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Volume 2007, Article ID 42530, 9 pagesdoi:10.1155/2007/42530 Research Article Existence of Periodic and Subharmonic Solutions for Second-Order p-Laplacian Difference Equations Peng Chen

Volume 2007, Article ID 42530, 9 pages

doi:10.1155/2007/42530

Research Article

Existence of Periodic and Subharmonic Solutions for

Second-Order p-Laplacian Difference Equations

Peng Chen and Hui Fang

Received 26 December 2006; Accepted 13 February 2007

Recommended by Kanishka Perera

We obtain a sufficient condition for the existence of periodic and subharmonic solutions

of second-orderp-Laplacian difference equations using the critical point theory.

Copyright © 2007 P Chen and H Fang This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited

1 Introduction

In this paper, we denote byN,Z,Rthe set of all natural numbers, integers, and real numbers, respectively Fora,b ∈ Z, defineZ(a) = { a,a + 1, },Z(a,b) = { a,a + 1, ,b }

whena ≤ b.

Consider the nonlinear second-order difference equation

Δϕ p

Δx n−1



+fn,x n+1,x n,x n−1



where Δ is the forward difference operator Δx n = x n+1 − x n, Δ2x n = Δ(Δx n), ϕ p(s) is p-Laplacian operator ϕ p(s) = | s | p−2s (1 < p < ∞), and f : Z × R3→ R is a continuous functional in the second, the third, and fourth variables and satisfies f (t + m,u,v,w) =

f (t,u,v,w) for a given positive integer m.

We may think of (1.1) as being a discrete analogue of the second-order functional

differential equation



ϕ p(x )

+ ft,x(t + 1),x(t),x(t −1)

which includes the following equation:

c2y (x) = v 

y(x + 1) − y(x)− v 

y(x) − y(x −1)

Equations similar in structure to (1.3) arise in the study of the existence of solitary waves

of lattice differential equations, see [1] and the references cited therein

Some special cases of (1.1) have been studied by many researchers via variational methods, see [2–7] However, to our best knowledge, no similar results are obtained in the literature for (1.1) Since f in (1.1) depends onx n+1andx n−1, the traditional ways of establishing the functional in [2–7] are inapplicable to our case The main purpose of this paper is to give some sufficient conditions for the existence of periodic and subharmonic solutions of (1.1) using the critical point theory

2 Some basic lemmas

To apply critical point theory to study the existence of periodic solutions of (1.1), we will state some basic notations and lemmas (see [5,8]), which will be used in the proofs of our main results

LetS be the set of sequences, x =( ,x −n, ,x −1,x0,x1, ,x n, ) = { x n }+∞

−∞, that is,

S = { x = { x n }:x n ∈ R,n ∈ Z} For a given positive integerq and m, E qmis defined as a subspace ofS by

E qm =x = { x n } ∈ S | x n+qm = x n,n ∈ Z. (2.1) For anyx, y ∈ S, a,b ∈ R,ax + by is defined by

ax + by =ax n+by n +∞

ThenS is a vector space Clearly, E qmis isomorphic toRqm,E qmcan be equipped with inner product

 x, y  E qm =

qm



j=1

by which the norm can be induced by

x

qm j=1

x2

j

1/2

It is obvious thatE qmwith the inner product in (2.3) is a finite dimensional Hilbert space and linearly homeomorphic toRqm

On the other hand, we define the norm ponE qmas follows:

x p =

qm i=1

x i p 1/p

for allx ∈ E qmandp > 1 Clearly, x x 2 Since pand 2are equivalent, there exist constantsC1,C2, such thatC2≥ C1> 0, and

Define the functionalJ on E qmas follows:

J(x) =

qm



n=1

1

p Δx n p

where

f (t,u,v,w) = F 

2(t −1,v,w) + F 

3(t,u,v),

F 

2(t −1,v,w) = ∂F(t −1,v,w)

∂v , F3(t,u,v) = ∂F(t,u,v)

(2.8)

then

fn,x n+1,x n,x n−1



= F 

3



n,x n+1,x n

+F 

2



n −1,x n,x n−1



Clearly,J ∈ C1(E qm,R) and for anyx = { x n } n∈Z ∈ E qm, by usingx0= x qm,x1= x qm+1, we can compute the partial derivative as

