Periodic and subharmonic solutions for fourth-order nonlinear difference equations

Applied Mathematics and Computation 236 (2014) 613–620

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Periodic and subharmonic solutions for fourth-order nonlinear difference equations q Xia Liu a,b, Yuanbiao Zhang c, Haiping Shi d,⇑, Xiaoqing Deng e a

Oriental Science and Technology College, Hunan Agricultural University, Changsha 410128, China Science College, Hunan Agricultural University, Changsha 410128, China c Packaging Engineering Institute, Jinan University, Zhuhai 519070, China d Modern Business and Management Department, Guangdong Construction Vocational Technology Institute, Guangzhou 510450, China e School of Mathematics and Statistics, Hunan University of Commerce, Changsha 410205, China b

a r t i c l e

i n f o

a b s t r a c t

Keywords: Periodic and subharmonic solutions Fourth-order Nonlinear difference equations Discrete variational theory

By using the critical point theory, some new criteria are obtained for the existence and multiplicity of periodic and subharmonic solutions to fourth-order nonlinear difference equations. The main approach used in our paper is a variational technique and the Linking Theorem. Our results generalize and improve the existing ones. Ó 2014 Elsevier Inc. All rights reserved.

1. Introduction Let N; Z and R denote the sets of all natural numbers, integers and real numbers respectively. For a; b 2 Z, define ZðaÞ ¼ fa; a þ 1; . . .g; Zða; bÞ ¼ fa; a þ 1; . . . ; bg when a 6 b. ⁄ denotes the transpose of a vector. In this paper, we consider the following forward and backward difference equation

  D2 r n2 D2 un2 ¼ f ðn; unþ1 ; un ; un1 Þ;

n 2 Z;

ð1:1Þ 2

where D is the forward difference operator Dun ¼ unþ1  un ; D un ¼ DðDun Þ, rn is real valued for each n 2 Z; f 2 CðZ  R3 ; RÞ, r n and f ðn; v 1 ; v 2 ; v 3 Þ are T-periodic in n for a given positive integer T. We may think of (1.1) as a discrete analogue of the following fourth-order functional differential equation 00

½rðtÞu00 ðtÞ ¼ f ðt; uðt þ 1Þ; uðtÞ; uðt  1ÞÞ;

t 2 R:

ð1:2Þ

Eq. (1.2) includes the following equation

uð4Þ ðtÞ ¼ f ðt; uðtÞÞ;

t 2 R;

ð1:3Þ

which is used to model deformations of elastic beams [7,26]. Equations similar in structure to (1.2) arise in the study of the existence of solitary waves of lattice differential equations, see Smets and Willem [28]. Difference equations occur widely in numerous settings and forms, both in mathematics itself and in its applications to statistics, computing, electrical circuit analysis, dynamical systems, economics, biology and other fields. For the general

q This project is supported by the Specialized Research Fund for the Doctoral Program of Higher Eduction of China (No. 20114410110002), National Natural Science Foundation of China (No. 11101098), Natural Science Foundation of Guangdong Province (No. S2013010014460) and Scientific Research Fund of Hunan Provincial Education Department (Grant No. 13C487). ⇑ Corresponding author. E-mail address: [email protected] (H. Shi).

http://dx.doi.org/10.1016/j.amc.2014.03.086 0096-3003/Ó 2014 Elsevier Inc. All rights reserved.

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X. Liu et al. / Applied Mathematics and Computation 236 (2014) 613–620

background of difference equations, one can refer to monographs [1,3,18]. Since the last decade, there has been much progress on the qualitative properties of difference equations, which included results on stability and attractivity [12,18,21,36] and results on oscillation and other topics, see [1–4,15–17,19,20,32–35]. The motivation of this paper is as follows. The widely used tools for the existence of periodic solutions of difference equations are the various fixed point theorems in cones. See, for example, [1,3,18] and references therein. It is well known that critical point theory is a powerful tool that deals with the problems of differential equations [5,7,10,11,22,26,30]. Only since 2003, critical point theory has been employed to establish sufficient conditions on the existence of periodic solutions of difference equations. By using the critical point theory, Guo and Yu [15–17] and Shi et al. [27] have successfully proved the existence of periodic solutions of second-order nonlinear difference equations. Compared to first-order or second-order difference equations, the study of higher-order equations, and in particular, fourth-order equations, has received considerably less attention (see, for example, [1,8,9,13,18,24,25,29,31] and the references contained therein). Yan and Liu [31] in 1997 and Thandapani, Arockiasamy [29] in 2001 studied the following fourth-order difference equation of form,

