Abstract

By using minimax methods in critical point theory, a new existence theorem of infinitely many periodic solutions is obtained for a class of second-order 𝑝-Laplacian systems with impulsive effects. Our result generalizes many known works in the literature.

1. Introduction

Consider the following 𝑝-Laplacian system with impulsive effects: 𝑑||||𝑑𝑡̇𝑢(𝑡)𝑝2||||̇𝑢(𝑡)𝐿(𝑡)𝑢(𝑡)𝑝2𝑢(𝑡)+𝐹(𝑡,𝑢(𝑡))=0,a.e.Δ||𝑡𝑡,𝑢(0)𝑢(𝑇)=̇𝑢(0)̇𝑢(𝑇)=0,̇𝑢𝑗||𝑝2𝑡̇𝑢𝑗=|||𝑡̇𝑢+𝑗|||𝑝2𝑡̇𝑢+𝑗||𝑡̇𝑢𝑗||𝑝2𝑡̇𝑢𝑗=𝐼𝑗𝑢𝑡𝑗,𝑗=1,2,,𝑚,(1.1) where 𝑝>1, 𝑇>0, 0=𝑡0<𝑡1<𝑡2<<𝑡𝑚<𝑡𝑚+1=𝑇, and 𝐼𝑗𝑁𝑁(𝑗=1,2,,𝑚) are continuous and 𝐹×𝑁 is 𝑇-periodic in 𝑡 for all 𝑢𝑁, 𝐹(𝑡,𝑢) is the gradient of 𝐹(𝑡,𝑢) with respect to 𝑢. 𝐿𝐶(,𝑁×𝑁) is a 𝑇-periodic positive definite symmetric matrix.

Throughout this paper, we always assume the following condition holds.(A)𝐹(𝑡,𝑥) is measurable in 𝑡 for all 𝑥𝑁 and continuously differentiable in 𝑥 for a.e. 𝑡[0,𝑇], and there exist 𝑎𝐶(+,+), 𝑏𝐿1([0,𝑇];+) such that ||||||||𝐹(𝑡,𝑥)𝑎(|𝑥|)𝑏(𝑡),𝐹(𝑡,𝑥)𝑎(|𝑥|)𝑏(𝑡)(1.2) for all 𝑥𝑁 and a.e. 𝑡[0,𝑇].

For the sake of convenience, in the sequel, we define 𝐵={1,2,,𝑚}.

When 𝑝=2, 𝐼𝑗0, 𝑗𝐵, problem (1.1) becomes the following second-order Hamiltonian system: ̈𝑢(𝑡)𝐿(𝑡)𝑢(𝑡)+𝐹(𝑡,𝑢(𝑡))=0,a.e.𝑡.(1.3)

There are many papers concerning the existence of periodic solutions or homoclinic solutions for problem (1.3) by minimax methods. Here for identifying a few, we only mention [18].

For 𝐼𝑗0, 𝑗𝐵, problem (1.1) involves impulsive effects. Impulsive differential equations are suitable for the mathematical simulation of evolutionary processes in which the parameters undergo relatively long periods of smooth variation followed by a short-term rapid change (that is jumps) in their values. Since these processes are subject to short-term perturbations whose duration is negligible in comparison with the duration of the processes, it is natural to suppose that these perturbations act instantaneously, that is, in the form of impulse. Processes of this type are often investigated in various fields of science and technology, for example, many biological phenomena involving thresholds, bursting rhythm models in medicine and biology, optimal control models in economics, pharmacokinetics and frequency modulated systems, and so on. For more details of impulsive differential equations, we refer the readers to the books [9, 10].

There are many methods for finding periodic solutions of impulsive differential equations, such as the monotone-iterative technique, a numerical-analytical method, the method of upper and lower solutions, and the method of bilateral approximations. For more information about periodic solutions of impulsive differential equations, one can refer to the papers [1118]. However, there are few papers [1925] concerning periodic solutions of impulsive differential equations by variational methods. So it is a novel method to employ variational methods to investigate the existence of periodic solutions for impulsive differential equations.

Motivated by the above papers, we study the existence of subharmonic solutions for problem (1.1) by applying minimax methods in critical point theory. Our result is new, which seems not to be found in the literature.

Throughout this paper, let 𝑞(1,+) satisfy 1/𝑝+1/𝑞=1.

2. Preliminaries

In this section, we recall some basic facts which will be used in the proofs of our main results. In order to apply the critical point theory, we construct a variational structure. With this variational structure, we can reduce the problem of finding solutions of problem (1.1) to that of seeking the critical points of the corresponding functional.

Let 𝑘 be a positive integer and 𝑊1,𝑝𝑘𝑇 the Sobolev space defined by𝑊1,𝑝𝑘𝑇=𝑢𝑁𝑢isabsolutelycontinuous,𝑢(𝑡)=𝑢(𝑡+𝑘𝑇),̇𝑢𝐿𝑝[]0,𝑘𝑇;𝑁(2.1) with the norm 𝑢=0𝑘𝑇||||𝑢(𝑡)𝑝𝑑𝑡+0𝑘𝑇||||̇𝑢(𝑡)𝑝𝑑𝑡1/𝑝.(2.2)

