Oscillation results for secondorder nonlinear neutral differential equations
 Tongxing Li^{1},
 Yuriy V Rogovchenko^{2}Email author and
 Chenghui Zhang^{1}
DOI: 10.1186/168718472013336
© Li et al.; licensee Springer. 2013
Received: 17 February 2013
Accepted: 24 October 2013
Published: 21 November 2013
Abstract
We obtain several oscillation criteria for a class of secondorder nonlinear neutral differential equations. New theorems extend a number of related results reported in the literature and can be used in cases where known theorems fail to apply. Two illustrative examples are provided.
MSC:34K11.
Keywords
oscillation secondorder neutral differential equation integral averaging1 Introduction
where $t\ge {t}_{0}>0$, $\tau \ge 0$, and $\gamma \ge 1$ is a quotient of two odd positive integers. In what follows, it is always assumed that

(H_{1}) $r\in {\mathrm{C}}^{1}([{t}_{0},+\mathrm{\infty}),(0,+\mathrm{\infty}))$;

(H_{2}) $p,q\in \mathrm{C}([{t}_{0},+\mathrm{\infty}),[0,+\mathrm{\infty}))$ and $q(t)$ is not identically zero for large t;

(H_{3}) $f\in \mathrm{C}({\mathbb{R}}^{2},\mathbb{R})$ and $f(x,y)/{y}^{\gamma}\ge \kappa $ for all $y\ne 0$ and for some $\kappa >0$;

