Homoclinic solutions for a class of nonlinear difference systems with classical \((\phi {1}, \phi {2})\) Laplacian

R E S E A R C H

Open Access

Homoclinic solutions for a class of

nonlinear difference systems with classical

(

φ

_

,

φ

_

)-Laplacian

Xingyong Zhang

_{and Yun Wang}

*_{Correspondence:} [email protected] Department of Mathematics, Faculty of Science, Kunming University of Science and Technology, Kunming, Yunnan 650500, P.R. China

Abstract

In this paper, we consider the existence of homoclinic solutions for a class of nonlinear difference systems involving classical (φ1,

φ

inequalities in known literature. Then, by using the variational method, some new existence results are obtained. Finally, some examples are given to verify our results.

MSC: 37J45; 58E50; 34C25

Keywords: difference systems; classical (φ1,

φ

variational method

1 Introduction and main results

LetRdenote the real numbers andZthe integers. Givena<binZ. LetZ[a,b] ={a,a+ , . . . ,b}. LetT>  andNbe fixed positive integers.

In this paper, we investigate the existence of homoclinic solutions for the following non-linear difference systems involving classical (φ,φ)-Laplacian:

φ(u(t– )) +∇uV(t,u(t),u(t)) =f(t),

φ(u(t– )) +∇uV(t,u(t),u(t)) =f(t),

(.)

wheret∈Z,um(t)∈RN,m= , ,V(t,x,x) = –K(t,x,x) +W(t,x,x),K,W:Z×RN× RN_→Randφ

m,m= , , satisfy the following condition:

(A) φm is a homeomorphism fromRN ontoRN such thatφm() = ,φm=∇m, with m∈C(RN, [, +∞])strictly convex andm() = ,m= , .

Remark . Assumption (A) is given in [], which is used to characterize the classical

homeomorphism. If, furthermore,m:RN→Ris coercive (i.e.,m(x)→+∞as|x| → ∞), there existsδm>  such that

m(x)≥δm

|x|– , x∈RN, (.)

whereδm=min|x|=m(x),m= ,  (see []).

We callu= (u,u) a nontrivial homoclinic solution of system (.) ifusatisfies system (.),u=  andu(t)→ ast→ ∞.

It is well known that the variational method has become an important tool to study the existence and multiplicity of solutions for various difference systems. Lots of contribu-tions have been obtained (for example, see [–]). It is remarkable that, to the best of our knowledge, few people investigated system (.). Recently, in [] and [], by using the vari-ational approach, J Mawhin investigated the following second order nonlinear difference systems withφ-Laplacian:

φu(n– )=∇uF

n,u(n)+h(n) (n∈Z), (.)

whereφ=∇,strictly convex, is a homeomorphism ofRN _{onto the ballB}

a⊂RN or

ofBa ontoRN. By using the variational approach, under different conditions, the author

obtained that system (.) has at least one orN+  geometrically distinctT-periodic so-lutions. It is interesting that J Mawhin considered three kinds ofφ: ()φ:RN_→RN _{is a}

classical homeomorphism, for example,φ(x) =|x|p–_{xfor somep>  and allx∈R}N_{; ()φ:} RN_→B

a(a< +∞) is a bounded homeomorphism, for example,φ(x) = √x

+|x| ∈Bfor all x∈RN_{; ()φ:B}

a⊂RN→RN is a singular homeomorphism, for example,φ(x) =√x

–|x|

for allx∈B. Recently, in [], we generalized some results in [] for classical homeomor-phism and bounded homeomorhomeomor-phism to system (.), which seem to be the first results for system (.).

In , He and Chen [] investigated the existence of homoclinic solutions for the following discretep-Laplacian systems:

u(t– )p–u(t– )=∇Ft,u(t)+f(t), t∈Z,u∈RN, (.)

wherep> . They obtained homoclinic orbits as the limit of the subharmonics for system (.).

In this paper, motivated by [, , , ] and [], we first improve some inequalities in [] and then investigate the existence of homoclinic solutions for system (.) with classical homeomorphism. Next we make the following assumption:

(A) Letp> .Assume that there exist positive constantsd,d,d,dsuch that

d|x|p≤(x)≤d|x|p, d|y|p≤(y)≤d|y|p, ∀x,y∈RN

and

φ(x),x

≤p(x),

φ(y),y

≤p(y), ∀x,y∈RN.

