Mathematical Problems in Engineering

Volume 2017 (2017), Article ID 5041783, 5 pages

https://doi.org/10.1155/2017/5041783

## Multiple Periodic Solutions for a Class of Second-Order Neutral Impulsive Functional Differential Equations

^{1}College of Mathematics and Statistics, Jishou University, Jishou, Hunan 416000, China^{2}Department of Mathematics, Hunan Normal University, Changsha, Hunan 410081, China^{3}Department of Mathematics, Hunan First Normal University, Changsha, Hunan 410205, China

Correspondence should be addressed to Jingli Xie

Received 29 August 2016; Revised 1 December 2016; Accepted 20 December 2016; Published 12 January 2017

Academic Editor: Olfa Boubaker

Copyright © 2017 Jingli Xie et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

In this paper, we study a class of second-order neutral impulsive functional differential equations. Under certain conditions, we establish the existence of multiple periodic solutions by means of critical point theory and variational methods. We propose an example to illustrate the applicability of our result.

#### 1. Introduction and Main Results

In this paper we consider a class of second-order neutral impulsive functional differential equationswhere , , and . The operator is defined as , where denotes the right-hand (left-hand) limit of at . , is a constant with and is a given positive integer.

The necessity to study delay differential equations is due to the fact that these equations are useful mathematical tools in modeling many real processes and phenomena studied in biology, medicine, chemistry, physics, engineering, economics, and so forth [1, 2].

On the other hand, impulsive differential equation not only is richer than the corresponding theory of differential equations but also represents a more natural framework for mathematical modeling of real world phenomena. People generally consider impulses in positions and for the second-order differential equation . However it is well known that in the motion of spacecraft instantaneous impulses depend on the position which result in jump discontinuities in velocity, with no change in position.

Thus, it is more realistic to consider the case of combined effects: impulses and time delays. This motivates us to consider neutral impulsive functional differential system (1).

The existence of periodic solutions of delay differential equations has been focused on by many researchers [3–6]. Several available approaches to tackle them include Lyapunov method, Fourier analysis method, fixed point theory, and coincidence degree theory [7–10]. Recently, some researchers have studied the existence of solutions for delay differential equations via variational methods [11–13]. In recent years, some researchers, by using critical point theory, have studied the existence of solutions for boundary value problems, periodic solutions, and homoclinic orbits of impulsive differential systems [14–19].

In this paper, we aim to establish existence of multiple periodic solutions for the second-order neutral impulsive functional differential equation (1) by using critical point theory and variational methods.

For (1) with , Shu and Xu [20] obtained the following periodic solutions result.

**Theorem A**.* Assume that the following conditions are satisfied. *(H1).(H2)* There exists a function ** such that *(H3)* is **-periodic in **.*(H4)* satisfies ** and **.*(H5)* if and only if **, **.*(H6)*, where **, **.*(H7)* There exists a constant ** such that when **, *, .*Moreover, if there exists an integer ** such that ** satisfying **then the system **possesses at least ** nonzero solutions with the period **.*

Our main result is stated as follows.

Theorem 1. *Assume that (H1)–(H7) and the following condition are satisfied. *(H8)* is odd about , and there exists a constant such that , where .**Moreover, if there exists an integer such thatthen system (1) admits at least nonzero solutions with the period .*

Clearly, when , Theorem 1 generalizes Theorem A.

Note that the first equation of system (1) is equivalent to the following equation:where and .

The rest of this paper is organized as follows. In Section 2, we present some preliminaries, which will be used to prove our main result. In Section 3 we prove our main result and provide an example to illustrate the applicability of our results.

#### 2. Some Preliminaries

Let Then is a separable and reflexive Banach space and the inner product induces the norm

*Definition 2. *A function is a solution of system (1) if the function satisfies system (1).

Define a functional asThen is Fréchet differentiable at any . For any , by a simple calculation, we have From (H3), we get

Therefore, the corresponding Euler equation of functional is

Note that (6) is equivalent to system (13) and critical points of the functional are classical -periodic solutions of system (1).

*Definition 3 (see [21]). *Let be a real reflexive Banach space, and Define as follows: Then we say is the genus of .

Denote and , where .

Lemma 4 (see [22]). *Let be a real Banach space and with even functional and satisfying the Palais-Smale (PS) condition. Suppose and *(i)*if there exist an -dimensional subspace of and a constant such that* *where is an open ball of radius in centered at , then we have ;*(ii)*if there exists -dimensional subspace of such that* *then we have .** Moreover, if , then possesses at least distinct critical points.*

#### 3. Proof of Theorem 1 and an Example

We apply Lemma 4 to finish the proof. Under assumption (H4), it is easy to see that if function is a solution of system (1), then function is also a solution of system (1). Therefore, the solutions of system (1) are a set which is symmetric with respect to the origin in . It follows directly from (10), (H5), and (H8) that is even in and . The rest of the proof is divided into three steps.

*Step 1. *We show that the functional satisfies assumption (ii) of Lemma 4.

It follows from (H7) that there exists a constant such thatwhere . Combining (10) and (18), we getwhich implies that is bounded from below. By condition (ii) of Lemma 4, we have .

*Step 2. *We show that the functional satisfies the PS condition.

For any given sequence such that is bounded and , there exists a constant such thatwhere is the dual space of .

Combining (19) and (20), we haveIt follows that is bounded.

Since is a reflexive Banach space, so we may extract a weakly convergent subsequence, for simplicity, we also note again by , in . So we haveTherefore, by (22), we have . Hence the functional satisfies the PS condition.

*Step 3. *We show that the functional satisfies assumption (i) of Lemma 4.

