Oscillation and asymptotic behavior of third-order neutral differential equations with distributed deviating arguments
https://doi.org/10.1186/s13662-015-0604-6
© Tian et al. 2015
Received: 23 February 2015
Accepted: 11 August 2015
Published: 28 August 2015
Abstract
By employing a generalized Riccati transformation and integral averaging technique, two Philos-type criteria are obtained which ensure that every solution of a class of third-order neutral differential equations with distributed deviating arguments is either oscillatory or converges to zero. These results extend and improve related criteria reported in the literature. Two illustrative examples are provided.
Keywords
MSC
1 Introduction
- \((\mathrm{A}_{1})\) :
-
\(r(t)\in C^{1}([t_{0},\infty),(0,\infty))\), \(r'(t)\geq 0\), \(\int_{t_{0}}^{\infty}r^{-1/\alpha}(s)\,ds=\infty\);
- \((\mathrm{A}_{2})\) :
-
\(p(t,\xi)\in C([t_{0},\infty)\times[a,b],[0,\infty))\), \(0\leq\int_{a}^{b}p(t,\xi)\,d\xi\leq P<1\);
- \((\mathrm{A}_{3})\) :
-
\(\tau(t,\xi)\in C([t_{0},\infty)\times[a,b],R)\) is a nondecreasing function for ξ satisfying \(\tau(t,\xi)\leq t\) and \(\liminf_{t\rightarrow\infty}\tau(t,\xi)=\infty\) for \(\xi\in[a,b]\);
- \((\mathrm{A}_{4})\) :
-
\(q(t,\xi)\in C([t_{0},\infty)\times[c,d],[0,\infty))\);
- \((\mathrm{A}_{5})\) :
-
\(\sigma(t,\xi)\in C([t_{0},\infty)\times[c,d],R)\) is a nondecreasing function for ξ satisfying \(\sigma(t,\xi)\leq t\) and \(\liminf_{t\rightarrow\infty}\sigma(t,\xi)=\infty\) for \(\xi\in[c,d]\);
- \((\mathrm{A}_{6})\) :
-
\(f(x)\in C(R, R)\) and there exists a positive constant K such that \(f(x)/x^{\alpha}\geq K\) for all \(x\neq0\).
In the special case when \(\alpha=1\), (1.1) reduces to (1.2). Now the following question arises. Could we obtain new Philos-type oscillation criteria for (1.1) by using a generalized Riccati transformation which differs from that of [19]? Motivated by Li [11], Li et al. [12], and Li and Saker [13], our purpose in this paper is to give a positive answer to this question. In Section 2, four lemmas are given to prove the main results. In Section 3, we establish two Philos-type theorems for (1.1). In Section 4, two examples and some conclusions are presented to illustrate the main results. As customary, all functional inequalities considered in this paper are supposed to hold for all t large enough.
2 Some lemmas
Lemma 2.1
- (I)
\(z(t)>0\), \(z'(t)>0\), \(z''(t)>0\), \(z'''(t)\leq0\);
- (II)
\(z(t)>0\), \(z'(t)<0\), \(z''(t)>0\), \(z'''(t)\leq0\),
Proof
Lemma 2.2
Proof
Lemma 2.3
([5], Lemma 3)
Lemma 2.4
Proof
The proof is similar to that of Baculíková and Džurina ([5], Lemma 4), and hence it is omitted. □
3 Main results
- (i)
\(H(t,t)=0\), \(t\geq t_{0}\), \(H(t,s)>0\), \((t,s)\in D_{0}\);
- (ii)\(\partial H(t,s)/\partial s\leq0\), there exist \(\rho(t)\in C^{1}([t_{0},\infty),(0,\infty))\), \(b(t)\in C^{1}([t_{0},\infty),[0,\infty))\), and \(h(t,s)\in C(D_{0},R)\) satisfying$$-\frac{\partial H(t,s)}{\partial s}=H(t,s) \biggl[\frac{\rho'(s)}{\rho(s)}+(\alpha+1)b^{\frac{1}{\alpha }}(s) \biggr]+h(t,s). $$
Theorem 3.1
Proof
Assume that (1.1) has a nonoscillatory solution \(x(t)\). Without loss of generality, we may assume that \(x(t)\) is an eventually positive solution of (1.1). By Lemma 2.1, we observe that \(z(t)\) satisfies either (I) or (II) for \(t\geq t_{1}\). We consider each of two cases separately.
Assume now that \(z(t)\) has the property (II). By Lemma 2.2, we have \(\lim_{t\rightarrow\infty}x(t)=0\). The proof is complete. □
It may happen that assumption (3.1) in Theorem 3.1 fails to hold. Consequently, Theorem 3.1 cannot be applied. The following theorem provides a new oscillation criterion for (1.1).
Theorem 3.2
Proof
Suppose that \(z(t)\) has the property (II). By Lemma 2.2, we obtain \(\lim_{t\rightarrow\infty}x(t)=0\). This completes the proof. □
4 Examples and conclusions
Example 4.1
Example 4.2
Remark 4.1
With an appropriate choice of the function H, one can derive from Theorems 3.1 and 3.2 a number of oscillation criteria for (1.1). For example, consider a Kamenev-type function \(H(t,s)\) by \(H(t,s)=(t-s)^{n-1}\), \((t,s)\in D\), where \(n>2\) is an integer. The remainder of the details are left to the reader.
Remark 4.2
Theorems 3.1 and 3.2 reported in this paper reduce to ([19], Theorems 3.1 and 3.2), respectively, when letting \(\alpha=1\) and \(b(t)=0\).
Remark 4.3
Note that Theorems 3.1 and 3.2 ensure that every solution \(x(t)\) to (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 the derivative \(z'(t)\) is not fixed, it is not easy to establish sufficient conditions which guarantee that all solutions to (1.1) are just oscillatory and do not satisfy \(\lim_{t\rightarrow\infty}x(t)=0\). Neither is it possible to use the technique exploited in this paper for proving that all solutions of (1.1) satisfy \(\lim_{t\rightarrow\infty}x(t)=0\). Hence, these two interesting problems are left for future research.
Declarations
Acknowledgements
The authors are grateful to the editors and two anonymous referees for a very thorough reading of the manuscript and for pointing out several inaccuracies. This research was supported by NNSF of P.R. China (Grant Nos. 61174217, 61374074, and 61473133) and NSF of Shandong Province (Grant No. JQ201119).
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
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