 Research
 Open Access
Oscillation criteria for second order EmdenFowler functional differential equations of neutral type
 Yingzhu Wu^{1},
 Yuanhong Yu^{2}Email author,
 Jimin Zhang^{3} and
 Jinsen Xiao^{4}
https://doi.org/10.1186/s1366001612689
© The Author(s) 2016
 Received: 29 October 2016
 Accepted: 29 November 2016
 Published: 22 December 2016
Abstract
Keywords
 EmdenFowler equation
 oscillation criterion
 Riccati method
MSC
 34C10
 34K11
1 Introduction
 \((A_{1})\) :

\(r(t)\in C^{1}([t_{0},\infty),R)\), \(r(t)>0\), \(r^{\prime }(t)\geq0\);
 \((A_{2})\) :

\(p(t),q(t)\in C([t_{0},\infty),R)\), \(0\leq p(t)\leq 1\), \(q(t)\geq0\);
 \((A_{3})\) :

\(\tau(t)\in C([t_{0},\infty),R)\), \(\tau(t)\leq t\), \(\lim_{t\rightarrow\infty}\tau(t)=\infty\);
 \((A_{4})\) :

\(\sigma(t)\in C^{1}([t_{0},\infty),R)\), \(\sigma (t)>0\), \(\sigma^{\prime}(t)>0\), \(\sigma(t)\leq t\), \(\lim_{t\rightarrow \infty}\sigma(t)=\infty\).
A function \(x(t)\in C^{1}([t_{0},\infty),R)\), \(T_{x}\geq t_{0}\), is called a solution of equation (1) if it satisfies the property \(r(t)\vert z^{\prime}(t)\vert ^{\alpha1}z^{\prime}(t)\in C^{1}([T_{x},\infty),R)\) and equation (1) on \([T_{x},\infty)\). In this article we only consider the nontrivial solutions of equation (1), which ensure \(\sup{ \{\vert x(t)\vert :t\geq T \}}>0\) for all \(T\geq T_{x}\). A solution of equation (1) is said to be oscillatory if it has arbitrarily large zero point on \([T_{0},\infty)\); otherwise, it is called nonoscillatory. Moreover, equation (1) is said to be oscillatory if all its solutions are oscillatory.
Recently, there were a large number of papers devoted to the oscillation of the delay and neutral differential equations. We refer the reader to [1–20].
In 2011, Li et al. [4] considered further the oscillation criteria for equation (5), where \(\int^{\infty}_{t_{0}}\frac{1}{a(t)}\,dt<\infty\) and \(\alpha\geq1\). In fact, equations (2) and (5) cannot be contained in each other. So in 2012, Liu et al. [5] considered the oscillation criteria for second order generalized EmdenFowler equation (1) for the condition \(\alpha\geq\beta>0\).
In 2015, Zeng et al. [6] used the Riccati transformation technique to get some new oscillation criterion for equation (1) under the condition \(\alpha\geq\beta>0\) or \(\beta\geq \alpha>0\), which improves the related results reported in [5].
Now in this article we shall apply the generalized Riccati inequality to study of the oscillation criteria of equation (1) under a more general case, namely, for all \(\alpha> 0\) and \(\beta> 0\).
2 Results and proofs
Theorem 1
Proof
Remark 1
Theorems 15 of [1], Theorem 1 of [2] and [7] hold only for equation (1) with \(p(t)=0\) and \(\alpha =\beta\). Theorem 2.1 of [5] (or [6]) holds only for equation (1) with \(\alpha\geq\beta\), and Theorem 3.1 of [6] holds only for equation (1) with \(\beta\geq\alpha\). Hence our theorem improves and unifies the above results.
In the following, we shall use the generalized Riccati technique and the integral averaging technique to show a new Philos type oscillation criterion for equation (1).
 \((H_{1})\)::

\(H(t,t)=0\) for \(t\geq t_{0}\) and \(H(t,s)>0\) for all \((t,s)\in D_{0}\),
 \((H_{2})\)::

\(\frac{\partial H(t,s)}{\partial s}\geq0\) for all \((t,s)\in D\).
 \((H_{3})\)::

\(\frac{\partial H(t,s)}{\partial s}+\frac{\rho^{\prime }(s)}{\rho(s)}H(t,s)=h(t,s)H^{\frac{\lambda}{\lambda+1}}(t,s)\) for all \((t,s)\in D_{0}\).
Theorem 2
Proof
Corollary 1
Notice that by choosing specific functions ρ and H, it is possible to derive several oscillation criteria for equation (1) and its special cases, the halflinear equation (2) and the EmdenFowler equation (5).
Remark 2
Theorem 2.1 of [3] holds only for equation (1) with \(\alpha=1\) and \(\beta>1\), Theorem 2.2 of [5] holds only for equation (1) with \(\alpha\geq\beta\), Theorem 5 of [7] holds only for equation (1) with \(\beta\geq\alpha\). Hence, Theorem 2 improves and unifies above oscillation criteria.
Theorem 3
Proof
Remark 3
Theorem 2.2 of [2] holds only for equation (1) with \(p(t)=0\) and \(\alpha=\beta\), Theorem 2.12.3 of [4] hold only for \(\alpha=1\) and \(\beta\geq1\), Theorem 2.5 of [5] and Theorem 2.3 of [6] hold only for \(\alpha\geq\beta\). Our Theorem 3 holds for equation (1) with all \(\alpha>0\) and \(\beta>0\).
3 Examples
Now in this section we shall give two examples to illustrate our results.
Example 1
Example 2
Declarations
Acknowledgements
The first author is supported by the Guangdong Engineering Technology Research Center of Cloud Robot (Grant 2015B090903084), sponsored by Science and technology project of Guangdong Province, P.R. China. The fourth author is supported by the National Natural Science Foundation of China (Grant 11501131) and the Training Project for Young Teachers in Higher Education of Guangdong, China (Grant YQ2015117).
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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