For everys∈N, define

ls=

g:Z→RN,

+∞

t=–∞

with the norm

gls=

_+∞

t=–∞

g(t)s

/s

.

Letp>  be such that_p+_p =  and

C∗=

(T)–p/p+min

(T+ )p++Tp+–  (T)p_{(p+ )} ,

T

p/p

/p

.

Next, we present our main results.

Theorem . Assume that(A)holds,fi= ,i= , ,W and K satisfy the following

condi-tions:

(V) V(t,x,x) = –K(t,x,x) +W(t,x,x),whereK,W:Z×RN ×RN→R,K(t,x,x)

andW(t,x,x)areT-periodic and for everyt∈Z,K,W∈C_(Z×RN_×RN_,R);

(H) there existγ ∈(,p)anda,a> such that

K(t,x,x)≥a|x|γ+a|x|γ for all(t,x,x)∈Z[,T– ]×RN×RN;

(H) K(t, , )≡and

x,∇xK(t,x,x)

+x,∇xK(t,x,x)

≤pK(t,x,x) for all(t,x,x)∈Z[,T– ]×RN_×RN_;

(H) (i) there existr∈(, ], <b<aC∗γ–p,and <b<aC∗γ–psuch that

W(t,x,x)≤b|x|p+b|x|p,

∀t∈Z[,T– ],|x| ≤rC∗,|x| ≤rC∗; (.)

(ii) there existr> , <b<a(C∗r)γ–p,and <b<a(C∗r)γ–psuch that(.)

holds; (H)

lim

|x|+|x|→+∞

W(t,x,x) |x|p+|x|p

>d+d+ p–A for allt∈Z[,T– ],

where

A= max

|x|≤,|x|≤,t∈Z[,T–]

K(t,x,x);

(H) there exist positive constantsξ,η,ηandν∈[,γ– )such that

≤

p+ 

ξ+η|x|ν+η|x|ν

W(t,x,x)

≤∇xW(t,x,x),x

+∇xW(t,x,x),x

(H) f,f∈lp_∩lpp––νν– _and (i) whenr∈(, ],

maxf_lp,f_lp

<  p–min

d,d,aCγ∗–p–b,aCγ∗–p–b

rp–;

(ii) whenr∈(, +∞),

maxf_lp,f_lp

<  p–min

d,d,a(C∗r)γ–p–b,a(C∗r)γ–p–brp–.

Then system(.)possesses a nontrivial homoclinic solution.

Theorem . Assume that(A)holds,fi= , i= , ,W and K satisfy(V), (H)-(H)and

the following conditions:

(H) f,f∈l_and (i) whenr∈(, ],

maxfl,f_l

<  p–_C∗min

d,d,aC∗γ–p–b,aC∗γ–p–b

rp–;

(ii) whenr∈(, +∞),

maxfl,f_l

<  p–_C∗min

d,d,a(C∗r)γ–p–b,a(C∗r)γ–p–brp–.

Then system(.)possesses a nontrivial homoclinic solution.

Theorem . Assume that(A)holds,fi= ,i= , ,W and K satisfy(V), (H), (H), (H)

and the following conditions:

(H) there exista,a> such that

K(t,x,x)≥a|x|p+a|x|p for all(t,x,x)∈Z[,T– ]×RN×RN;

(H) there existr> and <b<a, <b<asuch that

W(t,x,x)≤b|x|p+b|x|p, ∀|x| ≤rC∗,|x| ≤rC∗;

(H) f,f∈lp

∩lpp––νν– _and

maxf_lp,f_lp

< 

p–min{d,d,a–b,a–b}r

p–_.

Theorem . Assume that(A)holds,fi= ,i= , ,W and K satisfy(V), (H), (H),

(H), (H), (H)and the following condition:

(H) f,f∈l_and

maxfl,f_l

< 

p–_C∗min{d,d,a–b,a–b}r

p–_.

Then system(.)possesses a nontrivial homoclinic solution.

