Skip to main content

Periodic and subharmonic solutions for a 2nth-order difference equationcontaining both advance and retardation with ϕ-Laplacian

Abstract

In this paper, by using critical point theory, we obtain some new sufficientconditions on the existence and multiplicity of periodic and subharmonic solutions toa 2n th-order nonlinear difference equation containing both advance andretardation with ϕ-Laplacian. Some previous results have beengeneralized.

1 Introduction

Let N, Z, and R denote the sets of all natural numbers,integers and real numbers, respectively. For a,bZ, define Z(a)={a,a+1,}, Z(a,b)={a,a+1,,b} when ab. tr denotes the transpose of a vector.

Consider the following 2n th-order difference equation containing both advanceand retardation with ϕ-Laplacian of the type:

n ( r k n ϕ ( n u k 1 ) ) = ( 1 ) n f(k, u k + 1 , u k , u k 1 ),kZ,
(1.1)

where nZ, is forward difference operator defined by u k = u k + 1 u k , n u k =( n 1 u k ), ϕC(R,R) satisfied ϕ(0)=0, fC(Z× R 3 ,R), r k >0 for each kZ, { r k } and {f(k, v 1 , v 2 , v 3 )} are T-periodic in k and T is agiven positive integer.

In this paper, given positive integer m, we will study the existence ofmT-periodic solutions for (1.1). As usual, such a mT-periodicsolution will be called a subharmonic solution.

We may think of (1.1) as a discrete analog of the following 2n th-orderfunctional differential equation:

d n d t n [ r ( t ) ϕ ( d n u ( t ) d t n ) ] = ( 1 ) n f ( t , u ( t + 1 ) , u ( t ) , u ( t 1 ) ) ,tR.
(1.2)

Equations similar in structure to (1.2) have been studied by many authors. For example,for the case where ϕ(x)=x, n=1, Smets and Willem [1] have considered solitary waves with prescribed speed on infinite lattices ofparticles with nearest neighbor interaction for the following forward and backwarddifferential difference equation:

c 2 u (t)= V ( u ( t + 1 ) u ( t ) ) V ( u ( t ) u ( t 1 ) ) ,tR.

For the case where ϕ(x)= | x | p 2 x, p>1, n=1, Wang [2] has studied the existence of positive solutions of the equation

( | u | p 2 u ) +a(t)f(u)=0,tR.

For the case where ϕ(x)= | x | p 2 x, p>1, n=2, Agarwal, Lu, and O’Regan [3] have studied the existence of positive solutions of the equation

( | u | p 2 u ) =λq(t)f(u),tR.

For the case where ϕ(x)= x 1 + x 2 , n=1, Bonheure and Habets [4] have studied classical and non-classical solutions of a prescribed curvatureequation

( u 1 + u 2 ) =λf(t,u),tR.

In recent years, many authors have studied the existence of periodic solutions ofdifference equations. To mention a few, see [58] for second-order difference equations and [9, 10] for higher-order equations. Since 2003, critical point theory has beenemployed to establish sufficient conditions on the existence of periodic solutions ofdifference equations. By using the critical point theory, Guo and Yu [1113] and Zhou et al.[14] established sufficient conditions on the existence of periodic solutions ofsecond-order nonlinear difference equations. In 2007, by using the Linking Theorem, Caiand Yu [15] obtained some criteria for the existence of periodic solutions of thefollowing equation:

n ( r k n n u k n ) +f(k, u k )=0,kZ,
(1.3)

for the case where f grows superlinearly at both 0 and ∞, wherenZ(3). In 2010, by using the Linking Theorem and the SaddlePoint Theorem, Zhou [16] extended f in (1.3) into sublinear or asymptotically linear andimproved the results of [15] when f is superlinear. In particular, a necessary and sufficientcondition for the existence of the unique periodic solution of (1.3) is also establishedin [16]. In 2013, by using the Linking Theorem, Deng [17] provided some sufficient conditions of the existence and multiplicity ofperiodic solutions and subharmonic solutions of the following equation:

n ( r k n φ p ( n u k 1 ) ) = ( 1 ) n f(k, u k + 1 , u k , u k 1 ),kZ,
(1.4)

where nN, φ p is the p-Laplacian operator given by φ p (u)= | u | p 2 u (1<p<) and where f satisfies some growth conditionsnear both 0 and ∞. In 2012, Mawhin [18] considered T-periodic solutions of systems of difference equationsof the form

ϕ [ u ( k 1 ) ] = u F [ k , u ( k ) ] +h(k),kZ,
(1.5)

under various conditions upon F:Z× R n R and h:Z R n , where nZ, ϕ=Φ, in which Φ: R n [0,) is continuously differentiable and strictly convex,satisfies ϕ(0)=0 and is a homeomorphism of R n onto the ball B a R n or of B a onto R n . By using direct variational method, he gave sufficientconditions for the existence of a minimizing sequence for the case of coercivepotential, or some averaged coercivity conditions of the Ahmad-Lazer-Paul type addingthe nonlinearity satisfies some growth conditions, or the convex potential. Using theSaddle Point Theorem, previously obtained results are extended to the case of anaveraged anticoercivity condition in [18]. However, the results on periodic solutions of higher-order nonlineardifference equations involving ϕ-Laplacian are very scarce in theliterature. Furthermore, since (1.1) contains both advance and retardation, there arevery few works dealing with this subject; see [10, 19]. The main purpose of this paper is to give some sufficient conditions for theexistence and multiplicity of periodic and subharmonic solutions of (1.1). Particularly,our results generalize the results in the literature [17, 20]; see Remark 3.4 and Remark 3.5 for details.

