Oscillation of secondorder nonlinear delay differential equations with nonpositive neutral coefficients
 Qi Li^{1},
 Rui Wang^{1},
 Feng Chen^{1} and
 Tongxing Li^{1}Email author
https://doi.org/10.1186/s136620150377y
© Li et al.; licensee Springer. 2015
Received: 20 October 2014
Accepted: 14 January 2015
Published: 31 January 2015
Abstract
We present several oscillation criteria for a secondorder nonlinear delay differential equation with a nonpositive neutral coefficient. Two examples are given to illustrate the main results.
Keywords
MSC
1 Introduction
 (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 secondorder and thirdorder delay differential equations; see, for instance, [2–15] and the references cited therein.
2 Lemmas
In this section, we give two lemmas that will be useful for establishing oscillation criteria for (1.1).
Lemma 2.1
 (C_{1}):

\(z(t)>0\), \(z'(t)>0\), \((r(t)(z'(t))^{\alpha})'\leq0\);
 (C_{2}):

\(z(t)<0\), \(z'(t)>0\), \((r(t)(z'(t))^{\alpha})'\leq0\),
Proof
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
Proof
3 Oscillation results
Theorem 3.1
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.
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
Theorem 3.2
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.
If z satisfies (C_{2}), then \(\lim_{t\rightarrow\infty}{x(t)}=0\) when using Lemma 2.2. This completes the proof. □
4 Examples and discussion
Example 4.1
Example 4.2
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 \(\lim_{t\rightarrow\infty}x(t)=0\).
Declarations
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.
Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.
Authors’ Affiliations
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