Remark . Theorem . and Theorem . show thatf,fcan be large whenris large.

2 Preliminaries

Similar to [] and [], we will obtain the homoclinic orbit of system (.) as a limit of solutions of a sequence of difference systems:

φ(u(t– )) +∇uV(t,u(t),u(t)) =f,k(t),

φ(u(t– )) +∇uV(t,u(t),u(t)) =f,k(t),

(.)

where fm,k :Z→RN is a kT-periodic extension of restriction of fm to the interval Z[–kT,kT– ],k∈N,m= , .

Next, we present some basic notations. We use| · |to denote the usual Euclidean norm inRN_{. Define}

V=u= (u,u)τ=u(t)|u(t) =u(t),u(t)τ ∈RN,

um=

um(t)

,um(t)∈RN,m= , ,t∈Z

.

His defined as a subspace ofVby

Hk=

u=u(t)∈V|u(t+ kT) =u(t),t∈Z. Define

Hm,k=

um=

um(t)

|um(t+ kT) =um(t),um(t)∈RN,t∈Z

, m= , .

ThenHk=H,k×H,k. Forum∈Hm,k, set

ums,k=

_kT–

t=–kT

um(t) s

/s

, m= , ,s> .

Moreover,l∞_kTdenote the space of all bounded real functions onZ[–kT,kT– ] endowed with the norm

uml∞_kT= max t∈Z[–kT,kT–]

um(t), m= , .

For  <p< +∞, onHm,k, we define

umHm,k=

_kT–

t=–kT

um(t) p

+

kT–

t=–kT

um(t) p

/p

Foru= (u,u)τ_∈H k, define

u_Hk=u_H,k+u_H,k.

Then (Hk,uHk), (H,k,uH,k) and (H,k,uH,k) are reflexive Banach spaces.

Lemma . Let a,b∈Z,a≥,b≥,> ,um∈Hm,k,m= , .Then,for every t∈Z,

um(t)≤(a+b+ )–/

_t+b

s=t–a

um(s)

/

+min

[(a+ )p+_{+ (b+ )}p+_{– ]}/p

(a+b+ )p_{(p+ )}/p ,

max{a,b}

(a+b+ )/p

·

_t+b

s=t–a

um(s) p

/p

, (.)

where m= , .

Proof Fixt∈Z. For everyτ∈Z[t–a,t– ], we have

um(t) =um(τ) + t–

s=τ

um(s) (.)

and for everyτ∈Z[t,t+b],

um(t) =um(τ) – τ–

s=t

um(s). (.)

Summing (.) overZ[t–a,t– ] and (.) overZ[t,t+b], we have

aum(t) = t–

τ=t–a

um(τ) + t–

τ=t–a t–

s=τ

um(s)

=

t–

τ=t–a

um(τ) + t–

s=t–a

(s–t+a+ )um(s) (.)

and

(b+ )um(t) = t+b

τ=t

um(τ) – t+b

τ=t τ–

s=t

um(s)

=

t+b

τ=t

um(τ) – t+b–

s=t

(t+b–s)um(s). (.)

Set

φ(s) =

s–t+a+ , t–a≤s≤t– ,

Combining (.) with (.) and using Hölder’s inequality, we obtain

Equation (.) implies that

(a+b+ )um(t)

Corollary . Let um∈Hm,k,m= , .Then

The proof is complete.

Corollary . Let um∈Hm,k,m= , .Then

Proof In Corollary ., let=pand then use Hölder’s inequality. Then the proof is

Remark . Asa≥, Lemma ., Corollary ., and Corollary . improve Lemma ., Corollary ., and Corollary . in [], respectively.

By Lemma . and Lemma . in [], we have the following two lemmas.

Lemma .(see []) For any u= (u,u),v= (v,v)∈Hk,the following two equalities

hold:

–

kT–

t=–kT

φ

u(t– ),v(t)=

kT–

t=–kT

φ

u(t),v(t), (.)

–

kT–

t=–kT

φ

u(t– ),v(t)=

kT–

t=–kT

φ

u(t),v(t). (.)