2 Preliminaries

Throughout this paper, we assume that,

(F1) there exists a functional F(k, v 1 , v 2 ) C 1 (Z× R 2 ,R) with F(k, v 1 , v 2 )0 and satisfies

F ( k + T , v 1 , v 2 ) = F ( k , v 1 , v 2 ) , F ( k 1 , v 2 , v 3 ) v 2 + F ( k , v 1 , v 2 ) v 2 = f ( k , v 1 , v 2 , v 3 ) .

In this section, we first establish the variational setting associated with (1.1).

Let S be the set of all two-sided sequences, that is,

S= { u = { u k } | u k R , k Z } .

Then S is a vector space with au+bv={a u k +b v k } for u,vS, a,bR. For any fixed positive integer m and T,we define the subspace E m of S as

E m = { u = { u k } S | u k + m T = u k , k Z } .

Obviously, E m is isomorphic to R m T and hence E m can be equipped with the inner product(,) and norm as

(u,v)= j = 1 m T u j v j ,u,v E m ,

and

u= ( j = 1 m T u j 2 ) 1 2 ,u E m .

On the other hand, we define the norm q on E m as follows:

u= ( j = 1 m T | u j | q ) 1 q ,

for all u E m and q1. By Hölder’ inequality and Jensen’inequality, we have

u 2 u q ( m T ) 2 q 2 q u 2 , 1 q < 2 , ( m T ) 2 q 2 q u 2 u q u 2 , 2 q .

Let

d 1 , q = { 1 , 1 q < 2 , ( m T ) 2 q 2 q , 2 q , d 2 , q = { ( m T ) 2 q 2 q , 1 q < 2 , 1 , 2 q .

Therefore,

d 1 , q u 2 u q d 2 , q u 2 ,u E m .
(2.1)

Clearly, u= u 2 . For all u E m , define the functional J on E m as follows:

J(u)= k = 1 m T r k 1 Φ ( n u k 1 ) k = 1 m T F(k, u k + 1 , u k ),
(2.2)

where

Φ(u)= 0 u ϕ(s)ds

is the primitive function of ϕ(u).

Clearly, J C 1 ( E m ,R) and for any u= { u k } k Z E m , by using u j = u m T + j for jZ(0,mT1), we can compute the partial derivative as

J u k = ( 1 ) n n ( r k n ϕ ( n u k 1 ) ) f(k, u k + 1 , u k , u k 1 ).

Thus, u is a critical point of J on E m if and only if

n ( r k n ϕ ( n u k 1 ) ) = ( 1 ) n f(k, u k + 1 , u k , u k 1 ),kZ(1,mT).

Due to the periodicity of u= { u k } k Z E m and f(k, v 1 , v 2 , v 3 ) in the first variable k, we reduce the existenceof periodic solutions of (1.1) to the existence of critical points of Jon  E m .

Let M be the mT×mT matrix defined by

M=( 2 1 0 0 1 1 2 1 0 0 0 1 2 0 0 0 0 0 2 1 1 0 0 1 2 ).

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

λ j =2 ( 1 cos 2 j m T ) ,j=0,1,2,,mT1.

Thus, λ 0 =0, λ 1 >0, λ 2 >0, …, λ m T 1 >0. Therefore,

λ min = min { λ 1 , λ 2 , , λ m T 1 } = 2 ( 1 cos 2 m T ) , λ max = max { λ 1 , λ 2 , , λ m T 1 } = { 4 , when  m T  is even , 2 ( 1 + cos 1 m T ) , when  m T  is odd .

For convenience, we identify u E m with u= ( u 1 , u 2 , , u m T ) tr . Let

E ¯ m = { u = ( u 1 , u 2 , , u m T ) tr E m | n 1 u j = 0 , j Z ( 1 , m T ) } .

Then

E ¯ m = { u E m | u = { a } , a R } .

Let E ˜ m be the direct orthogonal complement of E m to E ¯ m , i.e., E m = E ¯ m E ˜ m .

For u= ( u 1 , u 2 , , u m T ) tr E m and x= ( n 1 u 1 , n 1 u 2 , , n 1 u m T ) tr , we have

x 2 q = [ k = 1 m T ( n 2 u k + 1 n 2 u k ) 2 ] q 2 [ λ max k = 1 m T ( n 2 u k ) 2 ] q 2 λ max ( n 1 ) q 2 u 2 q .
(2.3)

For u= ( u 1 , u 2 , , u m T ) tr E ˜ m and x= ( n 1 u 1 , n 1 u 2 , , n 1 u m T ) tr , we have

x 2 q = [ k = 1 m T ( n 2 u k + 1 n 2 u k ) 2 ] q 2 [ λ min k = 1 m T ( n 2 u k ) 2 ] q 2 λ min ( n 1 ) q 2 u 2 q .
(2.4)

Let H be a Hilbert space and C 1 (H,R) denote the set of functionals that are Fréchetdifferentiable and their Fréchet derivatives are continuous on H. LetJ C 1 (H,R). A sequence { x j }H is called a Palais-Smale sequence (P. S. sequence forshort) for J if {J( x j )} is bounded and J ( x j )0 as j. We say J satisfies the Palais-Smale condition(P. S. condition for short) if any P. S. sequence for J possesses a convergentsubsequence.

