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fference EquationsVolume 2008, Article ID 247071, 11 pages doi:10.1155/2008/247071 Research Article Existence Theorems of Periodic Solutions for Second-Order Nonlinear Difference Equation

fference Equations

Volume 2008, Article ID 247071, 11 pages

doi:10.1155/2008/247071

Research Article

Existence Theorems of Periodic Solutions for

Second-Order Nonlinear Difference Equations

Xiaochun Cai 1 and Jianshe Yu 2

1 College of Statistics, Hunan University, Changsha, Hunan 410079, China

2 College of Mathematics and Information Science, Guangzhou University, Guangzhou 510405, China

Correspondence should be addressed to Xiaochun Cai, cxchn8@hnu.cn

Received 14 August 2007; Accepted 14 November 2007

Recommended by Patricia J Y Wong

The authors consider the second-order nonlinear difference equation of the type Δpn Δx n−1δ 

qnx δ  fn, x n , n ∈Z, using critical point theory, and they obtain some new results on the existence

of periodic solutions.

Copyright q 2008 X Cai and J Yu 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

We denote byN, Z, R the set of all natural numbers, integers, and real numbers, respectively For a, b ∈ Z, define Za  {a, a  1, }, Za, b  {a, a  1, , b} when a ≤ b.

Consider the nonlinear second-order difference equation

Δp n



Δx n−1

δ

 q n x δ n  fn, x n



where the forward difference operator Δ is defined by the equation Δxn  x n1 − x nand

Δ2x n−1  ΔΔx n−1   Δx n − Δx n−1 1.2

In 1.1, the given real sequences {p n }, {q n } satisfy p nT  p n > 0, q nT  q n for any n ∈ Z,

f : Z × R → R is continuous in the second variable, and fn  T, z  fn, z for a given positive integer T and for all n, z ∈ Z× R −1δ  −1, δ > 0, and δ is the ratio of odd positive integers.

By a solution of1.1, we mean a real sequence x  {x n }, n ∈ Z, satisfying 1.1

In1,2 , the qualitative behavior of linear difference equations of type

Δp n Δx n   q n x n 0 1.3

has been investigated In3 , the nonlinear difference equation

Δp n Δx n−1   q n x n  fn, x n 1.4 has been considered However, results on periodic solutions of nonlinear difference equations are very scarce in the literature, see 4, 5 In particular, in 6 , by critical point method, the existence of periodic and subharmonic solutions of equation

Δ2x n−1  fn, x n



has been studied Other interesting contributions can be found in some recent papers 7 11 and in references contained therein It is interesting to study second-order nonlinear difference equations1.1 because they are discrete analogues of differential equation

ptϕu ft, u  0. 1.6

In addition, they do have physical applications in the study of nuclear physics, gas aerody-namics, infiltrating medium theory, and plasma physics as evidenced in12,13

The main purpose here is to develop a new approach to the above problem by using critical point method and to obtain some sufficient conditions for the existence of periodic solutions of1.1

Let X be a real Hilbert space, I ∈ C1X, R, which implies that I is continuously Fr´echet

differentiable functional defined on X I is said to be satisfying Palais-Smale condition P-S condition if any sequence {Iun } is bounded, and Iu n  → 0 as n → ∞ possesses a conver-gent subsequence in X Let B ρ be the open ball in X with radius ρ and centered at 0, and let

∂B ρdenote its boundary

Lemma 1.1 mountain pass lemma, see 14  Let X be a real Hilbert space, and assume that I ∈

C1X, R satisfies the P-S condition and the following conditions:

I1 there exist constants ρ > 0 and a > 0 such that Ix ≥ a for all x ∈ ∂B ρ , where B ρ  {x ∈ X :

x X < ρ};

I2 I0 ≤ 0 and there exists x0∈ B ρ such that Ix0 ≤ 0.

Then c  inf h∈Γsups∈0,1 Ihs is a positive critical value of I, where

Γ h ∈ C

0, 1 , X: h0  0, h1  x0



Lemma 1.2 saddle point theorem, see 14,15  Let X be a real Banach space, X  X1 ⊕ X2, where

X1/  {0} and is finite dimensional Suppose I ∈ C1X, R satisfies the P-S condition and

I3 there exist constants σ, ρ > 0 such that I| ∂B ρ

X1 ≤ σ;

I4 there is e ∈ B pX1and a constant ω > σ such that I| eX2 ≥ ω.