Lemma .(see []) Let L:Z[–kT,kT– ]×RN_×RN_×RN_×RN_{→R, (t,x,x,y,y)→}

L(t,x,x,y,y)and assume that L is continuously differential in(x,x,y,y)for all t∈

Z[–kT,kT– ].Then the functionϕk:Hk→Rdefined by

ϕk(u) =ϕk(u,u) = kT–

t=–kT

Lt,u(t),u(t),u(t),u(t)

is continuously differentiable onHkand

ϕ_k(u),v=ϕ_k(u,u), (v,v)

=

kT–

t=–kT

DxL

t,u(t),u(t),u(t),u(t),v(t) +DyL

t,u(t),u(t),u(t),u(t),v(t) +DxL

t,u(t),u(t),u(t),u(t),v(t) +DyL

t,u(t),u(t),u(t),u(t),v(t),

where u,v∈Hk.

Let

L(t,x,x,y,y) =(y) +(y) +K(t,x,x) –W(t,x,x) +

f,k(t),x

+f,k(t),x

and defineηk:Hk→[, +∞) by

ηk(u) =ηk(u,u) = kT–

t=–kT



u(t)+

u(t)+Kt,u(t),u(t).

Then

ϕk(u) =ϕk(u,u)

=

kT–

t=–kT



u(t)+

–Wt,u(t),u(t)+f,k(t),u(t)

+f,k(t),u(t)

=ηk(u) + kT–

t=–kT

–Wt,u(t),u(t)+f,k(t),u(t)

+f,k(t),u(t)

. (.)

It follows from (A), (V) and Lemma . that

ϕ_k(u),v=ϕ_k(u,u), (v,v) =

kT–

t=–kT

φ

u(t)

,v(t)

+φ

u(t)

,v(t)

+∇uK

t,u(t),u(t),v(t)+∇uK

t,u(t),u(t),v(t) –∇uW

t,u(t),u(t),v(t)–∇uW

t,u(t),u(t),v(t) +f,k(t),v(t)

+f,k(t),v(t)

, ∀u,v∈Hk. (.)

By Lemma ., it is easy to see that critical points ofϕkinHkare kT-periodic solutions

of system (.).

We shall use one linking method in [] to obtain the critical points ofϕ(the details can be seen in []). Let (E,·) be a Banach space. Define a continuous map: [, ]×E→E

by(t,x) =(t)x, where(t) satisfies the following conditions:

() () =I, the identity map.

() For eacht∈[, ),(t)is a homeomorphism ofEontoEand

–_{(t)∈C(E×[, ),E).}

() ()Eis a single point inEand(t)Aconverges uniformly to()East→for each bounded setA⊂E.

() For eacht∈[, )and each bounded setA⊂E,

sup ≤t≤t

u∈A

(t)u+–(t)u<∞.

Letbe the set of all continuous mapsas defined above.

Definition .(see [], Definition .) We say thatAlinksB[hm] ifAandBare subsets ofEsuch thatA∩B=∅, and for each∈, there ist∈(, ] such that(t)A∩B=∅.

Example (see [], p.) LetBbe an open set inE, and letAconsist of two pointse,e

withe∈Bande∈ ¯/B. ThenAlinks∂B[hm].

We use the following theorem to prove our main results.

Theorem .(see [], Theorem . and Theorem .) Let E be a Banach space,ϕ∈

C_{(E,R)and A and B be two subsets of E such that A links B[hm].Assume that}

sup

A

ϕ≤inf

B ϕ

and

c:= inf

∈_ssup_∈[,] u∈A

Letψ(t)be a positive,nonincreasing,locally Lipschitz continuous function on[,∞) sat-isfying_∞ψ(r)dr=∞.Then there exists a sequence{un} ⊂E such thatϕ(un)→c and ϕ(un)/ψ(un)→,as n→ ∞.Moreover,if c=supAϕ,then there is a sequence{un} ⊂E

satisfyingϕ(un)→c,ϕ(un)→,and d(un,B)→,as n→ ∞.

Remark . SinceAlinksB, by Definition ., it is easy to know thatc≥infBϕ. By [],

if we letψ(r) =_+_r, the sequence{un}is the Cerami sequence that is{un}satisfying

ϕ(un)→c,

 +unϕ(un)→, asn→ ∞.