Take 𝑣𝑊1,𝑝𝑘𝑇 and multiply the two sides of the equality𝑑||||𝑑𝑡̇𝑢(𝑡)𝑝2||||̇𝑢(𝑡)𝐿(𝑡)𝑢(𝑡)𝑝2𝑢(𝑡)+𝐹(𝑡,𝑢(𝑡))=0(2.3) by 𝑣 and integrate from 0 to 𝑘𝑇; we have0𝑘𝑇||||̇𝑢(𝑡)𝑝2̇𝑢(𝑡),𝑣(𝑡)𝑑𝑡=0𝑘𝑇||||𝐿(𝑡)𝑢(𝑡)𝑝2𝑢(𝑡),𝑣(𝑡)𝑑𝑡0𝑘𝑇(𝐹(𝑡,𝑢(𝑡)),𝑣(𝑡))𝑑𝑡.(2.4) Moreover, by ̇𝑢(0)=̇𝑢(𝑇), one has0𝑘𝑇||||̇𝑢(𝑡)𝑝2̇𝑢(𝑡),𝑣(𝑡)𝑑𝑡=𝑘𝑇0||||̇𝑢(𝑡)𝑝2̇𝑢(𝑡),𝑣(𝑡)𝑑𝑡=𝑘𝑚𝑗=0𝑡𝑗+1𝑡𝑗||||̇𝑢(𝑡)𝑝2̇𝑢(𝑡),𝑣(𝑡)𝑑𝑡=𝑘𝑚𝑗=0|||𝑡̇𝑢𝑗+1|||𝑝2𝑡̇𝑢𝑗+1𝑣𝑡𝑗+1|||𝑡̇𝑢+𝑗|||𝑝2𝑡̇𝑢+𝑗𝑣𝑡+𝑗𝑡𝑗+1𝑡𝑗||||̇𝑢(𝑡)𝑝2̇̇𝑢(𝑡),𝑣(𝑡)𝑑𝑡=𝑘𝑚𝑗=0|||𝑡̇𝑢𝑗+1|||𝑝2𝑡̇𝑢𝑗+1𝑣𝑡𝑗+1|||𝑡̇𝑢+𝑗|||𝑝2𝑡̇𝑢+𝑗𝑣𝑡+𝑗0𝑘𝑇||||̇𝑢(𝑡)𝑝2̇||||̇𝑢(𝑡),𝑣(𝑡)𝑑𝑡=𝑘̇𝑢(𝑇)𝑝2||||̇𝑢(𝑇)𝑣(𝑇)𝑘̇𝑢(0)𝑝2̇𝑢(0)𝑣(0)𝑘𝑚𝑗=1𝐼𝑗𝑢𝑡𝑗𝑣𝑡𝑗0𝑘𝑇||||̇𝑢(𝑡)𝑝2̇̇𝑢(𝑡),𝑣(𝑡)𝑑𝑡=𝑘𝑚𝑗=1𝐼𝑗𝑢𝑡𝑗𝑣𝑡𝑗0𝑘𝑇||||̇𝑢(𝑡)𝑝2̇̇𝑢(𝑡),𝑣(𝑡)𝑑𝑡.(2.5) Together with (2.4), we get 0𝑘𝑇||||̇𝑢(𝑡)𝑝2̇̇𝑢(𝑡),𝑣(𝑡)𝑑𝑡+𝑘𝑚𝑗=1𝐼𝑗𝑢𝑡𝑗𝑣𝑡𝑗+0𝑘𝑇||||𝐿(𝑡)𝑢(𝑡)𝑝2=𝑢(𝑡),𝑣(𝑡)𝑑𝑡0𝑘𝑇(𝐹(𝑡,𝑢(𝑡)),𝑣(𝑡))𝑑𝑡.(2.6)

Definition 2.1. We say that a function 𝑢𝑊1,𝑝𝑘𝑇 is a weak solution of problem (1.1) if the identity 0𝑘𝑇||||̇𝑢(𝑡)𝑝2̇̇𝑢(𝑡),𝑣(𝑡)𝑑𝑡+𝑘𝑚𝑗=1𝐼𝑗𝑢𝑡𝑗𝑣𝑡𝑗+0𝑘𝑇||||𝐿(𝑡)𝑢(𝑡)𝑝2=𝑢(𝑡),𝑣(𝑡)𝑑𝑡0𝑘𝑇(𝐹(𝑡,𝑢(𝑡)),𝑣(𝑡))𝑑𝑡(2.7) holds for any 𝑣𝑊1,𝑝𝑘𝑇.

Define the functional 𝜙𝑘 on 𝑊1,𝑝𝑘𝑇 by 𝜙𝑘1(𝑢)=𝑝0𝑘𝑇||||̇𝑢(𝑡)𝑝+||||𝐿(𝑡)𝑢(𝑡)𝑝2𝑢(𝑡),𝑢(𝑡)𝑑𝑡0𝑘𝑇𝐹(𝑡,𝑢(𝑡))𝑑𝑡+𝑘𝑚𝑗=1𝐼𝑗𝑢𝑡𝑗=𝜑𝑘(𝑢)+𝜓𝑘(𝑢),𝑢𝑊1,𝑝𝑘𝑇,(2.8) where 𝜑𝑘1(𝑢)=𝑝0𝑘𝑇||||̇𝑢(𝑡)𝑝+||||𝐿(𝑡)𝑢(𝑡)𝑝2𝑢(𝑡),𝑢(𝑡)𝑑𝑡0𝑘𝑇𝜓𝐹(𝑡,𝑢(𝑡))𝑑𝑡,𝑘(𝑢)=𝑘𝑚𝑗=1𝐼𝑗𝑢𝑡𝑗.(2.9)

It follows from assumption (A) that the functional 𝜑𝑘 is continuously differentiable on 𝑊1,𝑝𝑘𝑇 and𝜑𝑘(𝑢),𝑣=0𝑘𝑇||||̇𝑢(𝑡)𝑝2̇+||||̇𝑢(𝑡),𝑣(𝑡)𝐿(𝑡)𝑢(𝑡)𝑝2𝑢(𝑡),𝑣(𝑡)(𝐹(𝑡,𝑢(𝑡)),𝑣(𝑡))𝑑𝑡(2.10) for 𝑢,𝑣𝑊1,𝑝𝑘𝑇. By the continuity of 𝐼𝑗, 𝑗𝐵, one has that 𝜓𝑘(𝑊1,𝑝𝑘𝑇,). Hence, 𝜙𝑘(𝑢)(𝑊1,𝑝𝑘𝑇,). For any 𝑣𝑊1,𝑝𝑘𝑇, we have 𝜙𝑘(𝑢),𝑣=0𝑘𝑇||||̇𝑢(𝑡)𝑝2̇+||||̇𝑢(𝑡),𝑣(𝑡)𝐿(𝑡)𝑢(𝑡)𝑝2𝑢(𝑡),𝑣(𝑡)(𝐹(𝑡,𝑢(𝑡)),𝑣(𝑡))𝑑𝑡+𝑘𝑚𝑗=1𝐼𝑗𝑢𝑡𝑗𝑣𝑡𝑗.(2.11)

By Definition 2.1, the weak solutions of problem (1.1) correspond to the critical points of the functional 𝜙𝑘.