Let , . By calculations, we obtain Define the -dimensional linear subspace as follows: It is clear to see that is a symmetric set. Take , when , where denotes boundary of , has expansion , , andBy (H6), for given with , there exists such that when , we have Combining (10), (25), and (26), when , we have Therefore . Consequently, system (1) admits at least nonzero -periodic solutions.

We conclude this section with the following example.

*Example 5. *Consider (1) with It is easy to see that and when , ; then (H1) and (H5) hold. Set , , , and then . By a simple computation, we have , , and . So conditions (H2)–(H4) hold. Clearly, the conditions (H6)–(H8) hold. Therefore system (1) admits at least nonzero solutions with the period .

#### Competing Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgments

This work was partially supported by Hunan Provincial Natural Science Foundation of China (no. 2016JJ6122), National Natural Science Foundation of China (nos. 11661037 and 11471109), and Jishou University Doctor Science Foundation (no. jsdxxcfxbskyxm201504).

#### References

- J. K. Hale,
*Theory of Functional Differential Equations*, Springer, New York, NY, USA, 1977. View at MathSciNet - Y. Kuang,
*Delay differential equations with applications in population dynamics*, Mathematics in Science and Engineering, Academic Press, Inc., Boston, MA, USA, 1993. View at MathSciNet - W. Wang, P. Fergola, and C. Tenneriello, “Global attractivity of periodic solutions of population models,”
*Journal of Mathematical Analysis and Applications*, vol. 211, no. 2, pp. 498–511, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - R. Olach, “Positive periodic solutions of delay differential equations,”
*Applied Mathematics Letters*, vol. 26, no. 12, pp. 1141–1145, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - J. Yu and H. Xiao, “Multiple periodic solutions with minimal period 4 of the delay differential equation $\dot{x}\left(t\right)=-f\left(t,x\left(t-1\right)\right)$,”
*Journal of Differential Equations*, vol. 254, no. 5, pp. 2158–2172, 2013. View at Publisher · View at Google Scholar - E. Serra, “Periodic solutions for some nonlinear differential equations of neutral type,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 17, no. 2, pp. 139–151, 1991. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - B. Yang, R. Ma, and C. Gao, “Positive periodic solutions of delayed differential equations,”
*Applied Mathematics and Computation*, vol. 218, no. 8, pp. 4538–4545, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - X. M. Li and X. P. Yuan, “Quasi-periodic solutions for perturbed autonomous delay differential equations,”
*Journal of Differential Equations*, vol. 252, no. 6, pp. 3752–3796, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - K. Wu and X. Wu, “Multiplicity results of periodic solutions for a class of first order delay differential equations,”
*Journal of Mathematical Analysis and Applications*, vol. 390, no. 2, pp. 427–438, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - J. Wu and Z. Wang, “Two periodic solutions of second-order neutral functional differential equations,”
*Journal of Mathematical Analysis and Applications*, vol. 329, no. 1, pp. 677–689, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - H. Xiao and Z. Guo, “Multiplicity and minimality of periodic solutions to delay differential system,”
*Electronic Journal of Differential Equations*, vol. 39, no. 115, pp. 1–12, 2014. View at Google Scholar · View at MathSciNet - X.-B. Shu, Y.-T. Xu, and L. H. Huang, “Infinite periodic solutions to a class of second-order Sturm-Liouville neutral differential equations,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 68, no. 4, pp. 905–911, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - Z.-M. Guo and Y.-T. Xu, “Existence of periodic solutions to a class of second-order neutral differential difference equations,”
*Acta Analysis Functionalis Applicata*, vol. 5, no. 1, pp. 13–19, 2003. View at Google Scholar · View at MathSciNet - J. J. Nieto and D. O'Regan, “Variational approach to impulsive differential equations,”
*Nonlinear Analysis: Real World Applications*, vol. 10, no. 2, pp. 680–690, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - Z. Zhang and R. Yuan, “An application of variational methods to Dirichlet boundary value problem with impulses,”
*Nonlinear Analysis. Real World Applications*, vol. 11, no. 1, pp. 155–162, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - L. Bai and B. X. Dai, “An application of variational method to a class of Dirichlet boundary value problems with impulsive effects,”
*Journal of the Franklin Institute*, vol. 348, no. 9, pp. 2607–2624, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - Y. Tian, J. Wang, and W. Ge, “Variational methods to mixed boundary value problem for impulsive differential equations with a parameter,”
*Taiwanese Journal of Mathematics*, vol. 13, no. 4, pp. 1353–1370, 2009. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - J. Xie and Z. Luo, “Subharmonic solutions with prescribed minimal period of an impulsive forced pendulum equation,”
*Applied Mathematics Letters*, vol. 52, pp. 169–175, 2016. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - J. Xie and Z. Luo, “Homoclinic orbits for Hamiltonian systems induced by impulses,”
*Mathematical Methods in the Applied Sciences*, vol. 39, no. 9, pp. 2239–2250, 2016. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - X. B. Shu and Y. T. Xu, “Multiple periodic solutions to a class of second-order functional differential equations of mixed type,”
*Acta Mathematicae Applicatae Sinica*, vol. 29, no. 5, pp. 821–831, 2006. View at Google Scholar · View at MathSciNet - J. Mawhin and M. Willem,
*Critical Point Theory and Hamiltonian Systems*, Springer, Berlin, Germany, 1989. View at Publisher · View at Google Scholar · View at MathSciNet - C. Q. Ching,
*Critical Point Theory and Its Applications*, Shanghai Scientific and Technical Publishers, Shanghai, China, 1986.