3 Proofs

Lemma . Suppose that(H)holds.Then

K(t,x,x)≤p–K

t, x |x|,

x

|x|

|x|p+|x|p for all t∈Z[,T– ],|x| ≥;

K(t,x,x)≥  K

t, x |x|,

x

|x|

|x|p+|x|p for all t∈Z[,T– ],|x| ≤.

Proof Define the functionξ∈(, +∞)→K(t,ξ–x,ξ–x)(ξp+ξp). Then we have

Kt,ξ–x,ξ–x

ξp+ξp_ξ = –∇xK

t,ξ–x,ξ–x

,ξ–x

ξp+ξp –∇xK

t,ξ–x,ξ–x

,ξ–x

ξp+ξp+Kt,ξ–x,ξ–x

pξp–+pξp– ≥–∇xK

t,ξ–x,ξ–x,ξ–xξp–+ξp– –∇xK

t,ξ–x,ξ–x,ξ–xξp–+ξp–+pKt,ξ–x,ξ–xξp–+ξp– ≥.

Hence the functionξ∈(, +∞)→K(t,ξ–x,ξ–x)(ξp+ξp) is nondecreasing. Moreover, note that

|x|p_+|x|p

 ≤ |x|

p_≤|x

|+|x|

p

≤p–|x|p+|x|p

.

Then the proof can be completed easily.

Lemma . Suppose that(H)holds.Then,for any u∈Hk,

ηk(u)≥min

dup_H ,k,aC

γ–p

∗ uγH,k

+mindup_H ,k,aC

γ–p

∗ uγH,k

, ∀k∈N.

Proof It follows from (A), (H),γ ∈(,p) and Corollary . that

ηk(u) = kT–

t=–kT



u(t)

+

u(t)

+Kt,u(t),u(t)

≥ kT–

t=–kT

≥

Proof of Theorem. We divide the proof into the following Lemmas .-..

Lemma . Under the assumptions of Theorem.,for every k∈N, system(.)has a

nontrivial solution ukinHk.

Proof We first constructAandBwhich satisfy the assumptions in Theorem ..

–f_lpuH,k–flpuH,k

≥mind,aC_∗γ–p–bup_H ,k+min

d,aC_∗γ–p–bup_H ,k

–f_lpuH,k–flpuH,k

≥mind,d,aCγ_∗–p–b,aCγ_∗–p–b 

p–

uH,k+uH,k

p

–maxf_lp,f_lp

uH,k+uH,k

. (.)

(H)(i) implies that there existsα>  such that

ϕk(u)≥α>  for allu∈HkwithuHk=r,∀k∈N.

(ii) Whenr∈(, +∞), by Corollary ., (H), (H)(ii), Hölder’s inequality andγ<p, for

u∈HkwithuHk=r, we have

ϕk(u)≥min

d,a(C∗r)γ–p–b

up_H,k+min

d,a(C∗r)γ–p–b

up_H,k

–flpuH,k–flquH,k

≥mind,d,a(C∗r)γ–p–b,a(C∗r)γ–p–b 

p–

u_H,k+u_H,kp

–maxflp,flp

uH,k+uH,k

. (.)

(H)(ii) implies that there existsα>  such that

ϕk(u)≥α>  for allu∈HkwithuHk=r,∀k∈N.

By Lemma . and theT-periodicity ofK, there exists a constantB>  such that

K(t,x,x)≤p–A|x|p+|x|p+B for all (t,x,x)∈Z×RN×RN, (.) where

A= max

|x|≤,|x|≤,t∈Z[,T–]

K(t,x,x).

By (H), we know that there existε>  andL>  such that

W(t,x,x)≥d+d+ p–A+ε

|x|p+|x|p

for allt∈Z[,T– ] and∀|x| ≥L. (.)

By (.) and theT-periodicity ofW, there exists a constantB>  such that

W(t,x,x)≥d+d+ p–A+ε

|x|p+|x|p–B (.)

for all (t,x,x)∈Z[,T– ]×RN×RN. For anyk∈N, definew(k)∈Hkby

w(k)(t) =w(_k)(t),w(_k)(t)=

(, , . . . , , , , . . . , ) ift= ,

where

w(_ik)(t) =

(, , . . . , ) ift= ,

 ift∈Z[–kT,kT– ]/{}, i= , .