For 𝑢𝑊1,𝑝𝑘𝑇, let 𝑢=(1/𝑘𝑇)0𝑘𝑇𝑢(𝑡)𝑑𝑡 and ̃𝑢(𝑡)=𝑢(𝑡)𝑢; then it follows from Proposition  1.1 in [26] that 𝑢=max[]𝑡0,𝑘𝑇||||𝑢(𝑡)(𝑘𝑇)1/𝑝+(𝑘𝑇)1/𝑞𝑢=𝑑𝑘𝑢,(2.12) where 𝑑𝑘=(𝑘𝑇)1/𝑝+(𝑘𝑇)1/𝑞, and if (1/𝑘𝑇)0𝑘𝑇𝑢(𝑡)𝑑𝑡=0, then ̃𝑢=max[]𝑡0,𝑘𝑇||||̃𝑢(𝑡)(𝑘𝑇)1/𝑞̇𝑢𝐿𝑝,(2.13)̃𝑢𝑝𝐿𝑝(𝑘𝑇)𝑝̇𝑢𝑝𝐿𝑝,(2.14) where 1/𝑝+1/𝑞=1. Let 𝑊1,𝑝𝑘𝑇={𝑢𝑊1,𝑝𝑘𝑇𝑢=0}; then 𝑊1,𝑝𝑘𝑇=𝑊1,𝑝𝑘𝑇𝑁. We will use the following lemma to prove our main results.

Lemma 2.2 (see [27]). Let 𝐸 be a real Banach space with 𝐸=𝑋1𝑋2, where 𝑋1 is finite dimensional. Suppose that 𝜑𝐶1(𝐸,) satisfies the (PS) condition, and (a)there exist constants 𝜌, 𝛼>0 such that 𝜑|𝜕𝐵𝜌𝑋2𝛼, where 𝐵𝜌={𝑢𝐸𝑢𝜌}, and 𝜕𝐵𝜌 denotes the boundary of 𝐵𝜌; (b)there exists an 𝑒𝜕𝐵1𝑋2 and 𝐿>𝜌 such that if 𝑄(𝐵𝐿𝑋1){𝑟𝑒0𝑟𝐿}, then 𝜑|𝜕𝑄0.
Then 𝜑 possesses a critical value 𝑐𝛼 which can be characterized as 𝑐=infΓmax𝑢𝑄𝜑((𝑢)), where Γ={𝐶(𝑄,𝐸)=idon𝜕𝑄}.

It is well known that a deformation lemma can be proved with the weaker condition (C) replacing the usual (PS) condition. So Lemma 2.2 holds true under condition (C).

3. Main Result and Proof

Theorem 3.1. Assume that (A) holds and 𝐹,𝐼𝑗 satisfy the following conditions: (I1)there exists 𝑐𝑗>0 such that 0𝐼𝑗(𝑐𝑥)𝑗𝑘|𝑥|𝑝,𝑗𝐵,𝑥𝑁;(3.1)(I2)for any 𝑗𝐵, 𝐼𝑗(𝑥)𝑥𝑝𝐼𝑗(𝑥),𝑥𝑁;(3.2)(H1)𝑇0𝐹(𝑡,𝑥)𝑑𝑡0, for all 𝑥𝑁;(H2)lim|𝑥|0(𝐹(𝑡,𝑥)/|𝑥|𝑝)=0 uniformly for a.e. 𝑡[0,𝑇];(H3)lim|𝑥|(𝐹(𝑡,𝑥)/|𝑥|𝑝)=+ uniformly for a.e. 𝑡[0,𝑇];(H4)there exists a positive constant 𝑀 such that limsup|𝑥|(𝐹(𝑡,𝑥)/|𝑥|𝑟)𝑀 uniformly for a.e. 𝑡[0,𝑇];(H5)there exists 𝑀1>0 such that liminf|𝑥|((𝐹(𝑡,𝑥),𝑥)𝑝𝐹(𝑡,𝑥))/|𝑥|𝜇𝑀1 uniformly for a.e. 𝑡[0,𝑇],
where 𝑟>𝑝 and 𝜇>𝑟𝑝. Then problem (1.1) has a sequence of distinct periodic solutions with period 𝑘𝑗𝑇 satisfying 𝑘𝑗 and 𝑘𝑗 as 𝑗.

Remark 3.2. As far as we know, there is no paper considering subharmonic solutions of impulsive differential equations. Our result is new.