  D2 r n D2 un þ f ðn; un Þ ¼ 0;

n 2 Z;

ð1:4Þ

the authors obtain criteria for the oscillation and nonoscillation of solutions for Eq. (1.4). In 2005, Cai et al. [6] have obtained some criteria for the existence of periodic solutions of the fourth-order difference equation

  D2 r n2 D2 un2 þ f ðn; un Þ ¼ 0;

n 2 Z:

ð1:5Þ

In 1995, Peterson and Ridenhour considered the disconjugacy of Eq. (1.5) when r n  1 and f ðn; un Þ ¼ qn un (see [24]). However, to the best of our knowledge, the results on periodic solutions of fourth-order nonlinear difference equations are very scarce in the literature. We found that [6] is the only paper which deals with the problem of periodic solutions to fourthorder difference equation (1.5). Furthermore, since (1.1) contains both advance and retardation, there are very few manuscripts dealing with this subject. The main purpose of this paper is to give some sufficient conditions for the existence and multiplicity of periodic and subharmonic solutions to fourth-order nonlinear difference equations. The proof is based on the Linking Theorem in combination with variational technique. In particular, our results not only generalize the results in the literature [6], but also improve them. In fact, one can see the following Remarks 1.2 and 1.4 for details. Let

r ¼ min frn g; n2Zð1;TÞ

r ¼ max fr n g: n2Zð1;TÞ

Our main results are as follows. Theorem 1.1. Assume that the following hypotheses are satisfied: ðrÞr n > 0; 8n 2 Z; ðF 1 Þ there exists a functional Fðn; v 1 ; v 2 Þ 2 C 1 ðZ  R2 ; RÞ with Fðn; v 1 ; v 2 Þ P 0 and it satisfies

Fðn þ T; v 1 ; v 2 Þ ¼ Fðn; v 1 ; v 2 Þ; @Fðn  1; v 2 ; v 3 Þ @Fðn; v 1 ; v 2 Þ þ ¼ f ðn; v 1 ; v 2 ; v 3 Þ; @v 2 @v 2

  ðF 2 Þ there exist constants d1 > 0; a 2 0; 14 rk2min such that

  Fðn; v 1 ; v 2 Þ 6 a v 21 þ v 22 ;

for n 2 Z and

ðF 3 Þ there exist constants q1 > 0; f > 0; b 2

  Fðn; v 1 ; v 2 Þ P b v 21 þ v 22  f;

1 4

v 21 þ v 22 6 d21 ;

 rk2max ; þ1 such that

for n 2 Z and

v 21 þ v 22 P q21 ;

where kmin ; kmax are constants which can be referred to (2.7). Then for any given positive integer m > 0, (1.1) has at least three mT-periodic solutions.

Remark 1.1. By ðF 3 Þ it is easy to see that there exists a constant f0 > 0 such that

ðF 03 Þ Fðn; v 1 ; v 2 Þ P b



v 21 þ v 22



 f0 ;

8ðn; v 1 ; v 2 Þ 2 Z  R2 :

    As a matter of fact, let f1 ¼ max Fðn; v 1 ; v 2 Þ  b v 21 þ v 22 þ f : n 2 Z; desired result.

v 21 þ v 22 6 q21



; f0 ¼ f þ f1 , we can easily get the

Corollary 1.1. Assume that ðrÞ and ðF 1 Þ–ðF 3 Þ are satisfied. Then for any given positive integer m > 0, (1.1) has at least two nontrivial mT-periodic solutions.

