Asymptotic behavior of third-order differential equations with nonpositive neutral coefficients and distributed deviating arguments
- Cuimei Jiang^{1},
- Ying Jiang^{1} and
- Tongxing Li^{2, 3}Email author
https://doi.org/10.1186/s13662-016-0833-3
© Jiang et al. 2016
Received: 12 November 2015
Accepted: 6 April 2016
Published: 12 April 2016
Abstract
Using the Riccati transformation technique, we present several sufficient conditions that guarantee that all solutions to a third-order differential equation with nonpositive neutral coefficients and distributed deviating arguments are either oscillatory or converge to zero asymptotically. In particular, we establish Hille and Nehari type criteria. Two examples are given to demonstrate the practicability of the main results.
Keywords
MSC
1 Introduction
We assume that solutions of (1.1) exist for any \(t\in[t_{0},\infty)\). Our attention is restricted to those solutions of (1.1) that are not identically zero for large t. As usual, a solution of (1.1) is called oscillatory if it has arbitrarily large zeros on the interval \([t_{0},\infty)\). Otherwise, it is termed nonoscillatory (i.e., it is either eventually positive or eventually negative).
2 Several lemmas
Lemma 2.1
- (i)
\(z(t)>0\), \(z'(t)>0\), \(z''(t)>0\), \((r(t)(z''(t))^{\alpha})'\leq0\);
- (ii)
\(z(t)>0\), \(z'(t)<0\), \(z''(t)>0\), \((r(t)(z''(t))^{\alpha})'\leq0\);
- (iii)
\(z(t)<0\), \(z'(t)<0\), \(z''(t)>0\), \((r(t)(z''(t))^{\alpha})'\leq0\);
- (iv)
\(z(t)<0\), \(z'(t)<0\), \(z''(t)<0\), \((r(t)(z''(t))^{\alpha})'\leq0\).
Proof
Case 2. Assume that \(z''(t)>0\). Then \(z'(t)\) is of one sign. If \(z'(t)>0\), then \(z(t)>0\). If \(z'(t)<0\), then \(z(t)>0\) or \(z(t)<0\). Hence, we have three possible cases (i), (ii), and (iii) when \(z''(t)>0\). The proof is complete. □
Lemma 2.2
Proof
Lemma 2.3
Proof
3 Main results
Theorem 3.1
Proof
Suppose to the contrary that (1.1) has a nonoscillatory solution \(x(t)\). Without loss of generality, we may assume that \(x(t)\) is eventually positive (since the proof of the case where \(x(t)\) is eventually negative is similar). By Lemma 2.1, we observe that, for \(t\geq t_{1}\geq t_{0}\), \(z(t)\) satisfies four possible cases (i), (ii), (iii), or (iv) (as those of Lemma 2.1). We consider each of the four cases separately.
Suppose that case (ii) is satisfied. By Lemma 2.3, \(\lim_{t\rightarrow \infty}x(t)=0\).
If case (iii) or case (iv) holds, then \(\lim_{t\rightarrow\infty }z(t)=c_{0}<0\) (possibly \(c_{0}=-\infty\)) or \(\lim_{t\rightarrow\infty}z(t)=-\infty\), respectively. Proceeding similarly as in the proof of Lemma 2.3, we conclude that \(x(t)\) and \(z(t)\) are bounded. Hence, \(c_{0}\) is finite, and case (iv) does not occur. Similar analysis to that in Lemma 2.3 leads to the conclusion that \(\lim_{t\rightarrow\infty}x(t)=0\). This completes the proof. □
Letting \(\rho(t)=t\) and \(\rho(t)=1\), we can derive the following results from Theorem 3.1.
Corollary 3.1
Corollary 3.2
In what follows, we establish Hille and Nehari type criteria for (1.1). To this end, we introduce the following lemma.
Lemma 3.1
- (I)Let \(\bar{p}<\infty\), \(\bar{q}<\infty\), and suppose that the corresponding \(z(t)\) satisfies case (i) in Lemma 2.1. Then$$ \bar{p}\leq\bar{r}-\bar{r}^{1+{1}/{\alpha}}\leq\frac{\alpha^{\alpha }}{(\alpha+1)^{\alpha+1}} \quad\textit{and}\quad \bar{p}+\bar{q}\leq 1. $$(3.12)
- (II)
If \(\bar{p}=\infty\) or \(\bar{q}=\infty\), then \(z(t)\) does not have property (i) in Lemma 2.1.
Proof
On the basis of Lemma 3.1, we easily derive the following result with a proof similar to that of Theorem 3.1.
Theorem 3.2
4 Examples
The following examples illustrate applications of the main results in this paper.
Example 4.1
Example 4.2
Remark 4.1
Declarations
Acknowledgements
The authors express their sincere gratitude to the editors and two anonymous referees for careful reading of the manuscript and valuable suggestions that helped to improve the paper. This research is supported by NNSF of P.R. China (Grant Nos. 61503171, 61403061, and 11447005), CPSF (Grant No. 2015M582091), NSF of Shandong Province (Grant No. ZR2012FL06), DSRF of Linyi University (Grant No. LYDX2015BS001), and the AMEP of Linyi University, P.R. China.
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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