∂J

∂x n = −Δϕ p

Δx n−1



+fn,x n+1,x n,x n−1



, n ∈ Z(1,qm). (2.10)

By the periodicity of{ x n }and f (t,u,v,w) in the first variable t, we reduce the existence

of periodic solutions of (1.1) to the existence of critical points ofJ on E qm That is, the functionalJ is just the variational framework of (1.1)

For convenience, we identifyx ∈ E qmwithx =(x1,x2, ,x qm)T

Let X be a real Hilbert space, I ∈ C1(X,R), which means thatI is a continuously Fr´echet di fferentiable functional defined on X I is said to satisfy Palais-Smale condition

(P-S condition for short) if any sequence{ u n } ⊂ X for which { I(u n)}is bounded and

I (u n)→0, asn → ∞, possesses a convergent subsequence inX.

LetB ρ be the open ball inX with radius ρ and centered at 0 and let ∂B ρ denote its boundary

Lemma 2.1 (linking theorem) [8, Theorem 5.3] Let X be a real Hilbert space, X = X1⊕ X2, where X1 is a finite-dimensional subspace of X Assume that I ∈ C1(X,R) satisfies the P-S condition and

(A1) there exist constants σ > 0 and ρ > 0, such that I | ∂B ρ ∩X2≥ σ;

(A2) there is an e ∈ ∂B1∩ X2 and a constant R1> ρ, such that I | ∂Q ≤ 0, where Q =

(B R1∩ X1)⊕ { re |0< r < R1}

Then, I possesses a critical value c ≥ σ, where

c =inf

h∈Γmax

u∈ Ih(u), Γ=h ∈ CQ,X| h | ∂Q =id

(2.11)

and id denotes the identity operator.

3 Main results

Theorem 3.1 Assume that the following conditions are satisfied:

(H1) f (t,u,v,w) ∈ C(R 4,R) and there exists a positive integer m, such that for every

(t,u,v,w) ∈ R4, f (t + m,u,v,w) = f (t,u,v,w);

(H2) there exists a functional F(t,u,v) ∈ C1(R 3,R) with F(t,u,v) ≥ 0 and it satisfies

F 

2(t −1,v,w) + F 

3(t,u,v) = f (t,u,v,w),

lim

ρ→0

F(t,u,v)

(H3) there exist constants β ≥ p + 1, a1> 0, a2> 0, such that

F(t,u,v) ≥ a1



u2+v2 β

Then, for a given positive integer q, ( 1.1 ) has at least two nontrivial qm-periodic solutions.

First, we prove two lemmas which are useful in the proof ofTheorem 3.1

Lemma 3.2 Assume that f (t,u,v,w) satisfies condition (H3) of Theorem 3.1 , then the func-tional J(x) =qm n=1[1/p | Δx n | p − F(n,x n+1,x n )] is bounded from above on E qm

Proof By (H3), there exista1> 0, a2> 0, β > p, such that for all x ∈ E qm,

J(x) =

qm



n=1

1

p Δx n p

− Fn,x n+1,x n ≤

qm



n=1

2p

p max x n+1 p

, x n p

− Fn,x n+1,x n

≤2p p

qm



n=1

 x n+1 p

+ x n p

− a1

qm



n=1



x2

n+1+x2

n

β

+a2qm

≤2p+1 p

qm



n=1

x n p

− a1

qm



n=1

x n β

+a2qm =2p+1 p x

p

p − a1 x β β+a2qm.

(3.3)

In view of (2.6), there exist constantsC1,C3, such that

So

J(x) ≤ 2p+1

pC1

p x p − a1



C3

Byβ > p and the above inequality, there exists a constant M > 0, such that for every x ∈

Lemma 3.3 Assume that f (t,u,v,w) satisfies condition (H3) of Theorem 3.1 , then the func-tional J satisfies P-S condition.