Let B r be the open ball in H with radius r andcenter 0, and let B r denote its boundary. Lemma 2.1 is taken from [21].

Lemma 2.1 (Linking Theorem)

Let H be a real Hilbert space andH= H 1 H 2 , where H 1 is a finite-dimensional subspaceof H. Assume thatJ C 1 (H,R)satisfies the P. S. condition and thefollowing conditions.

(J1) There exist constantsa>0andρ>0such thatJ | B ρ H 2 a;

(J2) There exist ane B 1 H 2 and a constant R 0 >ρsuch thatJ | Q 0whereQ=( B ¯ R 0 H 1 ){re|0<r< R 0 }.

Then J possesses a critical valueca. Moreover, c can be characterized as

c= inf h Γ sup x Q J ( h ( x ) ) ,

whereΓ={hC( Q ¯ ,H):h | Q =id | Q }andid | Q is the identity operator on ∂Q.

3 Main results

Let

r ̲ = min k Z ( 1 , T ) { r k }, r ¯ = max k Z ( 1 , T ) { r k }.

Here we give some conditions.

( Φ 1 ) There exist constants ϵ 1 >0, a 1 >0 and μ1 such that

Φ(u) a 1 | u | μ for |u| ϵ 1 .

( Φ 2 ) There exist constants δ 1 >0, b 1 >0, c 1 0 and ν1 such that

Φ(u) b 1 | u | ν + c 1 for |u| δ 1 .

(F2) There exist constants ϵ 2 >0, a 2 >0 and θ1 such that

F(k, v 1 , v 2 ) a 2 ( v 1 2 + v 2 2 ) θ for  v 1 2 + v 2 2 ϵ 2 .

(F3) There exist constants δ 2 >0, b 2 >0, c 2 >0 and ϑ1 such that

F(k, v 1 , v 2 ) b 2 ( v 1 2 + v 2 2 ) ϑ c 2 for  v 1 2 + v 2 2 δ 2 .

( H 1 , s ) μ=θ=s and a 1 a 2 ( d 1 , s d 2 , s ) s r ̲ λ min n s 2 2 s 2 >1.

( H 1 , p ) ν=ϑ=p and b 1 b 2 ( d 2 , p d 1 , p ) p r ¯ λ max n p 2 2 p 2 <1.

( H 2 , s ) μ<θ.

( H 2 , p ) ν<ϑ.

Remark 3.1 By ( Φ 2 ) it is easy to see that there exists a constant c 1 >0 such that

Φ(u) b 1 | u | ν + c 1 ,uR.
(3.1)

Remark 3.2 By (F3) it is easy to see that there exists a constant c 2 >0 such that

F(k, v 1 , v 2 ) b 2 ( v 1 2 + v 2 2 ) ϑ c 2 ,(k, v 1 , v 2 )Z× R 2 .
(3.2)

Remark 3.3 The p-Laplacian operator given by φ p (u)= | u | p 2 u (1<p<), the curvature-type operator given by ϕ q (u)= | u | q 2 u 1 + | u | q (2q<) and the identity operator given by ϕ I (u)=u satisfy ( Φ 1 ) and ( Φ 2 ).

Our main results are as follows.

Theorem 3.1 Assume that ( Φ 1 ), ( Φ 2 ), (F1), (F2), (F3)are satisfied. If one of the following four cases is satisfied:

  1. (1)

    Assume that ( H 2 , s ) and ( H 2 , p ) are satisfied.

  2. (2)

    Assume that ( H 1 , s ) and ( H 1 , p ) are satisfied.

  3. (3)

    Assume that ( H 1 , s ) and ( H 2 , p ) are satisfied.

  4. (4)

    Assume that ( H 2 , s ) and ( H 1 , p ) are satisfied.

Then for any given positive integer m, (1.1) has at leastthree mT-periodic solutions.

Remark 3.4 If ϕ(u)= | u | p 2 u (1<p<), r k =1 and n=1, Theorem 3.1 reduces to Theorem 3.1 in [20].

Remark 3.5 If ϕ(u)= | u | p 2 u (1<p<), Theorem 3.1 reduces to Theorem 1.1 in [17].

Corollary 3.1 Assume that (F1) and the followingconditions are satisfied.

( Φ 1 ) There exists constantμ1such that lim | u | 0 Φ ( u ) | u | μ =d>0.

( Φ 2 ) There exist constants δ 1 >0andν1such that

0<ϕ(u)uνΦ(u),|u| δ 1 .

( F 2 ) There exists constantθμsuch that

lim ( v 1 , v 2 ) ( 0 , 0 ) F ( k , v 1 , v 2 ) ( v 1 2 + v 2 2 ) θ 2 =0,(k, v 1 , v 2 )Z× R 2 .