Then I possesses a critical value c ≥ ω and

c  inf

u∈B ρ

X1

where Γ  {h ∈ CB ρX1, X|h| ∂B 

X  id}.

2 Preliminaries

In this section, we are going to establish the corresponding variational framework for1.1 LetΩ be the set of sequences

x 

x n

n∈Z , x −n , , x−1, x0, x1, , x n , 

that is,

Ω x 

x n



: x n ∈ R, n ∈ Z. 2.2

For any x, y ∈ Ω, a, b ∈ R, ax  by is defined by

ax  by :

ax n  by n

∞

ThenΩ is a vector space For given positive integer T, E T is defined as a subspace ofΩ by

E T x 

x n



∈ Ω : x nT  x n , n ∈ Z

Clearly, E T is isomorphic toRT, and can be equipped with inner product

T



i1

x i y i , ∀x, y ∈ E T , 2.5

by which the norm· can be induced by

x :



T



i1

x2i

1/2

It is obvious that E T with the inner product defined by 2.5 is a finite-dimensional Hilbert space and linearly homeomorphic toRT Define the functional J on E T as follows:

Jx  1

δ  1

T



n1

p n Δx n−1δ1− 1

δ  1

T



n1

q n x n δ1T

n1

Fn, x n , ∀x ∈ E T , 2.7

where Ft, z  0z ft, sds Clearly, J ∈ C1E T , R, and for any x  {x n}n∈Z ∈ ET, by using

x0 x T , x1 x T1, we can compute the partial derivative as

∂J

∂x n  −Δp n Δx n−1δ − q n x δ n  fn, x n , n ∈ Z1, T. 2.8

Thus x  {x n}n∈Zis a critical point of J on E T i.e., Jx  0 if and only if

Δp n Δx n−1δ  q n x n δ  fn, x n , n ∈ Z1, T. 2.9

By the periodicity of x n and fn, z in the first variable n, we have reduced the existence

of periodic solutions of 1.1 to that of critical points of J on E T In other words, the

func-tional J is just the variafunc-tional framework of 1.1 For convenience, we identify x ∈ E T with

x  x1, x2, , x TT Denote W  {x1, x2, , x TT ∈ ET : x i ≡ v, v ∈ R, i ∈ Z1, T} and

W⊥  Y such that E T  Y ⊕ W Denote other norm ·r on ET as follows see, e.g., 16 :

x r  T

i1 |x i|r1/r , for all x ∈ E T and r > 1 Clearly, x2  x Due to · r1 and ·r2

being equivalent when r1, r2 > 1, there exist constants c1, c2, c3, and c4such that c2 ≥ c1 > 0,

c4≥ c3> 0, and

c1x ≤ x δ1 ≤ c2x, 2.10

c3x ≤ x β ≤ c4x, 2.11

for all x ∈ E T , δ > 0 and β > 1.

3 Main results

In this section, we will prove our main results by using critical point theorem First, we prove two lemmas which are useful in the proof of theorems

Lemma 3.1 Assume that the following conditions are satisfied:

F1 there exist constants a1> 0, a2> 0, and β > δ  1 such that

z

0

fn, sds ≤ −a1|z| β  a2, ∀z ∈ R; 3.1

F2

Then the functional

Jx  1

δ  1

T



n1

p n Δx n−1δ1− 1

δ  1

T



n1

q n x δ1 n T

n1

Fn, x n 3.3

satisfies P-S condition.

Proof For any sequence {x l } ⊂ E T , with Jx l  being bounded and Jx l → 0 as

l → ∞, there exists a positive constant M such that |Jx l | ≤ M Thus, by F1,

−M ≤ Jx l  1

δ  1

T



n1

p n



x l n − x n−1 l δ1 − q nx l n

δ1

T

n1

F

n, x n l



≤ 1

δ  1

T



n1

p n2δ1

x l n

δ1

x l n−1δ1

− 1

δ  1

T



n1

q n



x l n

δ1

T

n1

F

n, x l n



≤ 2δ1

δ  1

T



n1

p n  p n1x l n δ1

− 1

δ  1

T



n1

q n



x l n δ1

− a1

T



n1



x l n β

 a2T

 1

δ  1

T



n1



2δ1 p n  p n1  − q nx l n

δ1

− a1x lβ

β  a2T.