Proof. The proof is divided into three steps. In the following, 𝐶𝑖(𝑖=1,) denote different positive constants.Step 1. The functional 𝜙𝑘 satisfies condition (C). Let {𝑢𝑛}𝑊1,𝑝𝑘𝑇 satisfying (1+𝑢𝑛)𝜙𝑘(𝑢𝑛)0 as 𝑛 and 𝜙𝑘(𝑢𝑛) is bounded; then, there exists a constant 𝐶1 such that ||𝜙𝑘𝑢𝑛||𝐶1,𝑢1+𝑛𝜙𝑘𝑢𝑛𝐶1.(3.3) From (H4), there exists 𝑀2>0 such that 𝐹(𝑡,𝑥)𝑀|𝑥|𝑟|𝑥|𝑀2,a.e.[].𝑡0,𝑇(3.4) By assumption (A), for |𝑥|𝑀2, there exists 𝐶2=max|𝑥|𝑀2𝑎(|𝑥|)>0 such that ||||𝐹(𝑡,𝑥)𝐶2𝑏(𝑡),(3.5) which together with (3.4) implies that 𝐹(𝑡,𝑥)𝑀|𝑥|𝑟+𝐶2𝑏(𝑡),𝑥𝑁,a.e.[].𝑡0,𝑇(3.6) By (3.3) and (3.6), we have 𝜙𝑘𝑢𝑛+0𝑘𝑇𝐹𝑡,𝑢𝑛𝑑𝑡𝐶1+0𝑘𝑇𝑀||𝑢𝑛||(𝑡)𝑟+𝐶2𝑏(𝑡)𝑑𝑡=𝐶1+𝐶2𝑘𝑏𝐿1+𝑀0𝑘𝑇||𝑢𝑛||(𝑡)𝑟𝑑𝑡=𝐶3+𝑀0𝑘𝑇||𝑢𝑛||(𝑡)𝑟𝑑𝑡.(3.7) Since 𝐿(𝑡) is continuous 𝑇-periodic positive definite symmetric matrix on [0,𝑇], there exist constants 𝑐1,𝑐2>0 such that 𝑐1|𝑥|𝑝𝐿(𝑡)|𝑥|𝑝2𝑥,𝑥𝑐2|𝑥|𝑝,𝑥𝑁.(3.8) It follows from (3.8) and (I1) that 𝜙𝑘𝑢𝑛+0𝑘𝑇𝐹𝑡,𝑢𝑛1𝑑𝑡=𝑝0𝑘𝑇||̇𝑢𝑛||(𝑡)𝑝+||𝑢𝐿(𝑡)𝑛||(𝑡)𝑝2𝑢𝑛(𝑡),𝑢𝑛(𝑡)𝑑𝑡+𝑘𝑚𝑗=1𝐼𝑗𝑢𝑡𝑗1𝑝0𝑘𝑇||̇𝑢𝑛(||𝑡)𝑝+𝑐1||𝑢𝑛(||𝑡)𝑝1𝑑𝑡min𝑝,𝑐1𝑝𝑢𝑛𝑝=𝐶4𝑢𝑛𝑝.(3.9) By (3.7) and (3.9), we get 𝐶4𝑢𝑛𝑝𝐶3+𝑀0𝑘𝑇||𝑢𝑛||(𝑡)𝑟𝑑𝑡.(3.10) From (H5), there exists 𝑀3>0 such that (𝐹(𝑡,𝑥),𝑥)𝑝𝐹(𝑡,𝑥)𝑀1|𝑥|𝜇for|𝑥|𝑀3,a.e.[].𝑡0,𝑇(3.11) By assumption (A), for |𝑥|𝑀3, there exists 𝐶5=max|𝑥|𝑀3𝑎(|𝑥|)>0 such that ||||(𝐹(𝑡,𝑥),𝑥)𝑝𝐹(𝑡,𝑥)𝐶5𝑝+𝑀3𝑏(𝑡).(3.12) Thus, from (3.11) and (3.12), we have (𝐹(𝑡,𝑥),𝑥)𝑝𝐹(𝑡,𝑥)𝑀1|𝑥|𝜇𝑀1𝑀𝜇3𝐶5𝑝+𝑀3𝑏(𝑡)for𝑥𝑁,a.e.[],𝑡0,𝑇(3.13) which together with (3.3) and (I2) implies that (𝑝+1)𝐶1𝑝𝜙𝑘𝑢𝑛𝜙𝑘𝑢𝑛,𝑢𝑛=0𝑘𝑇𝐹𝑡,𝑢𝑛,𝑢𝑛𝑝𝐹𝑡,𝑢𝑛𝑑𝑡+𝑝𝑘𝑚𝑗=1𝐼𝑗𝑢𝑛𝑡𝑗𝑘𝑚𝑗=1𝐼𝑗𝑢𝑛𝑡𝑗𝑢𝑛𝑡𝑗𝑀10𝑘𝑇||𝑢𝑛||(𝑡)𝜇𝑑𝑡𝐶5𝑝+𝑀30𝑘𝑇𝑏(𝑡)𝑑𝑡𝑀1𝑀𝜇3𝑘𝑇=𝑀10𝑘𝑇||𝑢𝑛||(𝑡)𝜇𝑑𝑡𝐶6.(3.14) Hence, 0𝑘𝑇|𝑢𝑛(𝑡)|𝜇𝑑𝑡 is bounded. If 𝜇>𝑟, we have 0𝑘𝑇||𝑢𝑛||(𝑡)𝑟𝑑𝑡(𝑘𝑇)(𝜇𝑟)/𝜇0𝑘𝑇||𝑢𝑛||(𝑡)𝜇𝑑𝑡𝑟/𝜇,(3.15) which together with (3.10) implies that 𝑢𝑛 is bounded. If 𝜇𝑟, then from (2.12), we get 0𝑘𝑇||𝑢𝑛||(𝑡)𝑟𝑢𝑑𝑡𝑛𝑟𝜇0𝑘𝑇||𝑢𝑛||(𝑡)𝜇𝑑𝑡𝑟/𝜇𝑑𝑘𝑟𝜇𝑢𝑛𝑟𝜇0𝑘𝑇||𝑢𝑛||(𝑡)𝜇𝑑𝑡𝑟/𝜇.