X. Liu et al. / Applied Mathematics and Computation 236 (2014) 613–620

615

Remark 1.2. Corollary 1.1 reduces to Theorem 1.1 in [6]. Theorem 1.2. Assume that ðrÞ; ðF 1 Þ and the following conditions are satisfied: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðF 4 Þ limq!0 Fðn;vq12 ;v 2 Þ ¼ 0; q ¼ v 21 þ v 22 ; 8ðn; v 1 ; v 2 Þ 2 Z  R2 ; ðF 5 Þ there exist constants R1 > 0 and h > 2 such that for n 2 Z and v 21 þ v 22 P R21 ,

0 < hFðn; v 1 ; v 2 Þ 6

@Fðn; v 1 ; v 2 Þ @Fðn; v 1 ; v 2 Þ v1 þ v 2: @v 1 @v 2

Then for any given positive integer m > 0, (1.1) has at least three mT-periodic solutions. Remark 1.3. Assumption ðF 5 Þ implies that there exist constants a1 > 0 and a2 > 0 such that

ðF 05 Þ Fðn; v 1 ; v 2 Þ P a1

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffih v 21 þ v 22  a2 ;

8ðn; v 1 ; v 2 Þ 2 Z  R2 :

Corollary 1.2. Assume that ðrÞ and ðF 1 Þ; ðF 4 Þ; ðF 5 Þ are satisfied. Then for any given positive integer m > 0, (1.1) has atleast two nontrivial mT-periodic solutions. If f ðn; unþ1 ; un ; un1 Þ ¼ qn g ðun Þ, (1.1) reduces to the following fourth-order nonlinear equation,

  D2 r n2 D2 un2 ¼ qn g ðun Þ;

n 2 Z;

ð1:6Þ

where g 2 CðR; RÞ; qnþT ¼ qn > 0, for all n 2 Z. Then, we have the following results. Theorem 1.3. Assume that ðrÞ and the following hypotheses are satisfied: ðG1 Þ there exists a functional Gðv Þ 2 C 1 ðR; RÞ with Gðv Þ P 0 and it satisfies

G0 ðv Þ ¼ gðv Þ;   ðG2 Þ there exist constants d2 > 0; a 2 0; 12 rk2min such that

Gðv Þ 6 ajv j2 ;

for jv j 6 d2 ;

ðG3 Þ there exist constants q2 > 0; f > 0; b 2

Gðv Þ P bjv j2  f;

1 2

 rk2max ; þ1 such that

for jv j P q2 ;

where kmin ; kmax are constants which can be referred to (2.7). Then for any given positive integer m > 0, (1.6) has at least three mT-periodic solutions. Corollary 1.3. Assume that ðrÞ and ðG1 Þ–ðG3 Þ are satisfied. Then for any given positive integer m > 0, (1.6) has at least two nontrivial mT-periodic solutions. Remark 1.4. Corollary 1.3 reduces to Theorem 1.2 in [6]. The rest of the paper is organized as follows. Firstly, in Section 2, we shall establish the variational framework associated with (1.1) and transfer the problem of the existence of periodic solutions of (1.1) into that of the existence of critical points of the corresponding functional. Some related fundamental results will also be recalled. Then, in Section 3, we shall complete the proof of the results by using the critical point method. Finally, in Section 4, we shall give an example to illustrate the main result. About the basic knowledge for variational methods, please refer the reader to [14,22,23,26].

2. Variational structure and some lemmas In order to apply the critical point theory, we shall establish the corresponding variational framework for (1.1) and give some lemmas which will be of fundamental importance in proving our main results. Firstly, we state some basic notations.

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Let S be the set of sequences u ¼ ð. . . ; un ; . . . ; u1 ; u0 ; u1 ; . . . ; un ; . . .Þ ¼ fun gþ1 n¼1 , that is

S ¼ ffun gjun 2 R; n 2 Zg:

v 2 S;

For any u;

a; b 2 R; au þ bv is defined by þ1

au þ bv ¼ faun þ bv n gn¼1 : Then S is a vector space. For any given positive integers m and T; EmT is defined as a subspace of S by

EmT ¼ fu 2 SjunþmT ¼ un ; 8n 2 Zg: Clearly, EmT is isomorphic to RmT . EmT can be equipped with the inner product

hu; v i ¼

mT X uj v j ;

v 2 EmT ;