SinceK(t, , )≡ andW(t, , )≡, which are implied by (H) and (H), then by (.) and (.) we have

ϕk

ξw(k)=

kT–

t=–kT



ξ w(_k)(t)+

ξ w(_k)(t)+Kt,ξw_(k)(t),ξw(_k)(t)

–Wt,ξw_(k)(t),ξw(_k)(t)+ξf,k(t),w(k)(t)

+ξf,k(t),w(k)(t)

≤d|ξ|p

kT–

t=–kT

w(_k)(t)p+d|ξ|p

kT–

t=–kT

w(_k)(t)p

+K,ξw_(k)(),ξw(_k)()–W,ξw(_k)(),ξw(_k)() +ξf,k(),w(k)()

+ξf,k(),w(k)()

=d|ξ|pw(k)(–)

p

+w(k)()

p

+d|ξ|pw(k)(–)

p

+w(k)()

p

+K,ξw_(k)(),ξw(_k)()–W,ξw(_k)(),ξw(_k)() +ξf,k(),w(k)()

+ξf,k(),w(k)()

≤d|ξ|p+ d|ξ|p+ p–A|ξ|p+ p–A|ξ|p

+B–d+d+ p–A+ε

|ξ|p+|ξ|p +B+|ξ|f,k()+|ξ|f,k()

≤–ε|ξ|p+|ξ|f,k()+|ξ|f,k()+B+B. (.)

So there existsξ∈Rsuch thatξw(k)>randϕk(ξw(k)) < . Moreover, it is clear that ϕk() = . Lete=ξw(k)and

A={,e}, B=u∈Hk:uHk<r

.

Then ∈Bande∈ ¯/B. So by Example  in Section , we know thatAlinks∂B[hm]. So by Theorem . and Remark ., we have

ck= inf ∈_s∈sup_[,]

u∈A ϕk

(s)u≥inf

∂Bϕk>α> , (.)

and there exists a sequence{un= (u(n),u (n)

 )}∞n=⊂Hksuch that

ϕk(un)→ck,

 +unHkϕ

k(un)→.

Then there exists a constantCk>  such that

ϕk(un)≤Ck,

 +unHkϕ

The fact p – ν >  and the above inequality show that kT_t=––_kT|u(_n)(t)|p–ν _and

kT–

t=–kT|u

(n)

 (t)|p–νare bounded. By (A), (H), (H), (.), (.), (.), Hölder’s inequality

and Corollary ., we have

·

 H,k are bounded. SinceHis a finite-dimensional space, {u(n)= (u(n),u

nontrivial solution of system (.). The proof is complete.

·

Thus (.) and Lemma . imply that

+pηC_∗νMukν_H,k+pηC∗νMukν_H,k

+ (p– )ηC∗νukν_H+_,k

t∈Z f(t)

p/p

+ (p– )ηC_∗ν

ν ν+ uk

ν+ H,k+

 ν+ uk

ν+ H,k

t∈Z

f(t)p

/p

+ (p– )ηC_∗νukν_H+_,k

t∈Z

f(t)p

/p

+ (p– )ηC_∗ν

ν ν+ uk

ν+ H,k+

 ν+ uk

ν+ H,k

t∈Z

_f(t)p/p

≥mindup_H,k,aCγ∗–pu γ H,k

+mindup_H,k,aCγ∗–pu γ H,k

.

Note thatp>γ >ν+ . So (H) implies there existsM>  (independent ofk) such that

ukH,k≤M, ukH,k ≤M for everyk∈N.

By Corollary .,

ukl∞_kT≤C∗M, ukl∞_kT≤C∗M for everyk∈N.

LetM=max{C∗M,C∗M}. Thus the proof is complete.

Lemma . Let{uk}be determined by Lemma..Then there exists a subsequence{ukj=

(ukj,ukj)}of{uk}k∈Nconvergent to a certain function u∞= (u∞,u∞)and when f= and

f= ,u∞is a nontrivial solution of system(.)such that u∞(t)→andu∞(t– )→

as t→ ±∞.