(3.16) Since 𝜇>𝑟𝑝, it follows from (3.10) that 𝑢𝑛 is bounded too. Therefore, 𝑢𝑛 is bounded in 𝑊1,𝑝𝑘𝑇. Hence, there exists a subsequence, still denoted by {𝑢𝑛}, such that 𝑢𝑛𝑢0weaklyin𝑊1,𝑝𝑘𝑇,(3.17)𝑢𝑛𝑢0stronglyin𝐶[]0,𝑘𝑇;𝑁,(3.18)𝑢𝑛𝑢0stronglyin𝐿𝑝[]0,𝑘𝑇;𝑁.(3.19) From (2.11), we have 𝜙𝑘𝑢𝑛,𝑢𝑛𝑢0=0𝑘𝑇||̇𝑢𝑛||(𝑡)𝑝2̇𝑢𝑛(𝑡),̇𝑢𝑛(𝑡)̇𝑢0+||𝑢(𝑡)𝐿(𝑡)𝑛||(𝑡)𝑝2𝑢𝑛(𝑡),𝑢𝑛(𝑡)𝑢0(𝑡)𝑑𝑡0𝑘𝑇𝐹𝑡,𝑢𝑛(𝑡),𝑢𝑛(𝑡)𝑢0(𝑡)𝑑𝑡+𝑘𝑚𝑗=1𝐼𝑗𝑢𝑛𝑡𝑗,𝑢𝑛𝑡𝑗𝑢0𝑡𝑗.(3.20) From (3.3) and (3.18), we have ||𝜙𝑘𝑢𝑛,𝑢𝑛𝑢0||𝜙𝑘𝑢𝑛𝑢𝑛𝑢00as𝑛.(3.21) By (3.8), we know that 𝑐1𝐿𝑐2, which together with the boundedness of {𝑢𝑛} and (3.19) implies that 0𝑘𝑇||𝑢𝐿(𝑡)𝑛||(𝑡)𝑝2𝑢𝑛(𝑡),𝑢𝑛(𝑡)𝑢0𝑢(𝑡)𝑑𝑡𝐿𝑛𝐿𝑝1𝑝𝑢𝑛𝑢0𝐿𝑝0as𝑛.(3.22) From the boundedness of {𝑢𝑛}, the continuity of 𝐼𝑗, and (3.18), we have 𝑚𝑗=1𝐼𝑗𝑢𝑛𝑡𝑗,𝑢𝑛𝑡𝑗𝑢0𝑡𝑗0as𝑛.(3.23) It follows from (A), (3.18) and the boundedness of {𝑢𝑛} that 0𝑘𝑇𝐹𝑡,𝑢𝑛(𝑡),𝑢𝑛(𝑡)𝑢0(𝑡)𝑑𝑡0as𝑛,(3.24) which together with (3.20), (3.21), (3.22), and (3.23) implies that 0𝑘𝑇||̇𝑢𝑛||(𝑡)𝑝2̇𝑢𝑛(𝑡),̇𝑢𝑛(𝑡)̇𝑢0(𝑡)𝑑𝑡0as𝑛.(3.25) It is easy to see from the boundedness of {𝑢𝑛} and (3.18) that 0𝑘𝑇||𝑢𝑛||(𝑡)𝑝2𝑢𝑛(𝑡),𝑢𝑛(𝑡)𝑢0(𝑡)𝑑𝑡0as𝑛.(3.26) Let 𝑓(𝑢)=(1/𝑝)(0𝑘𝑇|𝑢(𝑡)|𝑝𝑑𝑡+0𝑘𝑇|̇𝑢(𝑡)|𝑝𝑑𝑡). Then, we have 𝑢𝑓𝑛,𝑢𝑛𝑢0=0𝑘𝑇||̇𝑢𝑛||(𝑡)𝑝2̇𝑢𝑛(𝑡),̇𝑢𝑛(𝑡)̇𝑢0+(𝑡)𝑑𝑡0𝑘𝑇||𝑢𝑛||(𝑡)𝑝2𝑢𝑛(𝑡),𝑢𝑛(𝑡)𝑢0(𝑡)𝑑𝑡,(3.27)𝑢𝑓0,𝑢𝑛𝑢0=0𝑘𝑇||̇𝑢0||(𝑡)𝑝2̇𝑢0(𝑡),̇𝑢𝑛(𝑡)̇𝑢0+(𝑡)𝑑𝑡0𝑘𝑇||𝑢0||(𝑡)𝑝2𝑢0(𝑡),𝑢𝑛(𝑡)𝑢0(𝑡)𝑑𝑡.(3.28) It follows from (3.25) and (3.26) that 𝑢𝑓𝑛,𝑢𝑛𝑢00as𝑛.(3.29) From (3.17), we get 𝑢𝑓0,𝑢𝑛𝑢00as𝑛.(3.30) By (3.27), (3.28), and Hölder's inequality, we have 𝑢𝑓𝑛𝑢𝑓0,𝑢𝑛𝑢0=0𝑘𝑇||̇𝑢𝑛||(𝑡)𝑝2̇𝑢𝑛(𝑡),̇𝑢𝑛(𝑡)̇𝑢0(𝑡)𝑑𝑡+0𝑘𝑇||𝑢𝑛||(𝑡)𝑝2𝑢𝑛(𝑡),𝑢𝑛(𝑡)𝑢0(𝑡)𝑑𝑡0𝑘𝑇||̇𝑢0(||𝑡)𝑝2̇𝑢0(𝑡),̇𝑢𝑛(𝑡)̇𝑢0(𝑡)𝑑𝑡0𝑘𝑇||𝑢0(||𝑡)𝑝2𝑢0(𝑡),𝑢𝑛(𝑡)𝑢0(=𝑢𝑡)𝑑𝑡𝑛𝑝+𝑢0𝑝0𝑘𝑇||̇𝑢𝑛(||𝑡)𝑝2̇𝑢𝑛(𝑡),̇𝑢0(𝑡)𝑑𝑡0𝑘𝑇||𝑢𝑛(||𝑡)𝑝2𝑢𝑛(𝑡),𝑢0(𝑡)𝑑𝑡0𝑘𝑇||̇𝑢0(||𝑡)𝑝2̇𝑢0(𝑡),̇𝑢𝑛(𝑡)𝑑𝑡0𝑘𝑇||𝑢0(||𝑡)𝑝2𝑢0(𝑡),𝑢𝑛(𝑢𝑡)𝑑𝑡𝑛𝑝+𝑢0𝑝𝑢𝑛𝐿𝑝1𝑝𝑢0𝐿𝑝+̇𝑢𝑛𝐿𝑝1𝑝̇𝑢0𝐿𝑝𝑢0𝐿𝑝1𝑝𝑢𝑛𝐿𝑝+̇𝑢0𝐿𝑝1𝑝̇𝑢𝑛𝐿𝑝𝑢𝑛𝑝+𝑢0𝑝𝑢𝑛𝑝𝐿𝑝+̇𝑢𝑛𝑝𝐿𝑝(𝑝1)/𝑝𝑢0𝑝𝐿𝑝+̇𝑢0𝑝𝐿𝑝1/𝑝𝑢0𝑝𝐿𝑝+̇𝑢0𝑝𝐿𝑝(𝑝1)/𝑝𝑢𝑛𝑝𝐿𝑝+̇𝑢𝑛𝑝𝐿𝑝1/𝑝=𝑢𝑛𝑝+𝑢0𝑝𝑢𝑛𝑝1𝑢0+𝑢0𝑝1𝑢𝑛=𝑢𝑛𝑝1𝑢0𝑝1𝑢𝑛𝑢0.(3.31) Hence, from (3.29) and (3.30), we obtain 𝑢0𝑛𝑝1𝑢0𝑝1𝑢𝑛𝑢0𝑓𝑢𝑛𝑓𝑢0,𝑢𝑛𝑢00as𝑛.