( F 3 ) There exist constants δ 2 >0andϑ>νsuch that

0<ϑF(k, v 1 , v 2 ) F ( k , v 1 , v 2 ) v 1 v 1 + F ( k , v 1 , v 2 ) v 2 v 2 , v 1 2 + v 2 2 δ 2 .

Then for any given positive integer m, (1.1) has at leastthree mT-periodic solutions.

4 Proof of the main results

Lemma 4.1 Assume that ( Φ 2 ), (F1), (F3), and( H 2 , p ) are satisfied. Then thefunctional J is bounded from above in E m .

Proof By (2.1), (2.3), (3.1), and (3.2), for any u E m , we have

J ( u ) = k = 1 m T r k 1 Φ ( n u k 1 ) k = 1 m T F ( k , u k + 1 , u k ) k = 1 m T r k 1 b 1 | n u k 1 | ν + m T c 1 k = 1 m T b 2 ( u k + 1 2 + u k 2 ) ϑ + m T c 2 b 1 r ¯ d 2 , ν ν ( k = 1 m T | n u k 1 | 2 ) ν 2 b 2 [ ( k = 1 m T ( u k + 1 2 + u k 2 ) ϑ ) 1 ϑ ] ϑ + m T ( c 1 + c 2 ) b 1 r ¯ d 2 , ν ν ( x tr M x ) ν 2 b 2 d 1 , ϑ ϑ ( 2 u 2 ) ϑ 2 + m T ( c 1 + c 2 ) b 1 r ¯ d 2 , ν ν λ max ν 2 x ν 2 ϑ 2 b 2 d 1 , ϑ ϑ u ϑ + m T ( c 1 + c 2 ) b 1 r ¯ d 2 , ν ν λ max n ν 2 u ν 2 ϑ 2 b 2 d 1 , ϑ ϑ u ϑ + m T ( c 1 + c 2 ) max u ρ 0 { b 1 r ¯ d 2 , ν ν λ max n ν 2 u ν 2 ϑ 2 b 2 d 1 , ϑ ϑ u ϑ } + m T ( c 1 + c 2 ) ,
(4.1)

where x= ( n 1 u 1 , n 1 u 2 , , n 1 u m T ) tr and ρ 0 = ( b 1 r ¯ d 2 , ν ν λ max n ν 2 2 ϑ 2 b 2 d 1 , ϑ ϑ ) 1 ϑ ν .

The proof of Lemma 4.1 is complete. □

Remark 4.1 The case mT=1 is trivial. For the case mT=2, M has a different form, namely,

M=( 2 2 2 2 ).

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

Lemma 4.2 Assume that ( Φ 2 ), (F1), (F3), and( H 2 , p ) are satisfied. Then thefunctional J satisfies the P. S. conditionin E m .

Proof Let { u ( j ) } be a P. S. sequence, then there exists a positive constant M 1 such that

M 1 J ( u ( j ) ) ,jN.

By (4.1), it is easy to see that

M 1 J ( u ( j ) ) b 1 r ¯ d 2 , ν ν λ max n ν 2 u ( j ) ν 2 ϑ 2 b 2 d 1 , ϑ ϑ u ( j ) ϑ +mT ( c 1 + c 2 ) ,jN.

Therefore,

2 ϑ 2 b 2 d 1 , ϑ ϑ u ( j ) ϑ b 1 r ¯ d 2 , ν ν λ max n ν 2 u ( j ) ν M 1 +mT ( c 1 + c 2 ) ,jN.

Since ϑ>ν, it is not difficult to see that { u ( j ) } is a bounded sequence in E m . As a consequence, { u ( j ) } possesses a convergence subsequence in E m . Thus the P. S. condition is verified. □

Lemma 4.3 Assume that ( Φ 2 ), (F1), (F3), and( H 1 , p ) are satisfied. Then thefunctional J is bounded from above in E m .

Proof Similar to the proof of Lemma 4.1, we have

J(u) b 1 r ¯ d 2 , p p λ max n p 2 u p 2 p 2 b 2 d 1 , p p u p +mT ( c 1 + c 2 ) ,
(4.2)

where x= ( n 1 u 1 , n 1 u 2 , , n 1 u m T ) tr . Since b 1 b 2 ( d 2 , p d 1 , p ) p r ¯ λ max n p 2 2 p 2 <1, we have

J(u)mT ( c 1 + c 2 ) .

The proof of Lemma 4.3 is complete. □

Lemma 4.4 Assume that ( Φ 2 ), (F1), (F3), and( H 1 , p ) are satisfied. Then thefunctional J satisfies the P. S. conditionin E m .

Proof Let { u ( j ) } be a P. S. sequence, then there exists a positive constant M 2 such that

M 2 J ( u ( j ) ) ,jN.

By (4.2), it is easy to see that

M 2 J ( u ( j ) ) b 1 r ¯ d 2 , p p λ max n p 2 u ( j ) p 2 p 2 b 2 d 1 , p p u ( j ) p +mT ( c 1 + c 2 ) ,jN.

Therefore,

2 p 2 b 2 d 1 , p p u ( j ) p b 1 r ¯ d 2 , p p λ max n p 2 u ( j ) p M 2 +mT ( c 1 + c 2 ) ,jN.