(H_{4}) $\sigma \in {\mathrm{C}}^{1}([{t}_{0},+\mathrm{\infty}),\mathbb{R})$, $\sigma (t)\le t$, ${\sigma}^{\prime}(t)>0$, and ${lim}_{t\to +\mathrm{\infty}}\sigma (t)=+\mathrm{\infty}$.
By a solution of equation (1) we mean a continuous function $x(t)$ defined on an interval $[{t}_{x},+\mathrm{\infty})$ such that $r(t){({(x(t)+p(t)x(t\tau ))}^{\prime})}^{\gamma}$ is continuously differentiable and $x(t)$ satisfies (1) for $t\ge {t}_{x}$. We consider only solutions satisfying $sup\{x(t):t\ge T\ge {t}_{x}\}>0$ and tacitly assume that equation (1) possesses such solutions. A solution of (1) is called oscillatory if it has arbitrarily large zeros on $[{t}_{x},+\mathrm{\infty})$; otherwise, it is called nonoscillatory. We say that equation (1) is oscillatory if all its continuable solutions are oscillatory.
During the past decades, a great deal of interest in oscillatory and nonoscillatory behavior of various classes of differential and functional differential equations has been shown. Many papers deal with the oscillation of neutral differential equations which are often encountered in applied problems in science and technology; see, for instance, Hale [1]. It is known that analysis of neutral differential equations is more difficult in comparison with that of ordinary differential equations, although certain similarities in the behavior of solutions of these two classes of equations are observed; see, for instance, the monographs [2–4], the papers [5–22] and the references cited there.
see also the paper by Han et al. [8] where inaccuracies in [20] were corrected and new oscillation criteria for (1) were obtained [[8], Theorems 2.1 and 2.2]. We conclude this brief review of the literature by mentioning that Li et al. [13] and Sun et al. [18] extended the results obtained in [8] to EmdenFowler neutral differential equations and neutral differential equations with mixed nonlinearities.
Our principal goal in this paper is to derive new oscillation criteria for equation (1) without requiring restrictive conditions (4) and (5). Developing further ideas from the paper by Hasanbulli and Rogovchenko [9] concerned with a particular case of equation (2) with $\gamma =1$, we study the oscillation of (1) in the case where $\gamma \ge 1$.
2 Oscillation criteria
 (i)
$H(t,t)=0$ and $H(t,s)>0$ for all $(t,s)\in {\mathbb{D}}_{0}$;
 (ii)H has a nonpositive continuous partial derivative with respect to the second variable satisfying$\frac{\partial}{\partial s}H(t,s)=h(t,s){(H(t,s))}^{\gamma /(\gamma +1)}$
for a locally integrable function $h\in {\mathcal{L}}_{\mathrm{loc}}(\mathbb{D},\mathbb{R})$.
In order to establish our main theorems, we need the following auxiliary result. The first inequality is extracted from the paper by Jiang and Li [[11], Lemma 5], whereas the second one is a variation of the wellknown Young inequality [23].
Lemma 1
 (i)Let $\gamma \ge 1$ be a ratio of two odd integers. Then${A}^{1+1/\gamma}AB{}^{1+1/\gamma}\le \frac{1}{\gamma}{B}^{1/\gamma}[(\gamma +1)AB]$(7)
for all $AB\ge 0$.
 (ii)For any two numbers $C,D\ge 0$ and for any $q>1$,${C}^{q}+(q1){D}^{q}qC{D}^{q1}\ge 0,$
the equality holds if and only if $C=D$.
Then equation (1) is oscillatory.
the function ${(r(t){({z}^{\prime}(t))}^{\gamma})}^{\prime}$ is nonincreasing for all $t\ge {t}_{1}$. Therefore, ${z}^{\prime}(t)$ does not change sign eventually, that is, there exists a ${t}_{2}\ge {t}_{1}$ such that either ${z}^{\prime}(t)>0$ or ${z}^{\prime}(t)<0$ for all $t\ge {t}_{2}$. We consider each of two cases separately.
which contradicts (8).
Proceeding as in the proof of Case 1, we obtain contradiction with our assumption (9). Therefore, equation (1) is oscillatory. □
equation (1) is oscillatory.
Proof Without loss of generality, assume again that (1) possesses a nonoscillatory solution $x(t)$ such that $x(t)>0$, $x(t\tau )>0$, and $x(\sigma (t))>0$ on $[{t}_{1},+\mathrm{\infty})$ for some ${t}_{1}\ge {t}_{0}$. From the proof of Theorem 2, we know that there exists a ${t}_{2}\ge {t}_{1}$ such that either ${z}^{\prime}(t)>0$ or ${z}^{\prime}(t)<0$ for all $t\ge {t}_{2}$.
which contradicts (33).
Case 2. Assume now that ${z}^{\prime}(t)<0$ for $t\ge {t}_{2}$. It has been established in Theorem 2 that (29) holds. Using (29) and proceeding as in Case 1 above, we arrive at the desired conclusion. □
As an immediate consequence of Theorem 3, we have the following result.
equation (1) is oscillatory.
3 Examples
Efficient oscillation tests can be easily derived from Theorems 24 with different choices of the functions H, ${\rho}_{1}$, ${\rho}_{2}$, ${\varphi}_{1}$, and ${\varphi}_{2}$. In this section, we illustrate possible applications with two examples.
Here, $r(t)={t}^{2}$, $p(t)=t/(2t+1)$, $\tau =1$, $q(t)=1$, $f(x(t),x(\sigma (t)))=(2+{x}^{4}(t))x(t/2)$, whereas $R(t)=1/t$.
Let $\gamma =1$, $\kappa =1$, $H(t,s)={(ts)}^{2}$, ${\rho}_{1}(t)=1/(2t)$, ${\rho}_{2}(t)=1/t$. Then ${h}^{2}(t,s)=4$, ${v}_{1}(t)={v}_{2}(t)={t}^{2}$, ${\psi}_{1}(t)={t}^{2}((t+2)/(2t+2)+1)$, ${\psi}_{2}(t)={t}^{2}(3({t}^{2}/((2t+2)(t2))))$, and a straightforward computation shows that all assumptions of Theorem 2 are satisfied. Hence, equation (42) is oscillatory.
Here, $r(t)={\mathrm{e}}^{t}$, $p(t)=1/3$, $\tau =\pi /4$, $q(t)=32\sqrt{65}{\mathrm{e}}^{t}/3$, $R(t)={\mathrm{e}}^{t}$, and $f(x(t),x(\sigma (t)))=x(t(arcsin(\sqrt{65}/65))/8)$.
Let $\gamma =1$, $\kappa =1$, $H(t,s)={(ts)}^{2}$, ${\rho}_{1}(t)={\rho}_{2}(t)=0$. Then ${h}^{2}(t,s)=4$, ${v}_{1}(t)={v}_{2}(t)=1$, ${\psi}_{1}(t)=(64\sqrt{65}/9){\mathrm{e}}^{t}$, ${\psi}_{2}(t)=(32\sqrt{65}/3)(1(1/3){\mathrm{e}}^{\pi /4}){\mathrm{e}}^{t}$. It is not difficult to verify that all assumptions of Theorem 2 hold. Hence, equation (43) is oscillatory. In fact, one such solution is $x(t)=sin8t$.
4 Conclusions
Most oscillation results reported in the literature for neutral differential equation (1) and its particular cases have been obtained under the assumption (2) which significantly simplifies the analysis of the behavior of $z(t)=x(t)+p(t)x(t\tau )$ for a nonoscillatory solution $x(t)$ of (1). In this paper, using a refinement of the integral averaging technique, we have established new oscillation criteria for secondorder neutral delay differential equation (1) assuming that (3) holds.
One of the principal difficulties one encounters lies in the fact that (44) does not hold when (3) is satisfied, cf. [8]. Since the sign of the derivative ${z}^{\prime}(t)$ is not known, our criteria for the oscillation of (1) include a pair of assumptions as, for instance, (8) and (9). On the other hand, we point out that, contrary to [8, 13, 18, 19], we do not need in our oscillation theorems quite restrictive conditions (4) and (5), which, in a certain sense, is a significant improvement compared to the results in the cited papers. However, this improvement has been achieved at the cost of imposing condition (6).
Therefore, two interesting problems for future research can be formulated as follows.

(P1) Is it possible to establish oscillation criteria for (1) without requiring conditions (4), (5), and (6)?

(P2) Suggest a different method to investigate (1) in the case where $\gamma <1$ (and thus inequality (7) does not hold).
Declarations
Acknowledgements
The research of TL and CZ was supported in part by the National Basic Research Program of PR China (2013CB035604) and the NNSF of PR China (Grants 61034007, 51277116, and 51107069). YR acknowledges research grants from the Faculty of Science and Technology of Umeå University, Sweden and from the Faculty of Engineering and Science of the University of Agder, Norway. TL would like to express his gratitude to Professors Ravi P. Agarwal and Martin Bohner for support and useful advices. Last but not least, the authors are grateful to two anonymous referees for a very thorough reading of the manuscript and for pointing out several inaccuracies.
Authors’ Affiliations
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