(3.32) That is, 𝑢𝑛𝑢0 as 𝑛. Since 𝑊1,𝑝𝑘𝑇 has the Kadec-Klee property, we have 𝑢𝑛𝑢0 in 𝑊1,𝑝𝑘𝑇. Therefore, the functional 𝜙𝑘 satisfies condition (C). Step 2. From (H2), for any small 𝜀=𝜀(𝑘)>0, there exists small enough 𝛿>0 such that 𝐹(𝑡,𝑢)𝜀|𝑢|𝑝for|𝑢|𝛿,a.e.[].𝑡0,𝑘𝑇(3.33) For 𝑊𝑢1,𝑝𝑘𝑇 and 𝑢𝑝=𝜌𝑝𝑘=𝛿𝑝/(𝑘𝑇)𝑝/𝑞, it follows from (2.13) that 𝑢𝑝(𝑘𝑇)𝑝/𝑞̇𝑢𝑝𝐿𝑝(𝑘𝑇)𝑝/𝑞𝑢𝑝=𝛿𝑝,(3.34) which implies that |𝑢(𝑡)|𝛿. Then from (I1), (3.8), and (3.33), we have 𝜑𝑘1(𝑢)=𝑝0𝑘𝑇||||̇𝑢(𝑡)𝑝1𝑑𝑡+𝑝0𝑘𝑇||||𝐿(𝑡)𝑢(𝑡)𝑝2𝑢(𝑡),𝑢(𝑡)𝑑𝑡0𝑘𝑇𝐹(𝑡,𝑢)𝑑𝑡+𝑘𝑚𝑗=1𝐼𝑗𝑢𝑡𝑗1𝑝0𝑘𝑇||||̇𝑢(𝑡)𝑝1𝑑𝑡+𝑝0𝑘𝑇𝑐1||||𝑢(𝑡)𝑝𝑑𝑡0𝑘𝑇𝜀||||𝑢(𝑡)𝑝1𝑑𝑡min𝑝,𝑐1𝑝𝑢𝑝𝑘𝑇𝜀𝛿𝑝=𝐶4𝑢𝑝𝑘𝑇𝜀𝛿𝑝.(3.35) Let 𝜀=𝜀(𝑘)(0,𝐶4/2(𝑘𝑇)𝑝); then from (3.24), we have 𝜑𝑘(𝑢)𝐶4𝜌𝑝𝑘𝑘𝑇𝜀𝛿𝑝𝐶42𝜌𝑝𝑘𝛼>0(3.36) for all 𝑊𝑢𝑇1,𝑝 and 𝑢=𝜌𝑘. This implies that condition (a) of Lemma 2.2 holds. Step 3. Let 𝑐=max{𝑐𝑗}, 𝑗𝐵. Choose 𝐶7>(𝑐2/𝑝)+(𝑚𝑐/𝑇); then from (H3), there exists 𝑀4>0 such that 𝐹(𝑡,𝑥)𝐶7|𝑥|𝑝,|𝑥|𝑀4,a.e.[].𝑡0,𝑇(3.37) By assumption (A), for |𝑥|𝑀4, there exists 𝐶8=max|𝑥|𝑀4𝑎(|𝑥|)>0 such that ||||𝐹(𝑡,𝑥)𝐶8𝑏(𝑡),a.e.[],𝑡0,𝑇(3.38) which together with (3.37) implies that 𝐹(𝑡,𝑥)𝐶7|𝑥|𝑝𝐶8𝑏(𝑡),𝑥𝑁,a.e.[].𝑡0,𝑇(3.39) Thus, from (H1), (I1), (3.8), and (3.39), we have 𝜙𝑘1(𝑢)=𝑝0𝑘𝑇𝐿(𝑡)|𝑢|𝑝2𝑢,𝑢𝑑𝑡0𝑘𝑇𝐹(𝑡,𝑢)𝑑𝑡+𝑘𝑚𝑗=1𝐼𝑗=𝑘(𝑢)𝑝𝑇0𝐿(𝑡)|𝑢|𝑝2𝑢,𝑢𝑑𝑡𝑘𝑇0𝐹(𝑡,𝑢)𝑑𝑡+𝑘𝑚𝑗=1𝐼𝑗𝑐(𝑢)2𝑘𝑝𝑇0|𝑢|𝑝𝑑𝑡𝑘𝑇0𝐶7|𝑢|𝑝𝑑𝑡+𝑘𝑇0𝐶8𝑏(𝑡)𝑑𝑡+𝑚𝑐|𝑢|𝑝for𝑢𝑁.(3.40) From (H3), we can choose 𝐶7 suitable large such that 𝜙𝑘(𝑢)0,𝑢𝑁.(3.41) Let 𝑊1,𝑝𝑘𝑇=span{𝑒𝑘}+𝑁, where 𝑒𝑘=(𝑘1sin(𝑘1𝜔𝑡)), 𝜔=2𝜋/𝑇. Since 𝑊𝑇1,𝑝 is finite dimensional, there exists a constant 𝑑>0 such that 𝑇0|𝑥|𝑝𝑑𝑡1/𝑝𝑑𝑇0|𝑥|2𝑑𝑡1/2,𝑥𝑊𝑇1,𝑝.(3.42) By (I1), we have ||𝜓𝑢+𝑟𝑒𝑘||=|||||𝑘𝑚𝑗=1𝐼𝑗𝑢+𝑟𝑒𝑘𝑡𝑗|||||𝑚𝑗=1𝑐𝑗||𝑢+𝑟𝑒𝑘𝑡𝑗||𝑝2𝑝𝑚𝑐|𝑢|𝑝+2𝑝𝑚𝑐𝑟𝑝||𝑒𝑘𝑡𝑗||𝑝2𝑝𝑚𝑐|𝑢|𝑝+2𝑝𝑚𝑐𝑟𝑝𝑘𝑝2𝑝𝑚𝑐|𝑢|𝑝+2𝑝𝑚𝑐𝑟𝑝,𝑢𝑁.(3.43) From (3.39), (3.42), and (3.43), we obtain 𝜙𝑘𝑢+𝑟𝑒𝑘=1𝑝0𝑘𝑇||𝑟̇𝑒𝑘||(𝑡)𝑝𝑑𝑡0𝑘𝑇𝐹𝑡,𝑢+𝑟𝑒𝑘+1(𝑡)𝑑𝑡𝑝0𝑘𝑇||𝐿(𝑡)𝑢+𝑟𝑒𝑘||(𝑡)𝑝2𝑢+𝑟𝑒𝑘(𝑡),𝑢+𝑟𝑒𝑘(𝑡)𝑑𝑡+𝑘𝑚𝑗=1𝐼𝑗𝑢+𝑟𝑒𝑘𝑡𝑗1𝑝𝑘2𝑝𝑟𝑝𝜔𝑝0𝑘𝑇||𝑘cos1||𝜔𝑡𝑝𝑐𝑑𝑡+2𝑝0𝑘𝑇||𝑢+𝑟𝑒𝑘||(𝑡)𝑝𝑑𝑡+0𝑘𝑇𝐶8𝑏(𝑡)𝑑𝑡0𝑘𝑇𝐶7||𝑢+𝑟𝑒𝑘||(𝑡)𝑝𝑑𝑡+2𝑝𝑚𝑐|𝑢|𝑝+2𝑝𝑚𝑐𝑟𝑝1𝑝𝑘2𝑝+1𝑟𝑝𝜔𝑝𝑇0||||cos(𝜔𝑡)𝑝𝑑𝑡𝑘𝑇0𝐶7𝑐2𝑝||𝑢+𝑟𝑒1(||𝑡)𝑝+𝑑𝑡𝑇0𝐶8𝑘𝑏(𝑡)𝑑𝑡+2𝑝𝑚𝑐|𝑢|𝑝+2𝑝𝑚𝑐𝑟𝑝𝑇𝑝𝑘2𝑝+1𝜔𝑝+2𝑝𝑟𝑚𝑐𝑝𝑘𝑑𝑝𝐶7𝑐2𝑝𝑇0||𝑢+𝑟𝑒1||(𝑡)2𝑑𝑡𝑝/2+2𝑝𝑚𝑐|𝑢|𝑝+𝐶9𝑘𝑇𝑝𝑘2𝑝+1𝜔𝑝+2𝑝𝑟𝑚𝑐𝑝𝑘𝑑𝑝𝐶7𝑐2𝑝𝑇0(|𝑢|2+𝑟2||𝑒1||(𝑡)2𝑑)𝑝/2+2𝑝𝑚𝑐|𝑢|𝑝+𝐶9𝑘𝑇𝑝𝑘2𝑝+1𝜔𝑝+2𝑝𝑟𝑚𝑐𝑝𝑘𝑑𝑝𝐶7𝑐2𝑝𝑇|𝑢|2+𝑇𝑟22𝑝/2+2𝑝𝑚𝑐|𝑢|𝑝+𝐶9𝑘,𝑟0,𝑢𝑁.