Abstract and Applied Analysis

Volume 2012 (2012), Article ID 457163, 9 pages

doi:10.1155/2012/457163

Research Article

Subharmonic Solutions with Prescribed Minimal Periodic for a Class of Second-Order Impulsive Functional Differential Equations

Qi Wang, Dicheng Lu, and Zhaoyue Zheng

School of Mathematical Sciences, Anhui University, Hefei 230039, China

Received 19 May 2012; Revised 25 June 2012; Accepted 25 June 2012

Academic Editor: Yong Zhou

Copyright © 2012 Qi Wang 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.

By using critical point theory and variational methods, we investigate the subharmonic solutions with prescribed minimal period for a class of second-order impulsive functional differential equations. The conditions for the existence of subharmonic solutions are established. In the end, we provide an example to illustrate our main results.

1. Introduction

During the last 40 years, the theory and applications of impulsive differential equations have been developed, see [1–28]. Recently, some researchers studied the minimal period problem or homoclinic solution for some classes of Hamiltonian systems and classical pendulum equations [29–35]. In [30, 31], using the variational methods and decomposition technique, Yu got some sufficient conditions for the existence of periodic solutions with minimal period for the following nonautonomous Hamiltonian systems:

and a classical forced pendulum equation:

respectively. In [35], by using critical point theory and variational methods, Luo et al. considered the existence results of subharmonic solutions with prescribed minimal period for a class of second-order impulsive differential equations:

where , , , , , , , , and if , while if .

Motivated by [30, 31, 35], in this paper, we consider the existence results of subharmonic solutions with prescribed minimal period for a class of second-order impulsive functional differential equations:

where , , , , , , , , and if , while if .

We make the following assumptions.

  is -periodic in for any , where is a positive integer.  is -periodic in and continuously differentiable for any such that and , where and are -periodic functions in .  There are constants , , , such that

 where .  Suppose is rational. If is a periodic function with minimal period , and is a periodic function with minimal period , then is necessarily an integer.

From , we have

Therefore, under the assumptions -, the existence of subharmonic solutions with minimal period for (1.4) has been changed into the existence of subharmonic solutions with minimal period for

The outline of the paper is as follows. In Section 2, some preliminaries and basic results are established. In Section 3, by using critical point theory, we give sufficient conditions for the existence of of subharmonic solutions with minimal period for the impulsive systems. In Section 4, we give an example to illustrate the application of our main result

2. Preliminaries and Basic Results

In the following, we introduce some notations and some necessary definitions.

Let . The norm in is denoted by . Denote the Sobolov space by

with the inner product

which induces the norm

It is easy to verify that is a reflexive Banach space.

Consider the functional defined on by

where .

We should caution that the solutions minimal periods may not be . Define , and as the smallest prime factor of .

Define , a subspace of the Sobolev space . For any has a Fourier series expansion . Moreover, if and only if .

We will show that the classic -solutions of (1.4) or (1.4)′ is equivalent to finding the critical points of .

Similar to the proof [13, 36, 37], we have two lemmas as following.

Lemma 2.1. Suppose that are continuous. Then, the following statements are equivalent:

(1) is a critical point of ;(2) is a classical solution of (1.4) or (1.4)′.

Lemma 2.2. If is a critical point of on , then is also a critical point of on . And the minimal period of is an integer multiple of .

Now we state some results on nonlinear functional analysis and critical point theory. Suppose that is a Banach space and . Say that is weakly lower semicontinuous if means and is coercive if .

Lemma 2.3 (see [38]). Let be a real reflexive Banach space and weak sequentially closed. is weakly lower semicontinuous and coercive. Then, has a critical point with .

Similar to the proof of [35, Lemma 2.3], we have the following lemma.

Lemma 2.4. Suppose that - hold. is a weak sequentially closed and is coercive and weakly lower semicontinuous on .

3. Main Results

Theorem 3.1. Suppose that hold. If

then (1.4) has at least one classical periodic solution with the minimal period .

Proof. It follows from Lemmas 2.3 and 2.4 that has a critical point with . Next, we show the minimal period of is . For the sake of a contradiction, let the minimal period of be for some integer . By Lemma 2.2, we know that is a factor of , and so .

