 Research
 Open Access
Asymptotic behavior of impulsive neutral delay differential equations with positive and negative coefficients of Euler form
 Fangfang Jiang^{1}Email author,
 Jianhua Shen^{2} and
 Zhicheng Ji^{3}
https://doi.org/10.1186/s1366201815034
© The Author(s) 2018
 Received: 27 July 2017
 Accepted: 23 January 2018
 Published: 6 March 2018
Abstract
Keywords
 Impulse
 Neutral differential equation
 Unbounded delay
 Positive and negative coefficients of Euler form
 Constant jump
MSC
 34K45
 34D05
 34K20
1 Introduction
According to the order of derivative, differential equations can be classified into integerorder and fractional differential equations. Fractional differential equations are a generalization of arbitrary nonintegerorder equations. Both of them are unified and widely used in mathematical modeling of practical applications in the real world. For more detail on the theory, see, for example, [1–3] and references therein. However, many dynamical systems possess an impulsive dynamical behavior due to abrupt changes at certain instants during the evolution process. The mathematical description of these phenomena leads to impulsive differential equations [4]. Indeed, they appeared as a more natural framework for mathematical modeling of many realworld phenomena often and occur in applied science and engineering [4–8], for example, in as physics, population dynamics, ecology systems, optimal control, industrial robotic, etc. The idea of impulsive differential equations has been a subject of interest not only among mathematicians, but also among physicists and engineers.
This paper is organized as follows. In Section 2, we present some preliminaries. In Section 3, we state and prove our main results. In Section 4, we give an example to illustrate the obtained results. Conclusion is outlined in Section 5.
2 Preliminaries
 \((H1)\) :

There exist \(M_{2}\geq M_{1}>0\) such that \(M_{1}\leq\frac {f(x)}{x}\leq M_{2}\) for \(x\neq0\).
 \((H2)\) :

There exist \(0< N_{1}\leq N_{2}\leq1\) such that \(N_{1}\leq \frac{g(x)}{x}\leq N_{2}\) for \(x\neq0\).
 \((H3)\) :

The integral \(\int_{t}^{+\infty}h(s)\,ds\) is convergent for \(t\geq t_{0}\).
 \((H4)\) :