(3.44) From (H3), we can choose 𝐶7 suitable such that 𝑑𝑝𝐶7𝑐2𝑝𝑇2𝑝/223𝑝𝑑𝑚𝑐>0,𝑝𝐶7𝑐2𝑝𝑇𝑝/22𝑝+1𝑚𝑐>0.(3.45) If 𝑘2(𝑇𝑝)1/2𝑝𝜔1/2/[𝑑𝑝(𝐶7𝑐2/𝑝)(𝑇/2)𝑝/223𝑝𝑚𝑐]=𝐶10, then we get 𝑘1𝜙𝑘𝑢+𝑟𝑒𝑘𝑇𝑝𝑘2𝑝𝑟𝑝𝜔𝑝+2𝑝𝑚𝑐𝑘𝑑𝑝𝐶7𝑐2𝑝𝑇2𝑝/2𝑟𝑝+𝐶9𝑇𝑘2𝑝𝜔𝑝+2𝑝𝑚𝑐𝑑𝑝𝐶7𝑐2𝑝𝑇2𝑝/2𝑟𝑝+𝐶912𝑑𝑝𝐶7𝑐2𝑝𝑇2𝑝/2𝑟𝑝+𝐶9,𝑘1𝜙𝑘𝑢+𝑟𝑒𝑘12𝑑𝑝𝐶7𝑐2𝑝𝑇𝑝/2|𝑢|𝑝+𝐶9.(3.46) It follows from (3.46) that 𝜑𝑘𝑢+𝑟𝑒𝑘0,either𝑟𝑟1or|𝑢|𝑟2,(3.47) where 𝑟1=2(2𝐶9)1/𝑝/(𝐶7𝑐2/𝑝)1/𝑝𝑑𝑇1/2, 𝑟2=(2𝐶9)1/𝑝/𝑑(𝐶7𝑐2/𝑝)1/𝑝𝑇1/2. Notice that for any 𝑢𝑁, we have 𝑢=𝑢𝐿𝑝=0𝑘𝑇|𝑢|𝑝𝑑𝑡1/𝑝=(𝑘𝑇)1/𝑝𝐶|𝑢|10𝑇1/𝑝𝑟2=𝑟3.(3.48) Hence, (3.47) holds for all 𝑢𝑟3 whenever 𝑢𝑁. Set 𝑄𝑘=𝑟𝑒𝑘0𝑟𝑟1,𝑒𝑘𝑊1,𝑝𝑘𝑇𝑢𝑁𝑢𝑟3;(3.49) then 𝜕𝑄𝑘=𝑄1𝑘𝑄2𝑘𝑄3𝑘, where 𝑄1𝑘=𝑢𝑁𝑢𝑟3,𝑄2𝑘=𝑢+𝑟𝑒𝑘𝑢=𝑟3,𝑟0,𝑟1,𝑒𝑘𝑊1,𝑝𝑘𝑇,𝑄3𝑘=𝑢+𝑟𝑒𝑘𝑢𝑟3,𝑟=𝑟1,𝑒𝑘𝑊1,𝑝𝑘𝑇.(3.50) By (3.41) and (3.47), we have 𝜑(𝑢)0,𝑢𝜕𝑄𝑘=𝑄1𝑘𝑄2𝑘𝑄3𝑘.(3.51) Furthermore, for all 𝑢+𝑟𝑒𝑘𝑄𝑘, it follows from (H1), (3.8), and (3.43) that 𝜙𝑘𝑢+𝑟𝑒𝑘=1𝑝0𝑘𝑇||𝑟̇𝑒𝑘||(𝑡)𝑝𝑑𝑡0𝑘𝑇𝐹𝑡,𝑢+𝑟𝑒𝑘+1(𝑡)𝑑𝑡𝑝0𝑘𝑇||𝐿(𝑡)𝑢+𝑟𝑒𝑘||(𝑡)𝑝2𝑢+𝑟𝑒𝑘(𝑡),𝑢+𝑟𝑒𝑘(𝑡)𝑑𝑡+𝑘𝑚𝑗=1𝐼𝑗𝑢+𝑟𝑒𝑘𝑡𝑗1𝑝𝑟𝑝0𝑘𝑇||̇𝑒𝑘||(𝑡)𝑝𝑐𝑑𝑡+2𝑝0𝑘𝑇||𝑢+𝑟𝑒𝑘||(𝑡)𝑝𝑑𝑡+2𝑝𝑚𝑐|𝑢|𝑝+2𝑝𝑚𝑐𝑟𝑝1𝑝𝑘2𝑝𝑟𝑝𝜔𝑝0𝑘𝑇||𝑘cos1||𝜔𝑡𝑝2𝑑𝑡+𝑝1𝑐2𝑝0𝑘𝑇|𝑢|𝑝+𝑟𝑝𝑘𝑝||𝑘sin1||𝜔𝑡𝑝𝑑𝑡+2𝑝𝑚𝑐|𝑢|𝑝+2𝑝𝑚𝑐𝑟𝑝1𝑝𝑘2𝑝+1𝑟𝑝𝜔𝑝𝑇0||||cos(𝜔𝑡)𝑝2𝑑𝑡+𝑝1𝑐2𝑝𝑢𝑝+𝑟𝑝𝑘𝑝+1𝑇0||||sin(𝜔𝑡)𝑝+2𝑑𝑡𝑝𝑚𝑐𝑇𝑢𝑝+2𝑝𝑚𝑐𝑟𝑝𝑇𝑝𝑘2𝑝+1𝑟𝑝𝜔𝑝+2𝑝1𝑐2𝑝𝑢𝑝+𝑟𝑝𝑘𝑝+1𝑇+2𝑝𝑚𝑐𝑇𝑢𝑝+2𝑝𝑚𝑐𝑟𝑝𝑇𝑝𝑟𝑝1𝜔𝑝+2𝑝1𝑐2𝑝𝑟𝑝3+𝑟𝑝1𝑇+2𝑝1𝑚𝑐𝑇𝑟𝑝3+𝑟𝑝1.(3.52) Then by Lemma 2.2, 𝜙𝑘 has at least a critical point 𝑢𝑘 whose critical value 𝑐𝑘 satisfies 0<𝛼𝑐𝑘=𝜙𝑘𝑢𝑘𝑇𝑝𝑟𝑝1𝜔𝑝+2𝑝1𝑐2𝑝𝑟𝑝3+𝑟𝑝1𝑇+2𝑝1𝑚𝑐𝑇𝑟𝑝3+𝑟𝑝1.(3.53) Similar to the proof of [28], let 𝑢𝑘1 be a 𝑘1𝑇-periodic solution; we can prove that there exists a positive integer 𝑘2>𝑘1 such that 𝑢𝑘𝑘1𝑢𝑘1 for all 𝑘𝑘1𝑘2. Otherwise, 𝜑𝑘(𝑢𝑘𝑘1)=𝑘𝜑𝑘(𝑢𝑘1) as 𝑘, which contradicts to (3.53). Repeating this process, we can obtain a sequence {𝑢𝑘𝑗} of distinct periodic solutions of problem (1.1). From (3.41), we know that 𝑢𝑘𝑗 is nonconstant. The proof is complete.