8u;

ð2:1Þ

j¼1

by which the norm k  k can be induced by

kuk ¼

mT X u2j

!12

8u 2 EmT :

;

ð2:2Þ

j¼1

It is obvious that EmT with the inner product (2.1) is a finite dimensional Hilbert space and linearly homeomorphic to RmT . On the other hand, we define the norm k  ks on EmT as follows:

kuks ¼

mT X juj js

!1s ð2:3Þ

;

j¼1

for all u 2 EmT and s > 1. Since kuks and kuk2 are equivalent, there exist constants c1 ; c2 such that c2 P c1 > 0, and

c1 kuk2 6 kuks 6 c2 kuk2 ;

8u 2 EmT :

ð2:4Þ

Clearly, kuk ¼ kuk2 . For all u 2 EmT , define the functional J on EmT as follows:

JðuÞ ¼

mT mT  2 X 1X rn1 D2 un1  Fðn; unþ1 ; un Þ; 2 n¼1 n¼1

ð2:5Þ

where

@Fðn  1; v 2 ; v 3 Þ @Fðn; v 1 ; v 2 Þ þ ¼ f ðn; v 1 ; v 2 ; v 3 Þ: @v 2 @v 2 Clearly, J 2 C 1 ðEmT ; RÞ and for any u ¼ fun gn2Z 2 EmT , by using u0 ¼ umT ; u1 ¼ umTþ1 , we can compute the partial derivative as

  @J ¼ D2 r n2 D2 un2  f ðn; unþ1 ; un ; un1 Þ: @un Thus, u is a critical point of J on EmT if and only if

  D2 r n2 D2 un2 ¼ f ðn; unþ1 ; un ; un1 Þ;

8n 2 Zð1; mTÞ:

Due to the periodicity of u ¼ fun gn2Z 2 EmT and f ðn; v 1 ; v 2 ; v 3 Þ in the first variable n, we reduce the existence of periodic solutions of (1.1) to the existence of critical points of J on EmT . That is, the functional J is just the variational framework of (1.1). Let P be the mT  mT matrix defined by

0

2 1 0    B B 1 2 1    B B B 0 1 2    B P¼B B B     B B B 0 0 0  @ 1

0

0

0 0 0  2

   1

1

1

C 0 C C C 0 C C C: C  C C C 1 C A 2

By matrix theory, we see that the eigenvalues of P are

X. Liu et al. / Applied Mathematics and Computation 236 (2014) 613–620

 2k kk ¼ 2 1  cos p ; mT

k ¼ 0; 1; 2; . . . ; mT  1:

Thus, k0 ¼ 0; k1 > 0; k2 > 0; . . . ; kmT1 > 0. Therefore,

  2 kmin ¼ minfk1 ; k2 ; . . . ; kmT1 g ¼ 2 1  cos mT p; ( 4; when mT is even;   kmax ¼ maxfk1 ; k2 ; . . . ; kmT1 g ¼ 1 2 1 þ cos mT p ; when mT is odd:

617

ð2:6Þ

9 > > = > > ;

ð2:7Þ

Let

W ¼ ker P ¼ fu 2 EmT jPu ¼ 0 2 RmT g: Then

W ¼ fu 2 EmT ju ¼ fcg; c 2 Rg: Let V be the direct orthogonal complement of EmT to W, i.e., EmT ¼ V  W. For convenience, we identify u 2 EmT with u ¼ ðu1 ; u2 ; . . . ; umT Þ . Let E be a real Banach space, J 2 C 1 ðE; RÞ, i.e., J is a continuously Fréchet-differentiable functionaldefined     on E. J is said to satisfy the Palais–Smale condition (P.S. condition for short) if any sequence uðkÞ E for which J uðkÞ is bounded and   J 0 uðkÞ ! 0ðk ! 1Þ possesses a convergent subsequence in E. Let Bq denote the open ball in E about 0 of radius q and let @Bq denote its boundary. Lemma 2.1 (Linking Theorem [26]). Let E be a real Banach space, E ¼ E1  E2 , where E1 is finite dimensional. Suppose that J 2 C 1 ðE; RÞ satisfies the P.S. condition and ðJ 1 Þ there exist constants a > 0 and q > 0 such that Jj@Bq \E2 P a; ðJ 2 Þ there exists an e 2 @B1 \ E2 and a constant R0 P q such that Jj@Q 6 0, where Q ¼ ðBR0 \ E1 Þ  fsej0 < s < R0 g. Then J possesses a critical value c P a, where

