Research Article | Open Access

Qi Wang, Dicheng Lu, Zhaoyue Zheng, "Subharmonic Solutions with Prescribed Minimal Periodic for a Class of Second-Order Impulsive Functional Differential Equations", *Abstract and Applied Analysis*, vol. 2012, Article ID 457163, 9 pages, 2012. https://doi.org/10.1155/2012/457163

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

**Academic Editor:**Yong Zhou

#### 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).

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#### Copyright

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.