Since b 1 b 2 ( d 2 , p d 1 , p ) p r ¯ λ max n p 2 2 p 2 <1, we know that { u ( j ) } is a bounded sequence in E m . As a consequence, { u ( j ) } possesses a convergence subsequence in E m . Thus the P. S. condition is verified. □

Proof of Theorem 3.1 Assumptions (F1) and (F2) imply thatF(k,0)=0 and f(k,0)=0 for kZ. Adding ϕ(0)=0, then u=0 is a trivial mT-periodic solution of (1.1).

By Lemma 4.1 or Lemma 4.3, J is bounded from above on E m . We define α 0 = sup u E m J(u). Equation (4.1) implies lim u J(u)=. This means that −J is coercive. By thecontinuity of J, there exists u ¯ E m such that J( u ¯ )= α 0 . Clearly, u ¯ is a critical point of J.

Case 1. Assume that ( H 2 , s ) and ( H 2 , p ) are satisfied. We claim that α 0 >0.

Let

ρ=min { ϵ 1 λ max n 2 , ϵ 2 , 1 2 ( a 1 r ̲ d 1 , μ μ λ min n μ 2 2 θ 2 a 2 d 2 , θ θ ) 1 θ μ } .

By ( Φ 1 ), (F2), and ( H 2 , s ), for any u E ˜ m , uρ, we have

J ( u ) = k = 1 m T r k 1 Φ ( n u k 1 ) k = 1 m T F ( k , u k + 1 , u k ) k = 1 m T r k 1 a 1 | n u k 1 | μ k = 1 m T a 2 ( u k + 1 2 + u k 2 ) θ a 1 r ̲ d 1 , μ μ ( k = 1 m T | n u k 1 | 2 ) μ 2 a 2 [ ( k = 1 m T ( u k + 1 2 + u k 2 ) θ ) 1 θ ] θ a 1 r ̲ d 1 , μ μ ( x tr M x ) μ 2 a 2 d 2 , θ θ ( 2 u 2 ) θ 2 a 1 r ̲ d 1 , μ μ λ min μ 2 x μ 2 θ 2 a 2 d 2 , θ θ u θ a 1 r ̲ d 1 , μ μ λ min n μ 2 u μ 2 θ 2 a 2 d 2 , θ θ u θ ,
(4.3)

where x= ( n 1 u 1 , n 1 u 2 , , n 1 u m T ) tr .

Take σ= a 1 r ̲ d 1 , μ μ λ min n μ 2 ρ μ 2 θ 2 a 2 d 2 , θ θ ρ θ . Then σ 1 2 a 1 r ̲ d 1 , μ μ λ min n μ 2 ρ μ >0 and

J(u)σ,u E ˜ m B ρ .
(4.4)

Therefore, α 0 = sup u E m J(u)σ>0. From (4.4), we have also proved that J satisfiesthe condition (J1) of the Linking Theorem.

For all u E ¯ m , we have

J(u)= k = 1 m T F(k, u k + 1 , u k )0.

Thus, the critical point u ¯ of J corresponding to the critical value α 0 is a nontrivial mT-periodic solution of (1.1). Inthe following, we will verify the condition (J2).

Take e B 1 E ˜ m , for any z E ¯ m and r>0, let u=re+z. Then

J ( u ) = k = 1 m T r k 1 Φ ( n u k 1 ) k = 1 m T F ( k , u k + 1 , u k ) = k = 1 m T r k 1 Φ ( r n e k 1 ) k = 1 m T F ( k , u k + 1 , u k ) k = 1 m T r k 1 b 1 | r n e k 1 | ν + m T c 1 k = 1 m T b 2 ( u k + 1 2 + u k 2 ) ϑ + m T c 2 b 1 r ¯ r ν d 2 , ν ν ( k = 1 m T | n e k 1 | 2 ) ν 2 b 2 [ ( k = 1 m T ( u k + 1 2 + u k 2 ) ϑ ) 1 ϑ ] ϑ + m T ( c 1 + c 2 ) b 1 r ¯ r ν d 2 , ν ν ( y tr M y ) ν 2 b 2 d 1 , ϑ ϑ ( 2 u 2 ) ϑ 2 + m T ( c 1 + c 2 ) b 1 r ¯ r ν d 2 , ν ν λ max ν 2 y ν 2 ϑ 2 b 2 d 1 , ϑ ϑ ( r ϑ + z ϑ ) + m T ( c 1 + c 2 ) b 1 r ¯ d 2 , ν ν λ max n ν 2 r ν 2 ϑ 2 b 2 d 1 , ϑ ϑ r ϑ 2 ϑ 2 b 2 d 1 , ϑ ϑ z ϑ + m T ( c 1 + c 2 ) ,
(4.5)

where y= ( n 1 e 1 , n 1 e 2 , , n 1 e m T ) tr .

Let g 1 (t)= b 1 r ¯ d 2 , ν ν λ max n ν 2 t ν 2 ϑ 2 b 2 d 1 , ϑ ϑ t ϑ , g 2 (t)= 2 ϑ 2 b 2 d 1 , ϑ ϑ t ϑ +mT( c 1 + c 2 ). We have lim t + g 1 (t)= and lim t + g 2 (t)=, and g 1 (t), g 2 (t) are bounded from above, and J(z)0 for z E ¯ m . Thus there exists a constant R 0 >ρ such that J | Q 0 where Q=( B ¯ R 0 E ˜ m ){re|0<r< R 0 }.