c ¼ inf sup JðhðuÞÞ h2C u2Q

n o and C ¼ h 2 CðQ ; EÞjhj@Q ¼ id , where id denotes the identity operator. Lemma 2.2. Assume that ðrÞ; ðF 1 Þ and ðF 3 Þ are satisfied. Then the functional J is bounded from above in EmT . Proof. By ðF 03 Þ and (2.4), for any u 2 EmT , mT mT mT mT   X   X 1X 1X rn1 D2 un1 ; D2 un1  Fðn; unþ1 ; un Þ ¼ r n D2 un ; D2 un  Fðn; unþ1 ; un Þ 2 n¼1 2 n¼1 n¼1 n¼1

JðuÞ ¼

mT X 2

r r  x Px  bðunþ1 þ u2n Þ  c0 6 kmax kxk22  2bkuk22 þ mTf0 ; 2 2 n¼1

6

where x ¼ ðDu1 ; Du2 ; . . . ; DumT Þ . Since

kxk22 ¼

mT X ðunþ1  un ; unþ1  un Þ ¼ u Pu 6 kmax kuk22 ; n¼1

we have

JðuÞ 6

 r 2 kmax  2b kuk22 þ mTf0 6 mTf0 : 2

The proof of Lemma 2.2 is complete.

h

Remark 2.1. The case mT ¼ 1 is trivial. For the case mT ¼ 2, P has a different form, namely,

P¼



2

2

2

2

 :

However, in this special case, the argument need not to be changed and we omit it.

ð2:8Þ

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X. Liu et al. / Applied Mathematics and Computation 236 (2014) 613–620

Lemma 2.3. Assume that ðrÞ; ðF 1 Þ and ðF 3 Þ are satisfied. Then the functional J satisfies the P.S. condition.   Proof. Let J uðkÞ be a bounded sequence from the lower bound, i.e., there exists a positive constant M 1 such that

  M 1 6 J uðkÞ ;

8k 2 N:

By the proof of Lemma 2.2, it is easy to see that

  M 1 6 J uðkÞ 6



  2 r 2 kmax  2b uðkÞ 2 þ mTf0 ; 2

8k 2 N:

Therefore,

  2 r 2b  k2max uðkÞ 2 6 M1 þ mTf0 : 2

    Since b > 14 rk2max , it is not difficult to know that uðkÞ is a bounded sequence in EmT . As a consequence, uðkÞ possesses a convergence subsequence in EmT . Thus the P.S. condition is verified. h 3. Proof of the main results In this Section, we shall prove our main results by using the critical point method. 3.1. Proof of Theorem 1.1 Assumptions ðF 1 Þ and ðF 2 Þ imply that Fðn; 0Þ ¼ 0 and f ðn; 0Þ ¼ 0 for n 2 Z. Then u ¼ 0 is a trivial mT-periodic solution of (1.1). By Lemma 2.2, J is bounded from the upper on EmT . We define c0 ¼ supu2EmT JðuÞ. The proof of Lemma 2.2 implies  2 EmT such that Jðu  Þ ¼ c0 . limkuk2 !þ1 JðuÞ ¼ 1. This means that JðuÞ is coercive. By the continuity of JðuÞ, there exists u  is a critical point of J. Clearly, u We claim that c0 > 0. Indeed, by ðF 2 Þ, for any u 2 V; kuk2 6 d1 , we have

JðuÞ ¼ P

mT mT mT mT   X   X 1X 1X rn1 D2 un1 ; D2 un1  Fðn; unþ1 ; un Þ ¼ r n D2 un ; D2 un  Fðn; unþ1 ; un Þ 2 n¼1 2 n¼1 n¼1 n¼1 mT X 1  1 rx Px  a ðu2nþ1 þ u2n Þ P rkmin kxk22  2akuk22 ; 2 2 n¼1