3.4

Set

A0 max

Then A0> 0 Also, by the above inequality, we have

−M ≤ Jx l ≤ A0

δ  1x lδ1

δ1 − a1x lβ

β  a2T. 3.6

In view of

T



n1

x l

n δ1 ≤ T β−δ−1/β

 T



n1

x l

n β

we have

x lβ

β ≥ T δ1−β/δ1x lβ

Then we get

−M ≤ Jx l ≤ A0

δ  1x lδ1

δ1 − a1T δ1−β/δ1x lβ

δ1  a2T. 3.9

Therefore, for any l ∈ N,

a1T δ1−β/δ1x lβ

δ1− A0

δ  1x lδ1

δ1 ≤ M  a2T. 3.10

Since β > δ 1, the above inequality implies that {x l } is a bounded sequence in E T Thus {x l} possesses convergent subsequences, and the proof is complete

Theorem 3.2 Suppose that F1 and following conditions hold:

F3 for each n ∈ Z,

lim

z→0

fn, z

F4

Then there exist at least two nontrivial T-periodic solutions for 1.1.

Proof We will useLemma 1.1to proveTheorem 3.2 First, byLemma 3.1, J satisfies P-S

condi-tion Next, we will prove that conditionsI1 and I2 hold In fact, by F3, there exists ρ > 0

such that for any|z| < ρ and n ∈ Z1, T,

|Fn, z| ≤ − qmax

2δ  1z δ1 , 3.13

where qmax  maxn∈ Z1,T q n < 0 Thus for any x ∈ E T , x ≤ ρ for all n ∈ Z1, T, we have

Jx ≥ − qmax

δ  1

T



n1

x δ1 n  qmax

2δ  1

T



n1

x δ1 n

 −2δ  1qmax x δ1

δ1

≥ − qmax

2δ  1c δ11 x δ1

3.14

Taking a  −c δ1

1 qmax/2δ  1ρ δ1 , we have

and the assumptionI1 is verified Clearly, J0  0 For any given w ∈ E T withw  1 and a constant α > 0,

Jαw  1

δ  1

T



n1

p n αw n − αw n−1δ1 − q n αw nδ1 T

n1

Fn, αw n

≤ 1

δ  1

T



n1

p n 2α δ1 − q n α δ1 − a1

T



n1

|αw n|β  a2T

≤ 1

δ  1

T



n1

2δ1 p n − q n α δ1 α δ1 − a1T 2−β/2 α β  a2T

−→ − ∞, α −→  ∞.

3.16

Thus we can easily choose a sufficiently large α such that α > ρ and for x  αw ∈ ET , Jx < 0.

Therefore, byLemma 1.1, there exists at least one critical value c ≥ a > 0 We suppose that  x is

a critical point corresponding to c, that is, J  x  c, and Jx  0 By a similar argument to the

proof ofLemma 3.1, for any x ∈ E T , there exists  x ∈ E T such that Jx  cmax Clearly, x / 0 If

x / x, and the proof is complete; otherwise, x  x and c  cmax ByLemma 1.1,

c  inf

h∈Γ

sup

s∈0,1

J

hs

whereΓ  {h ∈ C0, 1 , E T  | h0  0, h1  x} Then for any h ∈ Γ, cmax  maxs∈0,1 Jhs.

By the continuity of Jhs in s, J0 ≤ 0 and Jx < 0 show that there exists some s0 ∈

0, 1 such that Jhs0  cmax If we choose h1, h2 ∈ Γ such that the intersection {h1s |

s ∈ 0, 1}

{h2s | s ∈ 0, 1} is empty, then there exist s1, s2 ∈ 0, 1 such that Jh1s1 

Jh2s2  cmax Thus we obtain two different critical points x1  h1s1, x2  h2s2 of J in

E T In this case, in fact, we may obtain at least two nontrivial critical points which correspond

to the critical value cmax The proof ofTheorem 3.2is complete When fn, x n  ≡ h n, we have the following results

Theorem 3.3 Assume that the following conditions hold:

G1

G2

1

c δ11



T



n1

h2n

T



n1

−q n  <pminλ δ1/22 − qmax

T n1

h n

δ1

, 3.19

where pmin  minn∈ Z1,T p n , qmax  maxn∈ Z1,T q n , c1is a constant in2.10, and λ2is the minimal positive eigenvalue of the matrix

A 

⎛

⎜

⎜

⎜

⎜

⎜

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

⎞

⎟

⎟

⎟

⎟

⎟

T×T

Then equation

Δp n Δx n−1δ  q n x δ n  h n , n ∈ Z, 3.21

possesses at least one T-periodic solution.