Case 2. Assume that ( H 1 , s ) and ( H 1 , p ) are satisfied. We claim that α 0 >0.

Let ρ=min{ ϵ 1 λ max n 2 , ϵ 2 }. By ( H 1 , s ), ( Φ 1 ), and (F2), for any u E ˜ m , uρ, we have

J ( u ) = k = 1 m T r k 1 Φ ( n u k 1 ) k = 1 m T F ( k , u k + 1 , u k ) k = 1 m T r k 1 a 1 | n u k 1 | s k = 1 m T a 2 ( u k + 1 2 + u k 2 ) s a 1 r ̲ d 1 , s s ( k = 1 m T | n u k 1 | 2 ) s 2 a 2 [ ( k = 1 m T ( u k + 1 2 + u k 2 ) s ) 1 s ] s a 1 r ̲ d 1 , s s ( x tr M x ) s 2 a 2 d 2 , s s ( 2 u 2 ) s 2 a 1 r ̲ d 1 , s s λ min s 2 x s 2 s 2 a 2 d 2 , s s u s a 1 r ̲ d 1 , s s λ min n s 2 u s 2 s 2 a 2 d 2 , s s u s ,
(4.6)

where x= ( n 1 u 1 , n 1 u 2 , , n 1 u m T ) tr .

Take σ= a 1 r ̲ d 1 , s s λ min n s 2 ρ s 2 s 2 a 2 d 2 , s s ρ s . Then σ0 and

J(u)σ,u E ˜ m B ρ .
(4.7)

Therefore, α 0 = sup u E m J(u)σ>0. From (4.7), we have also proved that J satisfiesthe condition (J1) of the Linking Theorem.

For all u E ¯ m , we have

J(u)= k = 1 m T F(k, u k + 1 , u k )0.

Thus, the critical point u ¯ of J corresponding to the critical value α 0 is a nontrivial mT-periodic solution of (1.1). Inthe following, we will verify the condition (J2).

Take e B 1 E ˜ m , for any z E ¯ m and r>0, let u=re+z. Then

J ( u ) = k = 1 m T r k 1 Φ ( n u k 1 ) k = 1 m T F ( k , u k + 1 , u k ) = k = 1 m T r k 1 Φ ( r n e k 1 ) k = 1 m T F ( k , u k + 1 , u k ) k = 1 m T r k 1 b 1 | r n e k 1 | p + m T c 1 k = 1 m T b 2 ( u k + 1 2 + u k 2 ) p + m T c 2 b 1 r ¯ r p d 2 , p p ( k = 1 m T | n e k 1 | 2 ) p 2 b 2 [ ( k = 1 m T ( u k + 1 2 + u k 2 ) p ) 1 p ] p + m T ( c 1 + c 2 ) b 1 r ¯ r p d 2 , p p ( y tr M y ) p 2 b 2 d 1 , p p ( 2 u 2 ) p 2 + m T ( c 1 + c 2 ) b 1 r ¯ r p d 2 , p p λ max p 2 y p 2 p 2 b 2 d 1 , p p ( r p + z p ) + m T ( c 1 + c 2 ) ( b 1 r ¯ d 2 , p p λ max n p 2 2 p 2 b 2 d 1 , p p ) r p 2 p 2 b 2 d 1 , p p z p + m T ( c 1 + c 2 ) ,
(4.8)

where y= ( n 1 e 1 , n 1 e 2 , , n 1 e m T ) tr .

Since b 1 b 2 ( d 2 , p d 1 , p ) p r ¯ λ max n p 2 2 p 2 <1, and J(z)0 for z E ¯ m , thus there exists a constant R 0 >ρ such that J | Q 0 where Q=( B ¯ R 0 E ˜ m ){re|0<r< R 0 }.

Case 3. Assume that ( H 1 , s ) and ( H 2 , p ) are satisfied. Similar to Case 1, by (4.6), we see that α 0 >0. Similar to Case 2, by (4.5), we see that there exists aconstant R 0 >ρ such that J | Q 0 where Q=( B ¯ R 0 E ˜ m ){re|0<r< R 0 }. We have also proved that J satisfies thecondition (J1) and (J2) of the Linking Theorem.

Case 4. Assume that ( H 2 , s ) and ( H 1 , p ) are satisfied. Similar to Case 1, by (4.3), we see that α 0 >0. Similar to Case 2, by (4.8), we see that there exists aconstant R 0 >ρ such that J | Q 0 where Q=( B ¯ R 0 E ˜ m ){re|0<r< R 0 }. We have also proved that J satisfies thecondition (J1) and (J2) of the Linking Theorem.