Proof Let x(k) ∈ E qm, for allk ∈ N, be such that { J(x(k))}is bounded Then there exists

M1> 0, such that

− M1≤ Jx(k)

≤2p+1

pC1p

x(k)p

− a1

C3βx(k)β

that is,

a1

C3βx(k)β

−2p+1

pC1p

x(k)p

Byβ > p, there exists M2> 0 such that for every k ∈ N, x(k) M2

Thus,{ x(k) }is bounded onE qm SinceE qmis finite dimensional, there exists a subse-quence of{ x(k) }, which is convergent inE qmand the P-S condition is verified 

Proof of Theorem 3.1 The proof ofLemma 3.2implies lim x J(x) = −∞, then− J is

coercive Letc0=supx∈E qm J(x) By continuity of J on E qm, there existsx ∈ E qm, such that

J(x) = c0, andx is a critical point of J We claim that c0> 0 In fact, we have

J(x) = 1p

qm



n=1

Δx n p 1/p p

−

qm



n=1

Fn,x n+1,x n

≥ 1p

 1

C2

p qm



n=1

Δx n 2

 1/2 p

−

qm



n=1

Fn,x n+1,x n

= 1p

 1

C2

pqm

n=1

2x2

n − x n x n+1p/2

−

qm



n=1

Fn,x n+1,x n

= 1p

 1

C2

p

x T Axp/2 −

qm



n=1

Fn,x n+1,x n

,

(3.8)

wherex =(x1,x2, ,x qm)T,

A =

⎛

⎜

⎜

⎜

⎜

−1 2 −1 ··· 0 0

. . . .

−1 0 0 ··· −1 2

⎞

⎟

⎟

⎟

⎟

qm×qm

Clearly, 0 is an eigenvalue ofA and ξ =(v,v, ,v) T ∈ E qmis an eigenvector ofA

cor-responding to 0, wherev =0,v ∈ R Letλ1,λ2, ,λ qm−1be the other eigenvalues ofA By

matrix theory, we haveλ j > 0, for all j ∈ Z(1,qm −1)

DenoteZ = {(v,v, ,v) T ∈ E qm | v ∈ R}andY = Z ⊥, such thatE qm = Y ⊕ Z.

Set

λmin= min

j∈Z(1,qm−1)λ j > 0, λmax= max

j∈Z(1,qm−1)λ j > 0. (3.10)

By condition (H2), we have

lim

ρ→0

F(t,u,v)

Chooseε =2−p/2−2(1/p)λminp/2(C1/C2)p, there existsδ > 0, such that

F(t,u,v) ≤2−p/2−21

p λ

p/2

min

C1

C2

p

Therefore, for anyx =(x1,x2, ,x qm)Twith x δ, x ∈ Y, we have

J(x) ≥ 1p

 1

C2

p

x T Axp/2 −

qm



n=1

Fn,x n+1,x n

≥ 1p λ

p/2

min

 1

C2

p

x p −2−p/2−21p λ

p/2

min

C1

C2

p qm

n=1



2p/2max x n+1 p

, x n p

≥ 1p λ

p/2

min



1

C2

p

x p −2−p/2−21

p λ

p/2

min

C1

C2

p qm

n =1



2p/2 x n+1 p

+ x n p

= 1p λ

p/2

min

 1

C2

p

x p −2−p/2−21p λ

p/2

min

C1

C2

p

2p/2+1 x p p

≥ 1p λ

p/2

min

 1

C2

p

x p − 1

2p λ

p/2

min

C1

C2

p 1

C1

p

x p = 1

2p

 1

C2

p

λminp/2 x p

(3.13) Takeσ =1/2p(1/C2)p λ p/2minδ p, then

So

c0= sup

x∈E qm

which implies thatJ satisfies the condition (A1) of the linking theorem

Noting thatAx =0, for allx ∈ Z, we have

J(x) ≤1p

 1

C1

p

x T Axp/2 −

qm



n=1

Fn,x n+1,x n

Therefore, the critical point associated to the critical value c0 of J is a nontrivial

qm-periodic solution of (1.1) Now, we need to verify other conditions of the linking theorem

ByLemma 3.3,J satisfies P-S condition So, it suffices to verify condition (A2) Takee ∈

