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Oscillation of forced second-order neutral delay differential equations
Advances in Difference Equations volume 2015, Article number: 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:
- (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
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 [1–23] 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 (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,
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
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. □
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. 61403061), NSF of Shandong Province (Grant No. ZR2012FL06), and the AMEP of Linyi University, P.R. China.
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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
MSC
- 34C10
- 34K11
Keywords
- oscillation
- second-order
- forced term
- neutral differential equation