By one of the above four cases and the Linking Theorem, J possesses criticalvalue ασ>0. Moreover, α can be characterized as

α= inf h Γ sup x Q J ( h ( x ) ) ,

where Γ={hC( Q ¯ , E m ):h | Q =id | Q } and id | Q is the identity operator on ∂Q. Let u ˜ E m be a critical point associated to the critical valueα of J, i.e., J( u ˜ )=α. If u ˜ u ¯ , then The proof is complete. Otherwise, u ˜ = u ¯ . Then α 0 =J( u ¯ )=J( u ˜ )=α, i.e., sup u E m J(u)= inf h Γ sup x Q J(h(x)). Choosing h=id, we have sup u Q J(u)= α 0 . Take e B 1 E ˜ m . Similarly, there exists a positive number R 1 >ρ, J | Q 1 0, where Q 1 =( B ¯ R 1 E ˜ m ){re|0<r< R 1 }. Again, by the Linking Theorem, J possesses acritical value α σ>0. Moreover, α can be characterized as

α = inf h Γ 1 sup x Q 1 J ( h ( x ) ) ,

where Γ 1 ={hC( Q ¯ 1 , E m ):h | Q 1 =id | Q 1 } and id | Q 1 is the identity operator on Q 1 . If α α 0 , then the proof is finished. If α = α 0 , then sup u Q 1 J(u)= α 0 . Due to the fact that J | Q 0 and J | Q 1 0, J attains its maximum at some points in theinterior of sets Q and Q 1 . However, Q Q 1 E ¯ m and J | E ¯ m 0. Therefore, there must be a point u E m , u u ˜ and J( u )= α = α 0 .

The proof of Theorem 3.1 is complete. □

Proof of Corollary 3.1 By ( Φ 1 ), there exists constant ϵ 1 >0 such that Φ(u) d 2 | u | μ , for |u| ϵ 1 . Hence ( Φ 1 ) implies ( Φ 1 ). By ( Φ 2 ), there exist constants δ 1 >0, b 1 >0 and c 1 0 such that Φ(u) b 1 | u | ν + c 1 , for |u| δ 1 . So ( Φ 2 ) implies ( Φ 2 ).

By ( F 2 ), there exist constants ϵ 2 >0 and a 2 >0 such that

F(k, v 1 , v 2 ) a 2 ( v 1 2 + v 2 2 ) θ , v 1 2 + v 2 2 ϵ 2 .

So ( F 2 ) implies (F2).

By ( F 3 ), there exist constants δ 2 >0, b 2 >0 and c 2 >0 such that

F(k, v 1 , v 2 ) b 2 ( v 1 2 + v 2 2 ) ϑ c 2 , v 1 2 + v 2 2 δ 2 .

So ( F 3 ) implies (F3). Since ϑ>ν, ( F 3 ) implies ( H 2 , p ).

If θ>μ, then ( F 2 ) implies ( H 2 , s ). If θ=μ=s, then by ( F 2 ), there exist constants ϵ 2 >0 and a 2 = a 1 ( d 1 , s d 2 , s ) s r ̲ λ min n s 2 2 s + 2 2 such that

F(k, v 1 , v 2 ) a 2 ( v 1 2 + v 2 2 ) s , v 1 2 + v 2 2 ϵ 2 .

we have a 1 a 2 ( d 1 , s d 2 , s ) s r ̲ λ min n s 2 2 s 2 =2>1. So, if θ=μ=s, then ( F 2 ) implies ( H 1 , s ).

So, by Theorem 3.1, Corollary 3.1 holds. □

5 Example

As an application of Theorem 3.1, we give an example to illustrate our result.

Example 5.1 For a given positive integer T, consider the following2n th-order difference equation:

n ( n u k 1 1 + | n u k 1 | 2 ) = ( 1 ) n f(k, u k + 1 , u k , u k 1 ),nZ(1),kZ,
(5.1)

where

f(k, v 1 , v 2 , v 3 )= v 2 ( ( 2 + cos 2 π k T ) ( v 1 2 + v 2 2 ) + ( 2 + cos 2 π ( k 1 ) T ) ( v 2 2 + v 3 2 ) ) .

Let

F(k, v 1 , v 2 )= 2 + cos 2 π k T 4 ( v 1 2 + v 2 2 ) 2 .

It is easy to verify that all the assumptions of Theorem 3.1 are satisfied. So, for anygiven positive integer m, (5.1) has at least three mT-periodicsolutions.

References

  1. 1.

    Smets D, Willem M: Solitary waves with prescribed speed on infinite lattices. J. Funct. Anal. 1997, 149: 266–275. 10.1006/jfan.1996.3121

    MathSciNet  Article  Google Scholar 

  2. 2.

    Wang J: The existence of positive solutions for the one-dimensional p -Laplacian. Proc. Am. Math. Soc. 1997, 125: 2275–2283. 10.1090/S0002-9939-97-04148-8

    Article  Google Scholar 

  3. 3.

    Agarwal RP, Lu H, O’Regan D:Positive solutions for the boundary value problem ( | x ( t ) | p 2 x ( t ) ) =λq(t)f(x(t)). Mem. Differ. Equ. Math. Phys. 2003, 28: 33–44.

    MathSciNet  Google Scholar 

  4. 4.

    Bonheure D, Habets P, Obersnel F, Omari P: Classical and non-classical solutions of a prescribed curvature equation. J. Differ. Equ. 2007, 243: 208–237. 10.1016/j.jde.2007.05.031

    MathSciNet  Article  Google Scholar 

  5. 5.

    Xue YF, Tang CL: Existence of a periodic solution for subquadratic second-order discreteHamiltonian system. Nonlinear Anal. 2007, 67: 2072–2080. 10.1016/j.na.2006.08.038

    MathSciNet  Article  Google Scholar 

  6. 6.