∂B1∩ Y, for any z ∈ Z, r ∈ R, letx = re + z, then

J(x) = 1p

qm



n=1

Δx n p

−

qm



n=1

Fn,x n+1,x n

≤ 1p

 1

C1

p qm n=1

Δx n 2

p/2

−

qm



n=1

Fn,x n+1,x n

= 1p



1

C1

p

x T Axp/2 −

qm



n=1

Fn,x n+1,x n

= 1p

1

C1

pA(re + z),(re + z)p/2 −

qm



n=1

Fn,re n+1+z n+1,re n+z n

= 1p

1

C1

p

Are,rep/2 −

qm



n=1

Fn,re n+1+z n+1,re n+z n

≤ 1p



1

C1

p

λmaxp/2 r p − a1

qm



n=1



re n+1+z n+1 2

+

re n+z n 2 β

+a2qm

≤ 1p

1

C1

p

λmaxp/2 r p − a1

 1

C3

β qm

n=1

re

n+1+z n+1 2

+re n+z n 2

β/2

+a2qm

= 1p λ

p/2

max



1

C1

p

r p − a1

 1

C3

β

2r2+ 2 z 2 β/2

+a2qm

≤ 1p λ

p/2

max

 1

C1

p

r p − a1

 1

C3

β

2β/2 r β − a1

 1

C3

β

2β/2 z β+a2qm.

(3.17) Let

g1(r) = 1p λ

p/2

max

 1

C1

p

r p − a1

 1

C3

β

2β/2 r β, g2(t) = − a1

 1

C3

β

2β/2 t β+a2qm.

(3.18) Then

lim

r→+∞ g1(r) = −∞, lim

andg1(r) and g2(t) are bounded from above.

Thus, there exists a constantR2> δ, such that J(x) ≤0, for allx ∈ ∂Q, where

Q =B R2∩ Z⊕re |0< r < R2 . (3.20)

By the linking theorem,J possesses a critical value c ≥ σ > 0, where

c =inf

h∈Γmax

u∈ Jh(u),

Γ=h ∈ CQ,E qm

| h | ∂Q =id

The rest of the proof is similar to that of [5, Theorem 1.1], but for the sake of com-pleteness, we give the details

Let"x ∈ E qmbe a critical point associated to the critical valuec of J, that is, J( x)" = c.

Ifx"= x, then the proof is complete; if x"= x, then c0= J(x) = J( x)" = c, that is

sup

x∈E qm

J(x) =inf sup

Chooseh =id, we have supx∈ J(x) = c0 Since the choice ofe ∈ ∂B1∩ Y is arbitrary, we

can take− e ∈ ∂B1∩ Y By a similar argument, there exists a constant R3> δ, for any

x ∈ ∂Q1,J(x) ≤0, where

Q1=B R3∩ Z⊕− re |0< r < R3



Again, by using the linking theorem,J possesses a critical value c  ≥ σ > 0, where

c  =inf max

h∈Γ 1u∈ 1

Jh(u), Γ1=h ∈ CQ1,E qm

| h | ∂Q1=id

Ifc  = c0, then the proof is complete Ifc  = c0, then supx∈ 1J(x) = c0 Due to the fact thatJ | ∂Q ≤0,J | ∂Q1≤0,J attains its maximum at some points in the interior of the set Q

andQ1 Clearly,Q ∩ Q1= ∅, and for anyx ∈ Z, J(x) ≤0 This shows that there must be

a pointx#∈ E qm, such thatx#= " x and J(#x) = c  = c0

The above argument implies that whether or notc = c0, (1.1) possesses at least two nontrivialqm-periodic solutions.

Remark 3.4 when qm =1, (1.1) is reduced to trivial case; whenqm =2,A has the

fol-lowing form:

A =

In this case, it is easy to complete the proof ofTheorem 3.1

Finally, we give an example to illustrateTheorem 3.1

Example 3.5 Assume that

f (t,u,v,w) =2(p + 1)v1 + sin2πt

m



u2+v2 p

+



1 + sin2π(t −1)

m



v2+w2 p

(3.26) Take

F(t,u,v) =



1 + sin2πt m



Then,

F 

2(t −1,v,w) + F 

3(t,u,v)

=2(p + 1)v1 + sin2πt

m



u2+v2 p

+



1 + sin2π(t −1)

m



v2+w2 p (3.28)

It is easy to verify that the assumptions ofTheorem 3.1are satisfied and then (1.1)

Acknowledgment

This research is supported by the National Natural Science Foundation of China (no 10561004)

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Peng Chen: Department of Applied Mathematics, Faculty of Science, Kunming University of Science and Technology, Yunnan 650093, China

Email address:pengchen729@sina.com

Hui Fang: Department of Applied Mathematics, Faculty of Science, Kunming University of

Science and Technology, Yunnan 650093, China

Email address:huifang@public.km.yn.cn