ð2:9Þ

where x ¼ ðDu1 ; Du2 ; . . . ; DumT Þ . Since

kxk22 ¼

mT X ðunþ1  un ; unþ1  un Þ ¼ u Pu P kmin kuk22 ; n¼1

we have

 1 2 JðuÞ P rkmin  2a kuk22 : 2

Take

r¼

1 2

 rk2min  2a d21 . Then

JðuÞ P r;

8u 2 V \ @Bd1 :

Therefore, c0 ¼ supu2EmT JðuÞ P r > 0. At the same time, we have also proved that there exist constants r > 0 and d1 > 0 such that Jj@Bd \V P r. That is to say, J satisfies the condition ðJ 1 Þ of the Linking Theorem.  2 1 P 2 Noting that mT ¼ 0, for all u 2 W, we have n¼1 r n1 D un1

JðuÞ ¼

mT mT mT  2 X X 1X rn1 D2 un1  Fðn; unþ1 ; un Þ ¼  Fðn; unþ1 ; un Þ 6 0: 2 n¼1 n¼1 n¼1

 of J corresponding to the critical value c0 is a nontrivial mT-periodic solution of (1.1). Thus, the critical point u  , we need to use the conclusion of In order to obtain another nontrivial mT-periodic solution of (1.1) different from u Lemma 2.1. We have known that J satisfies the P.S. condition on EmT . In the following, we shall verify the condition ðJ 2 Þ. Take e 2 @B1 \ V, for any z 2 W and s 2 R, let u ¼ se þ z. Then

JðuÞ ¼

mT mT mT  mT   X  X r X 1X rn D2 un ; D2 un  Fðn; unþ1 ; un Þ 6 s2 D2 en ; D2 en  Fðn; senþ1 þ znþ1 ; sen þ zn Þ 2 n¼1 2 n¼1 n¼1 n¼1

6

mT n h mT i o r X X r 2  s y Py  b ðsenþ1 þ znþ1 Þ2 þ ðsen þ zn Þ2  f0 6 s2 kmax kyk22  2b ðsen þ zn Þ2 þ mTf0 2 2 n¼1 n¼1

¼

r 2 s kmax kyk22  2bs2  2bkzk22 þ mTf0 ; 2

ð2:10Þ

X. Liu et al. / Applied Mathematics and Computation 236 (2014) 613–620

619

where y ¼ ðDe1 ; De2 ; . . . ; DemT Þ . Since

kyk22 ¼

mT X ðenþ1  en ; enþ1  en Þ ¼ e Pe 6 kmax ; n¼1

we have

JðuÞ 6



 r 2 kmax  2b s2  2bkzk22 þ mTf0 6 2bkzk22 þ mTf0 : 2

Thus, there exists a positive constant R2 > d1 such that for any u 2 @Q , JðuÞ 6 0, where Q ¼ ðBR2 \ WÞ  fsej0 < s < R2 g. By the Linking Theorem, J possesses a critical value c P r > 0, where

c ¼ inf sup JðhðuÞÞ h2C u2Q

n o and C ¼ h 2 CðQ ; EmT Þjhj@Q ¼ id . ~ 2 EmT be a critical point associated to the critical value c of J, i.e., Jðu ~ Þ ¼ c. If u ~–u  , then the conclusion of Theorem 1.1 Let u ~¼u  . Then c0 ¼ Jðu  Þ ¼ Jðu ~ Þ ¼ c, that is supu2EmT JðuÞ ¼ inf h2C supu2Q JðhðuÞÞ. Choosing h ¼ id, we have holds. Otherwise, u supu2Q JðuÞ ¼ c0 . Since the choice of e 2 @B1 \ V is arbitrary, we can take e 2 @B1 \ V. Similarly, there exists a positive number R3 > d1 , for any u 2 @Q 1 , JðuÞ 6 0, where Q 1 ¼ ðBR3 \ WÞ  fsej0 < s < R3 g. Again, by the Linking Theorem, J possesses a critical value c0 P r > 0, where