Proof Note that

ukH,k≤M, ukH,k≤M for everyk∈N.

Then, similar to the argument in [] or [], one can prove that{umk}k∈Nhas a

conver-gent subsequence{umkj}such thatumkj→um∞ andum∞(t)→ andum∞(t– )→

ast→ ±∞, wherem= , . Letu∞= (u∞,u∞). By (.) and the continuity of m,

K(t,·,·),W(t,·,·) andϕ_k, similar to the argument in [] or [], the proof is easy to be

completed.

Proof of Theorem. The proof is easy to be completed by replacing

kT–

t=–kT

fm(t),um(t)

≤

_kT–

t=–kT

fm(t)p

/p

kT–

t=–kT

um(t)p

/p

≤ umHm,k

t∈Z fm(t)

p/p

with

kT–

t=–kT

fm(t),um(t)

≤ uml∞_kT kT–

t=–kT

fm(t)≤C∗umHm,k

t∈Z

fm(t), m= , ,

in the proofs of Lemma . and Lemma ..

Proofs of Theorem.and Theorem. We only note that in the proof of Lemma ., when γ =p, we do not need to consider the case thatr∈(, ] alone and it is sufficient thatr> . Other proofs are the same as those of Theorem . and Theorem ., respectively.

4 Examples

We first give two examples aboutwhich satisfy assumption (A). (I) An example withN= . Definem:R→RN,m= , , by

(x) =

α|x|p, x≥,

α|x|p, x< ,

(y) =

β|y|p, y≥,

β|y|p, y< ,

whereα,α∈[d,d],β,β∈[d,d]. Then it is easy to verify thatm,m= , , satisfies

(A).

(II) As described in [], the following classical case withp-Laplacian also satisfies the assumption (A). Definem:RN→RN,m= , , by

(x) =α|x|p, (y) =β|y|p,

whereα∈[d,d],β∈[d,d].

Next, we present some examples ofKandWwhich satisfy those assumptions in Theo-rem .. There are lots of examples ofK. For example, let

K(t,x,x) =a(t)|x|γ +a(t)|x|γ, (t,x,x)∈Z[,T– ]×RN×RN,

whereγ ∈(,p),ai,i= ,  :Z→R+areT-periodic. Letai=mint∈Z[,T–]ai(t). Then it is

easy to see thatKsatisfies (H) and (H). ForW, we assume that

W(t,x,x) =b(t)

|x|p+|x|p

ln|x|p+|x|p+ 

, (t,x,x)∈Z[,T– ]×RN×RN,

whereb:Z→R+_{isT-periodic. Letb}+_=max

t∈Z[,T–]{b(t)}. Then W(t,x,x)≤b+|x|p+|x|pln|rC∗|p+|rC∗|p+ 

for allt∈Z[,T– ],|x| ≤rC∗,|x| ≤rC∗.

Letb=b=b+ln(|rC∗|p+|rC∗|p+ ). Ifris sufficiently small, then (H)(i) holds. It is easy to see that

lim

|x|+|x|→+∞

W(t,x,x) |x|p+|x|p

So (H) holds. Letν∈(,γ– ). Note that

pξ|x|p+|x|p

≥ln|x|p+|x|p+ 

, pη|x|ν+η|x|ν

≥ln|x|p+|x|p+ 

for all (x,x)∈RN_×RN_{, when we choose sufficiently largeξ,η}

andη. Hence

pξ|x|p+|x|p+p|x|p+|x|pη|x|ν+η|x|ν

≥ln|x|p+|x|p+ +ln|x|p+|x|p+ |x|p+|x|p

⇐⇒ pξ+η|x|ν+η|x|ν

|x|p+|x|p

≥ln|x|p+|x|p+ |x|p+|x|p+  ⇐⇒ pξ+η|x|ν+η|x|ν

|x|p+|x|p

≥|x|p+|x|pln|x|p+|x|p+ |x|p+|x|p+  ⇐⇒ p(|x|p+|x|p)

|x|p_+|x|p_{+ } ≥

(|x|p+|x|p)ln(|x|p+|x|p+ ) ξ+η|x|ν+η|x|ν

⇐⇒ ∇xW(t,x,x),x

+∇xW(t,x,x),x

–pW(t,x,x)

≥ W(t,x,x) ξ+η|x|ν+η|x|ν

for all (t,x,x)∈Z[,T– ]×RN_×RN_{, which implies (H) holds.}

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The authors read and approved the final manuscript.

