higher-order quasilinear neutral differential equations. Abstr. Appl. Anal. 2014, 395368 (2014) MathSciNetGoogle Scholar
  15. Li, T, Rogovchenko, YuV, Zhang, C: Oscillation of second-order neutral differential equations. Funkc. Ekvacioj 56, 111-120 (2013) MathSciNetView ArticleMATHGoogle Scholar
  16. Shi, W, Wang, P: Oscillatory criteria of a class of second-order neutral functional differential equations. Appl. Math. Comput. 146, 211-226 (2003) MathSciNetView ArticleMATHGoogle Scholar
  17. Sun, S, Li, T, Han, Z, Li, H: Oscillation theorems for second-order quasilinear neutral functional differential equations. Abstr. Appl. Anal. 2012, 819342 (2012) MathSciNetGoogle Scholar
  18. Wang, P, Wu, Y, Caccetta, L: Oscillation criteria for boundary value problems of high-order partial functional differential equations. J. Comput. Appl. Math. 206, 567-577 (2007) MathSciNetView ArticleMATHGoogle Scholar
  19. Wong, JSW: Oscillation criteria for a forced second-order linear differential equation. J. Math. Anal. Appl. 231, 235-240 (1999) MathSciNetView ArticleMATHGoogle Scholar
  20. Ye, L, Xu, Z: Oscillation criteria for second-order quasilinear neutral delay differential equations. Appl. Math. Comput. 207, 388-396 (2009) MathSciNetView ArticleGoogle Scholar
  21. Zhang, C, Agarwal, RP, Bohner, M, Li, T: New oscillation results for second-order neutral delay dynamic equations. Adv. Differ. Equ. 2012, 227 (2012) MathSciNetView ArticleGoogle Scholar
  22. Zhang, Z, Wang, X, Lin, S, Yu, Y: Oscillation and nonoscillation criteria for second order nonlinear neutral delay differential equations. J. Syst. Sci. Math. Sci. 26, 325-334 (2006) (in Chinese) MathSciNetGoogle Scholar
  23. Zhong, J, Ouyang, Z, Zou, S: An oscillation theorem for a class of second-order forced neutral delay differential equations with mixed nonlinearities. Appl. Math. Lett. 24, 1449-1454 (2011) MathSciNetView ArticleMATHGoogle Scholar

Copyright

© Jiang et al. 2015

Advertisement