By the Wirtinger inequality and , we have

On the other hand, let . Then, is -periodic with minimal periodic . Since and are -periodic, we have

By the Wirtinger inequality and , we also have

If , then this is clearly in contradiction with the assumption for . Now, we are going to choose some positive number such that

Actually, we can choose . Then, we need to prove

This is true under the assumption (3.1). Hence, the proof is complete.

4. Example

Suppose

Then,

Let

Consider the following impulsive system:

where if , while if .

Proof. Let , , , , , , , . It is easy to check all the assumptions of Theorem 3.1 are satisfied. Thus, (4.4) has a periodic solution with the minimal period .

Acknowlegdment

Research was supported by Anhui Provincial Nature Science Foundation (090416237, 1208085MA13), Research Fund for Doctoral Station of Ministry of Education of China (20103401120002, 20113401110001), 211 Project of Anhui University (02303129, 02303303-33030011, 02303902-39020011, KJTD002B), and Foundation of Anhui Education Bureau (KJ2012A019).

References

V. Lakshmikantham, D. D. Baĭnov, and P. S. Simeonov, Theory of Impulsive Differential Equations, vol. 6 of Series in Modern Applied Mathematics, World Scientific, Teaneck, NJ, USA, 1989.

D. D. Bainov and P. S. Simeonov, Differential Equations: Periodic Solutions and Applications, Longman, Harlow, UK, 1993.

A. M. Samoĭlenko and N. A. Perestyuk, Impulsive Differential Equations, vol. 14 of World Scientific Series on Nonlinear Science. Series A: Monographs and Treatises, World Scientific, River Edge, NJ, USA, 1995.

L. F. Xi, B. Q. Yan, and Y. S. Liu, Introduction of Impulsive Differential Equations, Science Press, Beijing, China, 2005.

G. Tr. Stamov, “On the existence of almost periodic solutions for the impulsive Lasota-Wazewska model,” Applied Mathematics Letters, vol. 22, no. 4, pp. 516–520, 2009.

S. Ahmad and I. M. Stamova, “Asymptotic stability of an -dimensional impulsive competitive system,” Nonlinear Analysis, vol. 8, no. 2, pp. 654–663, 2007.

Q. Wang and B. Dai, “Existence of positive periodic solutions for a neutral population model with delays and impulse,” Nonlinear Analysis, vol. 69, no. 11, pp. 3919–3930, 2008.

F. Chen, “Periodic solutions and almost periodic solutions for a delay multispecies logarithmic population model,” Applied Mathematics and Computation, vol. 171, no. 2, pp. 760–770, 2005.

S. Gao, L. Chen, J. J. Nieto, and A. Torres, “Analysis of a delayed epidemic model with pulse vaccination and saturation incidence,” Vaccine, vol. 24, no. 35-36, pp. 6037–6045, 2006.

J. Chu and J. J. Nieto, “Impulsive periodic solutions of first-order singular differential equations,” Bulletin of the London Mathematical Society, vol. 40, no. 1, pp. 143–150, 2008.

B. Ahmad and J. J. Nieto, “Existence and approximation of solutions for a class of nonlinear impulsive functional differential equations with anti-periodic boundary conditions,” Nonlinear Analysis, vol. 69, no. 10, pp. 3291–3298, 2008.

J. Li, J. J. Nieto, and J. Shen, “Impulsive periodic boundary value problems of first-order differential equations,” Journal of Mathematical Analysis and Applications, vol. 325, no. 1, pp. 226–236, 2007.

J. J. Nieto and D. O'Regan, “Variational approach to impulsive differential equations,” Nonlinear Analysis, vol. 10, no. 2, pp. 680–690, 2009.

Z. Zhang and R. Yuan, “An application of variational methods to Dirichlet boundary value problem with impulses,” Nonlinear Analysis, vol. 11, no. 1, pp. 155–162, 2010.

J. Zhou and Y. Li, “Existence and multiplicity of solutions for some Dirichlet problems with impulsive effects,” Nonlinear Analysis, vol. 71, no. 7-8, pp. 2856–2865, 2009.

J. Sun and H. Chen, “Multiplicity of solutions for a class of impulsive differential equations with Dirichlet boundary conditions via variant fountain theorems,” Nonlinear Analysis, vol. 11, no. 5, pp. 4062–4071, 2010.