Acknowledgements

This project is supported by the National Natural Science Foundation of China (No. 11301235), Tianyuan Fund for Mathematics of the National Natural Science Foundation of China (No. 11226135) and the Fund for Fostering Talents in Kunming University of Science and Technology (No. KKSY201207032).

Received: 4 December 2014 Accepted: 10 April 2015

References

1. Mawhin, J: Periodic solutions of second order nonlinear difference systems withφ-Laplacian: a variational approach. Nonlinear Anal.75, 4672-4687 (2012)

2. Mawhin, J: Periodic solutions of second order Lagrangian difference systems with bounded or singularφ-Laplacian and periodic potential. Discrete Contin. Dyn. Syst., Ser. S6, 1065-1076 (2013)

3. Bonanno, G, Candito, P: Nonlinear difference equations investigated via critical points methods. Nonlinear Anal.70, 3180-3186 (2009)

4. Candito, P, Giovannelli, N: Multiple solutions for a discrete boundary value problem. Comput. Math. Appl.56, 959-964 (2008)

5. Guo, ZM, Yu, JS: The existence of periodic and subharmonic solutions to subquadratic second-order difference equations. J. Lond. Math. Soc.68, 419-430 (2003)

6. Guo, ZM, Yu, JS: Existence of periodic and subharmonic solutions for second-order superlinear difference equations. Sci. China Ser. A46, 506-515 (2003)

7. Zhou, Z, Yu, JS, Guo, ZM: Periodic solutions of higher-dimensional discrete systems. Proc. R. Soc. Edinb., Sect. A134, 1013-1022 (2004)

8. Xue, YF, Tang, CL: Existence of a periodic solution for subquadratic second-order discrete Hamiltonian system. Nonlinear Anal.67, 2072-2080 (2007)

9. Lin, XY, Tang, XH: Existence of infinitely many homoclinic orbits in discrete Hamiltonian systems. J. Math. Anal. Appl. 373, 59-72 (2011)

10. He, T, Chen, W: Periodic solutions of second order convex systems involving thep-Laplacian. Appl. Math. Comput. 206, 124-132 (2008)

12. Zhang, X, Tang, X: Existence of solutions for a nonlinear discrete system involving thep-Laplacian. Appl. Math.57, 11-30 (2012)

13. Zhang, X: Notes on periodic solutions for a nonlinear discrete system involving thep-Laplacian. Bull. Malays. Math. Soc.37, 499-509 (2014)

14. Zhang, Q, Tang, XH, Zhang, QM: Existence of periodic solutions for a class of discrete Hamiltonian systems. Discrete Dyn. Nat. Soc.2011, Article ID 463480 (2011)

15. Ma, M, Guo, ZM: Homoclinic orbits and subharmonics for nonlinear second order difference equations. Nonlinear Anal.67, 1737-1745 (2007)

16. He, X, Chen, P: Homoclinic solutions for second order discretep-Laplacian systems. Adv. Differ. Equ.2011, 57 (2011) 17. Wang, Y, Zhang, X: Multiple periodic solutions for a class of nonlinear difference systems with classical or bounded

(φ1,φ2)-Laplacian. Adv. Differ. Equ.2014, 218 (2014)

18. Cabada, A, Iannizzotto, A, Tersian, S: Multiple solutions for discrete boundary value problems. J. Math. Anal. Appl.356, 418-428 (2009)

19. Bereanu, C, Jebelean, P, Serban, C: Ground state and mountain pass solutions for discretep(·)-Laplacian. Bound. Value Probl.2012, 104 (2012)

20. Zhang, Q: Existence of homoclinic solutions for a class of difference systems involvingp-Laplacian. Adv. Differ. Equ. 2014, 291 (2014)