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Oscillation results for secondorder nonlinear neutral differential equations
Advances in Difference Equations volume 2013, Article number: 336 (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.
1 Introduction
In this paper, we are concerned with the oscillation of a class of nonlinear secondorder neutral differential equations
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.
Oscillation results for (1) have been reported in [2, 4, 6, 8, 11, 14, 18–20]. A commonly used assumption is
although several authors were concerned with the oscillation of equation (1) in the case where
In particular, Xu and Meng [[19], Theorem 2.3] established sufficient conditions for the oscillation of (1) assuming that
Further results in this direction were obtained by Ye and Xu [20] under the assumptions that
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
In what follows, all functional inequalities are tacitly assumed to hold for all t large enough, unless mentioned otherwise. As usual, we use the notation z(t):=x(t)+p(t)x(t\tau ) and {g}_{+}(t):=max\{g(t),0\}. Let
We say that a function H\in \mathrm{C}(\mathbb{D},[0,+\mathrm{\infty})) belongs to a class {\mathcal{W}}_{\gamma} if

(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 what follows, we assume that, for all t\ge {t}_{0},
where
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.
Theorem 2 Assume that conditions (H_{1})(H_{4}), (3), and (6) are satisfied. Suppose also that there exist two functions {\rho}_{1},{\rho}_{2}\in {\mathrm{C}}^{1}([{t}_{0},+\mathrm{\infty}),\mathbb{R}) such that, for some \beta \ge 1 and for some H\in {\mathcal{W}}_{\gamma},
and
where
and
Then equation (1) is oscillatory.
Proof Let x(t) be a nonoscillatory solution of (1). Since γ is a quotient of two odd positive integers, x(t) is also a solution of (1). Hence, without loss of generality, we may assume that there exists a {t}_{1}\ge {t}_{0} such that x(t)>0, x(t\tau )>0, and x(\sigma (t))>0 for all t\ge {t}_{1}. Then z(t)\ge x(t)>0, and by virtue of
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.
Case 1. Assume first that {z}^{\prime}(t)>0 for all t\ge {t}_{2}. Equation (1) and condition (H_{2}) yield
In view of (H_{4}), there exists a {t}_{3}\ge {t}_{2} such that, for all t\ge {t}_{3},
and
It follows from (14) and (15) that
Define a generalized Riccati substitution by
Differentiating (18) and using (16) and (17), one arrives at
Let
By virtue of Lemma 1, part (i), we have the following estimate:
It follows now from (19) and (20) that
where {\psi}_{1} is defined by (10). Replacing in (21) t with s, multiplying both sides by H(t,s) and integrating with respect to s from {t}_{3} to t, we have, for some \beta \ge 1 and for any t\ge {t}_{3},
Let q:=1+1/\gamma,
and
Application of Lemma 1, part (ii), yields
Hence, by the latter inequality and (22), we have
Using monotonicity of H, we conclude that, for all t\ge {t}_{3},
Thus,
and
which contradicts (8).
Case 2. Assume now that {z}^{\prime}(t)<0 for all t\ge {t}_{2}. It follows from the inequality {(r(t){({z}^{\prime}(t))}^{\gamma})}^{\prime}\le 0 that, for all s\ge t\ge {t}_{2},
Integrating this inequality from t to l, l\ge t\ge {t}_{2}, we have
Passing to the limit as l\to +\mathrm{\infty}, we conclude that
which yields
Hence, we have
It follows from (1) and the latter inequality that there exists a {t}_{4}\ge {t}_{2} such that
For t\ge {t}_{4}, define a generalized Riccati substitution by
Differentiating (25), we have
Letting in Lemma 1, part (i),
we have
It follows from (24) and (26) that
where {\psi}_{2} is defined by (12). Replacing in (27) t with s, multiplying both sides by H(t,s) and integrating with respect to s from {t}_{4} to t, we conclude that, for some \beta \ge 1 and for all t\ge {t}_{4},
Letting in Lemma 1, part (ii),
and
we conclude that
Using the latter inequality and (28), we have
Proceeding as in the proof of Case 1, we obtain contradiction with our assumption (9). Therefore, equation (1) is oscillatory. □
Theorem 3 Assume that conditions (H_{1})(H_{4}), (3), and (6) are satisfied. Suppose also that there exist functions H\in {\mathcal{W}}_{\gamma}, {\rho}_{1},{\rho}_{2}\in {\mathrm{C}}^{1}([{t}_{0},+\mathrm{\infty}),\mathbb{R}), {\varphi}_{1},{\varphi}_{2}\in \mathrm{C}([{t}_{0},+\mathrm{\infty}),\mathbb{R}) such that, for all T\ge {t}_{0} and for some \beta >1,
and
where {\psi}_{1}, {\psi}_{2}, {v}_{1}, and {v}_{2} are as in Theorem 2. If
and
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}.
Case 1. Assume first that {z}^{\prime}(t)>0 for all t\ge {t}_{2}. Proceeding as in the proof of Theorem 2, we arrive at inequality (23), which yields, for all t>{t}_{3} and for some \beta >1,
The latter inequality implies that, for all t>{t}_{3} and for some \beta >1,
Consequently,
and
Assume now that
Condition (30) implies existence of a \vartheta >0 such that
It follows from (37) that, for any positive constant η, there exists a {t}_{5}>{t}_{3} such that, for all t\ge {t}_{5},
Using integration by parts and (39), we have, for all t\ge {t}_{5},
By virtue of (38), there exists a {t}_{6}\ge {t}_{5} such that, for all t\ge {t}_{6},
which implies that
Since η is an arbitrary positive constant,
but the latter contradicts (36). Consequently,
and, by virtue of (35),
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.
Theorem 4 Let {\psi}_{1}, {\psi}_{2}, {v}_{1}, and {v}_{2} be as in Theorem 3, and assume that conditions (H_{1})(H_{4}), (3), and (6) are satisfied. Suppose also that there exist functions H\in {\mathcal{W}}_{\gamma}, {\rho}_{1},{\rho}_{2}\in {\mathrm{C}}^{1}([{t}_{0},+\mathrm{\infty}),\mathbb{R}), {\varphi}_{1},{\varphi}_{2}\in \mathrm{C}([{t}_{0},+\mathrm{\infty}),\mathbb{R}) such that (30), (33), and (34) hold. If, for all T\ge {t}_{0} and for some \beta >1,
and
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.
Example 5 For t\ge 1, consider the secondorder nonlinear neutral delay differential equation
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.
Example 6 For t\ge 1, consider the secondorder neutral delay differential equation
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.
We stress that the study of oscillatory properties of equation (1) in the case (3) brings additional difficulties. In particular, in order to deal with the case when {z}^{\prime}(t)<0 (which is simply eliminated if condition (2) holds), we have to impose an additional assumption p(t)<R(t)/R(t\tau )\le 1. In fact, it is well known (see, e.g., [6, 14]) that if x(t) is an eventually positive solution of (1), then
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).
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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.
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Li, T., Rogovchenko, Y.V. & Zhang, C. Oscillation results for secondorder nonlinear neutral differential equations. Adv Differ Equ 2013, 336 (2013). https://doi.org/10.1186/168718472013336
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DOI: https://doi.org/10.1186/168718472013336
Keywords
 oscillation
 secondorder
 neutral differential equation
 integral averaging