    Yu JS, Deng XQ, Guo ZM: Periodic solutions of a discrete Hamiltonian system with a change of sign in thepotential. J. Math. Anal. Appl. 2006, 324: 1140–1151. 10.1016/j.jmaa.2006.01.013

    MathSciNet  Article  Google Scholar 

  7. 7.

    He TS, Chen WG: Periodic solutions of second order discrete convex systems involving the p -Laplacian. Appl. Math. Comput. 2008, 206: 124–132. 10.1016/j.amc.2008.08.037

    MathSciNet  Article  Google Scholar 

  8. 8.

    Bin HH, Yu JS, Guo ZM: Nontrivial periodic solutions for asymptotically linear resonant differenceproblem. J. Math. Anal. Appl. 2006, 322: 477–488. 10.1016/j.jmaa.2006.01.028

    MathSciNet  Article  Google Scholar 

  9. 9.

    Agarwal RP: Difference Equations and Inequalities: Theory, Methods and Applications. Dekker, New York; 1992.

    Google Scholar 

  10. 10.

    Liu X, Zhang YB, Shi HP: Existence theorems of periodic solutions for fourth-order nonlinear functionaldifference equations. J. Appl. Math. Comput. 2013, 42: 51–67. 10.1007/s12190-012-0640-2

    MathSciNet  Article  Google Scholar 

  11. 11.

    Guo ZM, Yu JS: Existence of periodic and subharmonic solutions for second-order superlineardifference equations. Sci. China Math. 2003, 46: 506–515. 10.1007/BF02884022

    MathSciNet  Article  Google Scholar 

  12. 12.

    Guo ZM, Yu JS: The existence of periodic and subharmonic solutions of subquadratic second orderdifference equations. J. Lond. Math. Soc. 2003, 68: 419–430. 10.1112/S0024610703004563

    MathSciNet  Article  Google Scholar 

  13. 13.

    Guo ZM, Yu JS: Applications of critical point theory to difference equations. Fields Inst. Commun. 2004, 42: 187–200.

    MathSciNet  Google Scholar 

  14. 14.

    Zhou Z, Yu JS, Guo ZM: Periodic solutions of higher-dimensional discrete systems. Proc. R. Soc. Edinb., Sect. A 2004, 134: 1013–1022. 10.1017/S0308210500003607

    MathSciNet  Article  Google Scholar 

  15. 15.

    Cai XC, Yu JS: Existence of periodic solutions for a 2 n th-order nonlinear differenceequation. J. Math. Anal. Appl. 2007, 329: 870–878. 10.1016/j.jmaa.2006.07.022

    MathSciNet  Article  Google Scholar 

  16. 16.

    Zhou Z, Yu JS, Chen YM: Periodic solutions of a 2 n th-order nonlinear difference equation. Sci. China Math. 2010, 53: 41–50. 10.1007/s11425-009-0167-7

    MathSciNet  Article  Google Scholar 

  17. 17.

    Deng XQ, Liu X, Zhang YB, Shi HP: Periodic and subharmonic solutions for a 2 n th-order difference equationinvolving p -Laplacian. Indag. Math. 2013, 24: 613–625. 10.1016/j.indag.2013.04.003

    MathSciNet  Article  Google Scholar 

  18. 18.

    Mawhin J: Periodic solutions of second order nonlinear difference systems with ϕ -Laplacian: a variational approach. Nonlinear Anal. 2012, 75: 4672–4687. 10.1016/j.na.2011.11.018

    MathSciNet  Article  Google Scholar 

  19. 19.

    Liu X, Zhang YB, Zheng B, Shi HP: Periodic and subharmonic solutions for second order p -Laplaciandifference equations. Proc. Indian Acad. Sci. Math. Sci. 2011, 121: 457–468. 10.1007/s12044-011-0046-3

    MathSciNet  Article  Google Scholar 

  20. 20.

    Chen P, Fang H: Existence of periodic and subharmonic solutions for second-order p -Laplacian difference equations. Adv. Differ. Equ. 2007., 2007: Article ID 42530

    Google Scholar 

  21. 21.

    Rabinowitz PH: Minimax Methods in Critical Point Theory with Applications to DifferentialEquations. Am. Math. Soc., Providence; 1986.

    Google Scholar 

Download references

Acknowledgements

This work is supported by the Program for Changjiang Scholars and Innovative ResearchTeam in University (no. IRT1226), the National Natural Science Foundation of China(no. 11171078), the Specialized Fund for the Doctoral Program of Higher Education ofChina (no. 20114410110002), and the Project for High Level Talents of GuangdongHigher Education Institutes.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Zhan Zhou.

Additional information

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors jointly worked on the results, and they read and approved the finalmanuscript.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Reprints and Permissions

About this article

Cite this article

Lin, G., Zhou, Z. Periodic and subharmonic solutions for a 2nth-order difference equationcontaining both advance and retardation with ϕ-Laplacian. Adv Differ Equ 2014, 74 (2014). https://doi.org/10.1186/1687-1847-2014-74

Download citation

Keywords

  • periodic and subharmonic solutions
  • 2n th-order
  • critical point theory
  • difference equations
  • ϕ-Laplacian