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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
MSC
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
References
 Podlubny, I.: Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications. Academic Press, San Diego (1998) MATHGoogle Scholar
 Diethelm, K., Ford, N.J.: Analysis of fractional differential equations. J. Math. Anal. Appl. 265(2), 229–248 (2002) MathSciNetView ArticleMATHGoogle Scholar
 Baleanu, D., Hedayati, V., Rezapour, Sh.: Al Qurashi, M.M.: On two fractional differential inclusions. SpringerPlus 5, Article ID 882 (2016) View ArticleGoogle Scholar
 Lakshmikantham, V., Bainov, D.D., Simeonov, P.S.: Theory of Impulsive Differential Equations. World Scientific, Singapore (1989) View ArticleMATHGoogle Scholar
 Bainov, D.: Systems with Impulse Effect: Stability, Theory and Applications (1989) Google Scholar
 Yang, L., Tian, B.: Asymptotic properties of a stochastic nonautonomous competitive system with impulsive perturbations. Adv. Differ. Equ. (2017). https://doi.org/10.1186/s1366201712565 MathSciNetGoogle Scholar
 Wu, R., Zou, X., Wang, K.: Asymptotic behavior of a stochastic nonautonomous predator–prey model with impulsive perturbations. Commun. Nonlinear Sci. Numer. Simul. 20, 965–974 (2015) MathSciNetView ArticleMATHGoogle Scholar
 Yang, Y., He, Y., Wang, Y., et al.: Stability analysis for impulsive fractional hybrid systems via variational Lyapunov method. Commun. Nonlinear Sci. Numer. Simul. 45, 140–157 (2017) MathSciNetView ArticleGoogle Scholar
 Baleanu, D., Mousalou, A., Rezapour, Sh.: A new method for investigating approximate solutions of some fractional integrodifferential equations involving the Caputo–Fabrizio derivative. Adv. Differ. Equ. 2017(1), Article ID 51 (2017) MathSciNetView ArticleMATHGoogle Scholar
 Shabibi, M., Rezapour, Sh., Vaezpour, S.M.: A singular fractional integrodifferential equation. UPB Sci. Bull., Ser. A 79(1), 109–118 (2017) MathSciNetGoogle Scholar
 Shabibi, M., Postolache, M., Rezapour, Sh., Vaezpour, S.M.: Investigation of a multisingular pointwise defined fractional integrodifferential equation. J. Math. Anal. 7(5), 61–77 (2016) MathSciNetMATHGoogle Scholar
 Baleanu, D., Rezapour, Sh., Mohammadi, H.: Some existence results on nonlinear fractional differential equations. Philos. Trans. R. Soc. A, Math. Phys. Eng. Sci. 371(1990), Article ID 20120144 (2013) MathSciNetView ArticleMATHGoogle Scholar
 Baleanu, D., Mohammadi, H., Rezapour, Sh.: The existence of solutions for a nonlinear mixed problem of singular fractional differential equations. Adv. Differ. Equ. 2013(1), Article ID 359 (2013) MathSciNetView ArticleMATHGoogle Scholar
 Guan, K.Z., Shen, J.H.: Asymptotic behavior of solutions of a firstorder impulsive neutral differential equation in Euler form. Appl. Math. Lett. 24, 1218–1224 (2011) MathSciNetView ArticleMATHGoogle Scholar
 Pandian, S., Balachandran, Y.: Asymptotic behavior results for nonlinear impulsive neutral differential equations with positive and negative coefficients. Bonfring Int. J. Data Min. 2, 13–21 (2012) Google Scholar
 Shen, J.H., Liu, Y.J., Li, J.L.: Asymptotic behavior of solutions of nonlinear neutral differential equations with impulses. J. Math. Anal. Appl. 322, 179–189 (2007) MathSciNetView ArticleMATHGoogle Scholar
 Tariboon, J., Ntouyas, S., Thaiprayoon, C.: Asymptotic behavior of solutions of mixed type impulsive neutral differential equations. Adv. Differ. Equ. 2014, Article ID 327 (2014) MathSciNetView ArticleGoogle Scholar
 Wei, G.P., Shen, J.H.: Asymptotic behavior for a class of nonlinear impulsive neutral delay differential equations. J. Math. Phys. 30, 753–763 (2010) MathSciNetMATHGoogle Scholar
 Wei, G.P., Shen, J.H.: Asymptotic behavior of solutions of nonlinear impulsive delay differential equations with positive and negative coefficients. Math. Comput. Model. 44, 1089–1096 (2006) MathSciNetView ArticleMATHGoogle Scholar
 Jiang, F.F., Sun, J.T.: Asymptotic behavior of neutral delay differential equation of Euler form with constant impulsive jumps. Appl. Math. Comput. 219, 9906–9913 (2013) MathSciNetMATHGoogle Scholar
 Jiang, F.F., Shen, J.H.: Asymptotic behaviors of nonlinear neutral impulsive delay differential equations with forced term. Kodai Math. J. 35, 126–137 (2012) MathSciNetView ArticleMATHGoogle Scholar
 Liu, X.Z., Shen, J.H.: Asymptotic behavior of solutions of impulsive neutral differential equations. Appl. Math. Lett. 12, 51–58 (1999) MathSciNetView ArticleMATHGoogle Scholar
 Zhao, A., Yan, J.: Asymptotic behavior of solutions of impulsive delay differential equations. J. Math. Anal. Appl. 201, 943–954 (1996) MathSciNetView ArticleMATHGoogle Scholar