Open Access

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

Advances in Difference Equations20142014:74

https://doi.org/10.1186/1687-1847-2014-74

Received: 25 December 2013

Accepted: 12 February 2014

Published: 21 February 2014

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.

Keywords

periodic and subharmonic solutions2n th-ordercritical point theorydifference equationsϕ-Laplacian

1 Introduction

Let N, Z, and R denote the sets of all natural numbers,integers and real numbers, respectively. For a , b Z , define Z ( a ) = { a , a + 1 , } , Z ( a , b ) = { a , a + 1 , , b } when a b . 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 ) , k Z ,
(1.1)

where n Z , is forward difference operator defined by u k = u k + 1 u k , n u k = ( n 1 u k ) , ϕ C ( R , R ) satisfied ϕ ( 0 ) = 0 , f C ( Z × R 3 , R ) , r k > 0 for each k Z , { 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 ) ) , t R .
(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 ) ) , t R .
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 , t R .
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 ) = λ q ( t ) f ( u ) , t R .
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 ) = λ f ( t , u ) , t R .
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 , k Z ,
(1.3)
for the case where f grows superlinearly at both 0 and ∞, where n Z ( 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 ) , k Z ,
(1.4)
where n N , 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 ) , k Z ,
(1.5)

under various conditions upon F : Z × R n R and h : Z R n , where n Z , 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 a u + b v = { a u k + b v k } for u , v S , a , b R . 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 q 1 . 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 ) d s

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 j Z ( 0 , m T 1 ) , 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 ) , k Z ( 1 , m T ) .

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 m T × m T 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 , , m T 1 .
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. Let J 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 and H = H 1 H 2 , where H 1 is a finite-dimensional subspaceof H. Assume that J C 1 ( H , R ) satisfies the P. S. condition and thefollowing conditions.

(J1) There exist constants a > 0 and ρ > 0 such that J | B ρ H 2 a ;

(J2) There exist an e B 1 H 2 and a constant R 0 > ρ such that J | Q 0 where Q = ( B ¯ R 0 H 1 ) { r e | 0 < r < R 0 } .

Then J possesses a critical value c a . Moreover, c can be characterized as
c = inf h Γ sup x Q J ( h ( x ) ) ,

where Γ = { h C ( Q ¯ , H ) : h | Q = id | Q } and id | 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 μ 1 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 ν 1 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 θ 1 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 ϑ 1 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 , u R .
(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 ( 2 q < ) 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 constant μ 1 such that lim | u | 0 Φ ( u ) | u | μ = d > 0 .

( Φ 2 ) There exist constants δ 1 > 0 and ν 1 such that
0 < ϕ ( u ) u ν Φ ( u ) , | u | δ 1 .
( F 2 ) There exists constant θ μ such 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 > 0 and ϑ > ν such that
0 < ϑ F ( 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 m T = 1 is trivial. For the case m T = 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 ) ) , j N .
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 + m T ( c 1 + c 2 ) , j N .
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 + m T ( c 1 + c 2 ) , j N .

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 + m T ( 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 ) m T ( 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 ) ) , j N .
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 + m T ( c 1 + c 2 ) , j N .
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 + m T ( c 1 + c 2 ) , j N .

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 that F ( k , 0 ) = 0 and f ( k , 0 ) = 0 for k Z . 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 , u ρ , 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 = r e + 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 + m T ( 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 ) { r e | 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 , u ρ , 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 σ 0 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 = r e + 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 ) { r e | 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 ) { r e | 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 ) { r e | 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 α σ > 0 . Moreover, α can be characterized as
α = inf h Γ sup x Q J ( h ( x ) ) ,
where Γ = { h C ( 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 ) { r e | 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 = { h C ( 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 ) , n Z ( 1 ) , k Z ,
(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.

Declarations

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.

Authors’ Affiliations

(1)
School of Mathematics and Information Science, Guangzhou University
(2)
Key Laboratory of Mathematics and Interdisciplinary Sciences of Guangdong HigherEducation Institutes, Guangzhou University

References

  1. Smets D, Willem M: Solitary waves with prescribed speed on infinite lattices. J. Funct. Anal. 1997, 149: 266–275. 10.1006/jfan.1996.3121MathSciNetView ArticleGoogle Scholar
  2. Wang J: The existence of positive solutions for the one-dimensional p -Laplacian. Proc. Am. Math. Soc. 1997, 125: 2275–2283. 10.1090/S0002-9939-97-04148-8View ArticleGoogle Scholar
  3. Agarwal RP, Lu H, O’Regan D:Positive solutions for the boundary value problem ( | x ( t ) | p 2 x ( t ) ) = λ q ( t ) f ( x ( t ) ) . Mem. Differ. Equ. Math. Phys. 2003, 28: 33–44.MathSciNetGoogle Scholar
  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.2007.05.031MathSciNetView ArticleGoogle Scholar
  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.2006.08.038MathSciNetView ArticleGoogle Scholar
  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.2006.01.013MathSciNetView ArticleGoogle Scholar
  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.2008.08.037MathSciNetView ArticleGoogle Scholar
  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.2006.01.028MathSciNetView ArticleGoogle Scholar
  9. Agarwal RP: Difference Equations and Inequalities: Theory, Methods and Applications. Dekker, New York; 1992.Google Scholar
  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/s12190-012-0640-2MathSciNetView ArticleGoogle Scholar
  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/BF02884022MathSciNetView ArticleGoogle Scholar
  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/S0024610703004563MathSciNetView ArticleGoogle Scholar
  13. Guo ZM, Yu JS: Applications of critical point theory to difference equations. Fields Inst. Commun. 2004, 42: 187–200.MathSciNetGoogle Scholar
  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/S0308210500003607MathSciNetView ArticleGoogle Scholar
  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.2006.07.022MathSciNetView ArticleGoogle Scholar
  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/s11425-009-0167-7MathSciNetView ArticleGoogle Scholar
  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.2013.04.003MathSciNetView ArticleGoogle Scholar
  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.2011.11.018MathSciNetView ArticleGoogle Scholar
  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/s12044-011-0046-3MathSciNetView ArticleGoogle Scholar
  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 42530Google Scholar
  21. Rabinowitz PH: Minimax Methods in Critical Point Theory with Applications to DifferentialEquations. Am. Math. Soc., Providence; 1986.Google Scholar

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© Lin and Zhou; licensee Springer. 2014

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