Oscillation criteria for a class of nonlinear neutral differential equations
 Shuhui Wu^{1}Email author,
 Pargat Singh Calay^{2} and
 Zhanyuan Hou^{2}
https://doi.org/10.1186/s1366201504938
© Wu et al.; licensee Springer. 2015
Received: 30 October 2014
Accepted: 4 May 2015
Published: 14 May 2015
Abstract
Keywords
neutral differential equations bounded oscillation almost oscillation bounded almost oscillation1 Introduction
The differential equations that we study describe many phenomena and dynamical processes in various fields, and they have attracted a great deal of attention of researchers in physical sciences, mathematics, biology, and economy. In addition, these equations play an important role in numerical simulations of nonlinear partial differential equations, queuing problems, and discretization in solid state and quantum physics. For the application, please see [1].
 (H_{1}):

\(\int_{t}^{\infty}\frac{1}{a(s)}\,ds=\infty\) for all \(t\geq t_{0} \),
 (H_{2}):

\(\frac{f(t,u)}{u}\geq q(t\sigma)>0\) for \(u\neq0\) and \(0<\frac{g(t,v)}{v}\leq r(t\rho)\) for \(v\neq0\),
 (H_{3}):

\(0<\frac{f(t,u)}{u}\leq q(t\sigma)\) for \(u\neq0\) and \(\frac{g(t,v)}{v}\geq r(t\rho)>0 \) for \(v\neq 0\),
 (H_{4}):

\(\frac{1}{r(t)q(t)}\) is bounded, where \(q,r \in C([t_{0},\infty),R^{+})\).
We also assume that \(x(t)\) is a nontrivial solution of (1.1). The investigation of oscillatory behavior of solutions of various types of differential equations done by many researchers is motivated by many application problems in physics, biology, ecology, and so on. In particular, an increasing interest in obtaining oscillation criteria for different classes of differential and functional differential equations has been manifested recently. Please see [1–16].
The paper is organized as follows. We will first present criteria for (1.1) when \(\delta=+1\) in Section 2 and then for (1.1) when \(\delta=1\) in Section 3. Some examples will be given to illustrate the obtained criteria, respectively. The proofs of the main results are left to Section 4.
2 Statement of the main results when \(\delta=+1\)
Theorem 2.1
Suppose that conditions (H_{1}), (H_{2}) and (H_{4}) hold, \(q(t)>r(t)\), \(r(t)\) is bounded and \(\sigma\geq\rho\). Then (2.1) is bounded oscillatory.
Remark 2.2
Example 1
Theorem 2.3
Suppose that conditions (H_{1}), (H_{2}) and (H_{4}) hold, \(q(t)>r(t)\), \(q(t)\), \(1/a(t)\) are bounded and \(\sigma<\rho\). Then (2.1) is almost oscillatory.
Example 2
Theorem 2.4
Suppose that conditions (H_{1}), (H_{3}) and (H_{4}) hold, \(q(t)< r(t)\), \(r(t)\), \(1/a(t)\) are bounded and \(\sigma\geq\rho\). Then (2.1) is bounded almost oscillatory.
Theorem 2.5
Suppose that conditions (H_{1}), (H_{3}) and (H_{4}) hold, \(q(t)< r(t)\), \(q(t)\) is bounded and \(\sigma< \rho\). Then (2.1) is bounded almost oscillatory.
Example 3
3 Statement of the main results when \(\delta=1\)
Theorem 3.1
Suppose that conditions (H_{1}), (H_{3}) and (H_{4}) hold, \(p(t)\ge1\), \(q(t)< r(t)\), \(\sigma\le\rho\) and \(r(t)\) is bounded. Then (3.1) is bounded oscillatory.
Example 4
Theorem 3.2
Suppose that conditions (H_{1}), (H_{3}) and (H_{4}) hold, \(q(t)< r(t)\), \(\sigma\ge \rho\), \(0\le p(t)\le p_{1}<1\) or \(1< p_{2}\le p(t)\), \(r(t)\) and \(1/a(t)\) are bounded. Then (3.1) is bounded almost oscillatory.
Theorem 3.3
Suppose that conditions (H_{1}), (H_{2}) and (H_{4}) hold, \(q(t)>r(t)\), \(\sigma<\rho\), \(0\le p(t)\le p_{1}<1\) or \(1< p_{2}\le p(t)\), \(q(t)\) and \(1/a(t)\) are bounded. Then (3.1) is bounded almost oscillatory.
Example 5
Theorem 3.4
Suppose that conditions (H_{1}), (H_{2}) and (H_{4}) hold, \(q(t)>r(t)\), \(\sigma \ge\rho\), \(q(t)\) is bounded, \(0\le p(t)\le p_{1}<1\), or \(1/a(t)\) is bounded and \(1< p_{2}\le p(t)\). Then (3.1) is bounded almost oscillatory.
Example 6
4 Proofs of the main results
Here we will give the proofs of the main results.
Proof of Theorem 2.1
Proof of Theorem 2.3
Proof of Theorem 2.4
Proof of Theorem 2.5
Proof of Theorem 3.1
Proof of Theorem 3.2
(i) If \(0\le p(t)\le p_{1}<1\), then we have \((1p_{1})l \le 0\), which contradicts \(l>0\) and \(1p_{1}>0\).
Proof of Theorem 3.3
Proof of Theorem 3.4
If \(1/a(t)\) is bounded and \(1< p_{2}\le p(t)\), from the proof of Theorem 3.2, we have \(\lim_{t \to\infty}(x(t)p(t)x(t\tau))=0\) and thus \(\lim_{t\rightarrow\infty}x(t)=0\). Therefore, (3.1) is bounded almost oscillatory. □
Declarations
Acknowledgements
This research is supported by the NNSF of China via Grant 11171306 and the interdisciplinary research funding from Zhejiang University of Science and Technology via Grant 2012JC09Y.
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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