J. Sun, H. Chen, and L. Yang, “The existence and multiplicity of solutions for an impulsive differential equation with two parameters via a variational method,” Nonlinear Analysis, vol. 73, no. 2, pp. 440–449, 2010.

J. Sun, H. Chen, J. J. Nieto, and M. Otero-Novoa, “The multiplicity of solutions for perturbed second-order Hamiltonian systems with impulsive effects,” Nonlinear Analysis, vol. 72, no. 12, pp. 4575–4586, 2010.

J. J. Nieto, “Variational formulation of a damped Dirichlet impulsive problem,” Applied Mathematics Letters, vol. 23, no. 8, pp. 940–942, 2010.

L. Chen and J. Sun, “Nonlinear boundary value problem of first order impulsive functional differential equations,” Journal of Mathematical Analysis and Applications, vol. 318, no. 2, pp. 726–741, 2006.

D. Zhang and B. Dai, “Infinitely many solutions for a class of nonlinear impulsive differential equations with periodic boundary conditions,” Computers & Mathematics with Applications, vol. 61, no. 10, pp. 3153–3160, 2011.

H. Zhang and Z. Li, “Variational approach to impulsive differential equations with periodic boundary conditions,” Nonlinear Analysis, vol. 11, no. 1, pp. 67–78, 2010.

X. Han and H. Zhang, “Periodic and homoclinic solutions generated by impulses for asymptotically linear and sublinear Hamiltonian system,” Journal of Computational and Applied Mathematics, vol. 235, no. 5, pp. 1531–1541, 2011.

H. Zhang and Z. Li, “Periodic and homoclinic solutions generated by impulses,” Nonlinear Analysis, vol. 12, no. 1, pp. 39–51, 2011.

J. Sun, H. Chen, and J. J. Nieto, “Infinitely many solutions for second-order Hamiltonian system with impulsive effects,” Mathematical and Computer Modelling, vol. 54, no. 1-2, pp. 544–555, 2011.

J. Sun and H. Chen, “Multiplicity of solutions for a class of impulsive differential equations with Dirichlet boundary conditions via variant fountain theorems,” Nonlinear Analysis, vol. 11, no. 5, pp. 4062–4071, 2010.

J. Sun, H. Chen, and L. Yang, “The existence and multiplicity of solutions for an impulsive differential equation with two parameters via a variational method,” Nonlinear Analysis, vol. 73, no. 2, pp. 440–449, 2010.

J. Sun, H. Chen, J. J. Nieto, and M. Otero-Novoa, “The multiplicity of solutions for perturbed second-order Hamiltonian systems with impulsive effects,” Nonlinear Analysis, vol. 72, no. 12, pp. 4575–4586, 2010.

Q. Wang, Z.-Q. Wang, and J.-Y. Shi, “Subharmonic oscillations with prescribed minimal period for a class of Hamiltonian systems,” Nonlinear Analysis, vol. 28, no. 7, pp. 1273–1282, 1997.

J. Yu, “Subharmonic solutions with prescribed minimal period of a class of nonautonomous Hamiltonian systems,” Journal of Dynamics and Differential Equations, vol. 20, no. 4, pp. 787–796, 2008.

J. Yu, “The minimal period problem for the classical forced pendulum equation,” Journal of Differential Equations, vol. 247, no. 2, pp. 672–684, 2009.

E. Serra, M. Tarallo, and S. Terracini, “Subharmonic solutions to second-order differential equations with periodic nonlinearities,” Nonlinear Analysis, vol. 41, no. 5-6, pp. 649–667, 2000.

Y. Long, “Applications of Clark duality to periodic solutions with minimal period for discrete Hamiltonian systems,” Journal of Mathematical Analysis and Applications, vol. 342, no. 1, pp. 726–741, 2008.

T. An, “On the minimal periodic solutions of nonconvex superlinear Hamiltonian systems,” Journal of Mathematical Analysis and Applications, vol. 329, no. 2, pp. 1273–1284, 2007.

Z. Luo, J. Xiao, and Y. Xu, “Subharmonic solutions with prescribed minimal period for some second-order impulsive differential equations,” Nonlinear Analysis, vol. 75, no. 4, pp. 2249–2255, 2012.

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.

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.

J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems, vol. 74 of Applied Mathematical Sciences, Springer, New York, NY, USA, 1989.