4. Examples

In this section, we give an example to illustrate our result.

Example 4.1. Let 𝑝=3, 𝑟=5, 𝜇=4, and consider the following 𝑝-Laplacian system with impulsive effects 𝑑||||||||𝑑𝑡̇𝑢(𝑡)̇𝑢(𝑡)𝐿(𝑡)𝑢(𝑡)𝑢(𝑡)+𝐹(𝑡,𝑢(𝑡))=0,a.e.Δ||𝑡𝑡,𝑢(0)𝑢(𝑇)=̇𝑢(0)̇𝑢(𝑇)=0,̇𝑢𝑖||𝑡̇𝑢𝑖=||𝑡̇𝑢+𝑖||𝑡̇𝑢+𝑖||𝑡̇𝑢𝑖||𝑡̇𝑢𝑖=𝐼𝑖𝑢𝑡𝑖,𝑖=1,2,,𝑚.(4.1) Let 𝐿𝑘(𝑡)=diag1+exp1sin1𝑘𝜔𝑡,,1+exp1sin1,𝐼𝜔𝑡𝑖(𝑐𝑥)=𝑖𝑘|𝑥|𝑝,𝐹(𝑡,𝑥)=1+𝑒3𝑘2+sin1|𝜔𝑡𝑥|5,(4.2) where 𝑐𝑖>0, 𝑖𝐵. It is easy to check that 𝐹 satisfies (A), (H1), and (H2). By a direct computation, we have lim|𝑥|𝐹(𝑡,𝑥)|𝑥|3=+,limsup|𝑥|𝐹(𝑡,𝑥)|𝑥|51+𝑒,liminf|𝑥|(𝐹(𝑡,𝑥),𝑥)3𝐹(𝑡,𝑥)|𝑥|42(1+𝑒)3,(4.3) which show that (H3), (H4), and (H5) hold. On the other hand, 0𝐼𝑖𝑐(𝑥)𝑘|𝑥|𝑝,𝐼𝑖(𝑥)𝑥=𝑝𝑐𝑖𝑘|𝑥|𝑝=𝑝𝐼𝑖(𝑥),(4.4) where 𝑐=max{𝑐𝑖}, 𝑖𝐵. It is easy to see that 𝐼𝑖 satisfies (I1) and (I2). Hence, from Theorem 3.1, problem (4.1) has a sequence of distinct nonconstant periodic solutions with period 𝑘𝑗𝑇 satisfying 𝑘𝑗 and 𝑘𝑗 as 𝑗.

Acknowledgments

This paper is partially supported by the NNSF (no. 10771215) of China. W.-Z. Gong is supported by Guangxi Natural Science Foundation (2010GXNSFA013125) and X. H. Tang is supported by the NNSF (no. 10771215) of China.