c0 ¼ inf sup JðhðuÞÞ h2C1 u2Q 1

n

o and C1 ¼ h 2 CðQ 1 ; EmT Þjhj@Q 1 ¼ id . If c0 – c0 , then the proof is finished. If c0 ¼ c0 , then supu2Q 1 JðuÞ ¼ c0 . Due to the fact Jj@Q 6 0 and Jj@Q 1 6 0; J attains its maximum at some points in the interior of sets Q and Q 1 . However, Q \ Q 1 W and JðuÞ 0 for any u 2 W. Therefore, there must ~ and Jðu0 Þ ¼ c0 ¼ c0 . The proof of Theorem 1.1 is complete. h be a point u0 2 EmT ; u0 – u Remark 3.1. Similarly to above argument, we can also prove Theorems 1.2 and 1.3. For simplicity, we omit their proofs. Remark 3.2. Due to Theorems 1.1, 1.2 and 1.3, the conclusion of Corollaries 1.1, 1.2 and 1.3 is obviously true.

4. Examples As an application of Theorem 1.1, finally, we give two examples to illustrate our main result. Example 4.1. For all n 2 Z, assume that





      l1  2 l1 2 pn 2 pðn  1Þ un þ u2n1 2 ; D2 r n2 D2 un2 ¼ lun 9 þ sin u2nþ1 þ u2n 2 þ 9 þ sin T T where r n is real valued for each n 2 Z and r nþT ¼ r n > 0; We have

f ðn; v 1 ; v 2 ; v 3 Þ ¼ lv 2

ð4:1Þ

l > 2, T is a given positive integer.





    l l pðn  1Þ  2 2 pn 9 þ sin v 21 þ v 22 21 þ 9 þ sin2 v 2 þ v 23 21 T T

and

h  i l 2 pn Fðn; v 1 ; v 2 Þ ¼ 9 þ sin v 21 þ v 22 2 : T Then





    l l @Fðn  1; v 2 ; v 3 Þ @Fðn; v 1 ; v 2 Þ pðn  1Þ  2 2 pn þ ¼ lv 2 9 þ sin v 21 þ v 22 21 þ 9 þ sin2 v 2 þ v 23 21 : @v 2 @v 2 T T It is easy to verify all the assumptions of Theorem 1.1 are satisfied. Consequently, for any given positive integer m > 0, (4.1) has at least three mT-periodic solutions. Example 4.2. For all n 2 Z, assume that

h 3  3 i D4 un2 ¼ 2un u2nþ1 þ u2n 2 þ u2n þ u2n1 2 :

ð4:2Þ

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X. Liu et al. / Applied Mathematics and Computation 236 (2014) 613–620

We have

rn  1;

f ðn; v 1 ; v 2 ; v 3 Þ ¼ 2v 2

h

v 21 þ v 22

32

þ



v 22 þ v 23

32 i

and

Fðn; v 1 ; v 2 Þ ¼

5 2 2 v þ v 22 2 : 5 1

It is easy to verify all the assumptions of Theorem 1.1 are satisfied. Consequently, for any given positive integer m > 0, (4.2) has at least three 4m-periodic solutions un  0; un ¼ sinðp2 nÞ and un ¼ cosðp2 nÞ. References [1] R.P. Agarwal, Difference Equations and Inequalities: Theory, Methods and Applications, Marcel Dekker, New York, 1992. [2] R.P. Agarwal, K. Perera, D. O’regan, Multiple positive solutions of singular and nonsingular discrete problems via variational methods, Nonlinear Anal. 58 (1–2) (2004) 69–73. [3] R.P. Agarwal, P.J.Y. Wong, Advanced Topics in Difference Equations, Kluwer Academic Publishers, Dordrecht, 1997. [4] R.I. Avery, A.C. Pererson, Three positive fixed points of nonlinear operators on ordered Banach space, Comput. Math. Appl. 42 (3–5) (2001) 313–322. [5] V. Benci, P.H. Rabinowitz, Critical point theorems for indefinite functionals, Invent. Math. 52 (3) (1979) 241–273. [6] X.C. Cai, J.S. Yu, Z.M. 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