\(\tau(t_{k})\), \(k\in\mathbb{Z}_{+}\), are not impulsive points.
It is easy to show the global existence and uniqueness of solutions of the initial value problem (2.1)–(2.2). In the following, we give two relevant definitions.
Definition 2.1
 (1)
\(x(t)=\varphi(t)\) for \(t\in[t_{0}\gamma,t_{0}]\), and \(x(t)\) is continuous for \(t\geq t_{0}\), \(t\neq t_{k}\), \(k\in\mathbb{Z}_{+}\);
 (2)
\(x(t)C(t)g(x(\tau(t)))\) is continuously differentiable for \(t\geq t_{0}\), \(t\neq t_{k}\), \(t\neq t_{k}/\alpha\), \(t\neq t_{k}/\beta\), \(k\in \mathbb{Z}_{+}\), and satisfies (2.1);
 (3)
\(x(t_{k}^{+})\) and \(x(t_{k}^{})\) exist with \(x(t_{k})=x(t_{k}^{})\), \(k\in \mathbb{Z}_{+}\), and satisfy (2.1).
Definition 2.2
A solution \(x(t)\) is said to be eventually positive (negative) if it is positive (negative) for all sufficiently large t. It is called an oscillatory solution if it is neither eventually positive nor eventually negative. Otherwise, it is called as a nonoscillatory solution.
3 Main results
Theorem 3.1
Proof
Since \(0\leq \liminf_{t\to+\infty}C(t)\leq \limsup_{t\to +\infty}C(t)=C<1\), we have three possible cases.
Case I. If \(0<\liminf_{t\to+\infty}C(t)<\limsup_{t\to+\infty}C(t)=C<1\), then \(C(t)\) is eventually positive or eventually negative. Otherwise there exists a sequence \(\{\xi_{k}\}\) with \(\xi_{k}\to+\infty\) as \(k\to+\infty\) such that \(\lim_{k\to+\infty}C(\xi_{k})=0\), a contradiction. Hence we can find a sufficiently large T such that \(0<C(t)<1\) for \(t>T\).
 (1)\(1< C(t)<0\) for \(t>\max\{T, \tau(u_{n_{0}}), \tau (v_{n_{0}})\}\). We have that$$ \begin{gathered} \mu=\lim_{n\to+\infty}\bigl[x(u_{n})C(u_{n})x \bigl(\tau(u_{n})\bigr)\bigr]\leq\lim_{n\to +\infty}x(u_{n})+ \limsup_{n\to+\infty}\bigl[C(u_{n})x\bigl( \tau(u_{n})\bigr)\bigr]\leq C\eta, \\ \mu=\lim_{n\to+\infty}\bigl[x(v_{n})C(v_{n})x \bigl(\tau(v_{n})\bigr)\bigr]\geq\lim_{n\to +\infty}x(v_{n})+ \liminf_{n\to+\infty}\bigl[C(v_{n})x\bigl( \tau(v_{n})\bigr)\bigr]\geq\eta. \end{gathered} $$
 (2)\(0< C(t)<1\) for \(t>\max\{T, \tau(u_{n_{0}}), \tau(v_{n_{0}})\} \). We have that$$ \begin{gathered} 0=\lim_{n\rightarrow+\infty}x(u_{n})\geq \lim_{n\rightarrow+\infty }\bigl[x(u_{n})C(u_{n})x\bigl( \tau(u_{n})\bigr)\bigr]+\liminf_{n\rightarrow\infty } \bigl[C(u_{n})x\bigl(\tau( u_{n})\bigr)\bigr]\geq\mu, \\ \eta=\lim_{n\rightarrow+\infty}x(v_{n})\leq\lim _{n\rightarrow \infty}\bigl[x(v_{n})C(v_{n})x\bigl(\tau( v_{n}) \bigr)\bigr]+\limsup_{n\rightarrow\infty } \bigl[C(v_{n})x\bigl(\tau(v_{n} )\bigr)\bigr]\leq\mu+C\eta. \end{gathered} $$
Case II. If \(0=\liminf_{t\to+\infty}C(t)<\limsup_{t\to+\infty}C(t)=C\), then as in Case I, we get \(\lim_{t\to+\infty}x(t)=0\).
Case III. If \(\liminf_{t\to+\infty}C(t)=\limsup_{t\to+\infty}C(t)=C\), then the proof is as in Theorem 2.1 in [22] and so is omitted. The proof is complete. □
Theorem 3.2
Proof
4 Example
On one hand, \(\int_{2e}^{+\infty}\frac{1}{4t\ln t}\,dt=+\infty\) and \(\int_{\frac{t}{2e}}^{t}\frac{dt}{2t\ln t}<\frac{1}{2}\ln2<\frac {1CN_{2}}{M_{2}}\) for t sufficiently large. So by Theorem 3.1, every nonoscillatory solution of (4.1) tends to zero as \(t\to +\infty\).
On the other hand, by simple computations we have \(I_{1}(t)=\frac {1}{4}\ln\frac{\ln2et}{\ln\frac{t}{2e}}\) and \(I_{2}(t)=\frac {1}{4}\ln\frac{\ln\frac{t}{e}}{\ln\frac{t}{2e}}\). Furthermore, \(\limsup_{t\to+\infty}(I_{1}(t)+I_{2}(t))=0\). So by Theorem 3.2 every oscillatory solution of (4.1) tends to zero as \(t\to+\infty\). In conclusion, every solution of (4.1) tends to zero as \(t\to+\infty\).
5 Conclusion
In this paper, we have investigated asymptotic properties of solutions for an impulsive neutral differential equation with positive and negative coefficients, unbounded delays, forced term, and constant impulsive jumps. By constructing auxiliary functions, using analytical method and combining with the technique of considering asymptotic behaviors of nonoscillatory and oscillatory solutions, we have provided two criteria for tending to zero of every (non)oscillatory solution of the system as \(t\to+\infty\). Finally, as an application, we have given an example to illustrate the effectiveness of the obtained results.
Declarations
Acknowledgements
The authors express their gratitude to the Professor D. Baleanu and two reviewers for their constructive suggestions, which improved the final version of this paper. Research of FJ and JS is supported by the National Natural Science Foundation of China (11701224), the Provincial Youth Foundation of JiangSu Province (BK20170168), the China Postdoctoral Science Foundation (2017M611685), the Zhejiang Provincial Natural Science Foundation of China (LY14A010024), and the Fundamental Research Funds for the Central Universities (JUSRP11723).
Authors’ contributions
All authors have read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
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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