First, we proved the following lemma

Lemma 3.4 Assume that G1 holds, then the functional

Jx  1

δ  1

T



n1

p n



Δx n−1δ1− 1

δ  1

T



n1

q n x δ1 n T

n1

h n x n 3.22

satisfies P-S condition on E T

Proof For any sequence {x l } ⊂ E T with Jx l  being bounded and Jx l  → 0 as n → ∞, there exists a positive constant M such that |Jx l | ≤ M In view of G3 and

T



n1

|h n x l n | ≤

 T



n1

h2n

1/2 T



n1



x l n 2 1/2

we have

M ≥ Jx l  1

δ  1

T



n1

p n



Δx n−1 l δ1



− 1

δ  1

T



n1

q n



x l n δ1

T

n1

h n x l n

≥ − 1

δ  1

T



n1

q n



x n lδ1

−T

n1

h n x l

n 

≥ − 1

δ  1 qmax

T



n1



x n lδ1

−

 T



n1

h2n

1/2 T



n1



x n l2 1/2

 −qmax

δ  1x lδ1

δ1−

 T



n1

h2n

1/2

x l

≥ −qmax

δ  1 c

δ1

1 x lδ1−

T

n1

h2n

1/2

x l.

3.24

By δ  1 > 1, the above inequality implies that {x l } is a bounded sequence in E T Thus{x l} possesses a convergent subsequence, and the proof ofLemma 3.4is complete Now we prove

Proof of Theorem 3.3 For any w  z, z, , z T ∈ W, we have

Jw  − 1

δ  1

T



n1

q n z δ1T

n1

Take z   T

n1 h n / T

n1 q n1/δ and ρ  w  T 1/2|T

n1 h n / T

n1 q n|1/δ , then

Jw  δ

δ  1

T

n1 h n

δ1/δ



T n1 q n1/δ 3.26 Set

σ  δ

δ  1

T

n1 h nδ1/δ



T n1 q n1/δ , 3.27 then we have

Jw  σ, ∀w ∈ ∂B ρ

On the other hand, for any x ∈ Y, we have

Jx  1

δ  1

T



n1

p n



Δx n−1δ1− 1

δ  1

T



n1

q n x n δ1T

n1

h n x n

≥ pmin

δ  1

T



n1



Δx n−1δ1− qmax

δ  1

T



n1

x δ1 n −T

n1

|h n x n|

≥ pmin

δ  1 c

δ1

1

T



n1

Δx n−12

δ1/2

− qmax

δ  1 x δ1

δ1−



T



n1

h2n

1/2

x

 pmin

δ  1 c

δ1

δ  1 x δ1

δ1−T

n1

|h n x n |,

3.29

where x T  x1, x2, , x T .

Clearly, λ1  0 is an eigenvalue of the matrix A and ξ  v, v, , v T ∈ E T is an

eigen-vector of A corresponding to 0, where v /  0, v ∈ R Let λ2, λ3, , λ T be the other eigenvalues

of A By matrix theory, we have λ j > 0 for all j ∈ Z2, T Without loss of generality, we may

assume that 0 λ1< λ2≤ · · · ≤ λ T , then for any x ∈ Y,

Jx ≥ pmin

δ  1 c

δ1

1 λ δ1/22 x δ1− qmax

δ  1 x δ1

δ1−

T

n1

h2n

1/2

x

 pmin

δ  1 c

δ1

1 λ δ1/22 − qmax

δ  1 c

δ1

1



x δ1−

T

n1

h2n

1/2

x

≥ − δ

δ  1

T

n1

h2n

1/2 T

n1 h2n1/2

pminc δ11 λ δ1/22 − qmaxc δ11

1/δ

,

3.30

as one finds by minimizing with respect tox That is

Jx ≥ − δ

δ  1

T

n1 h2

n

δ1/2δ

1/c1δ1/δ



pminλ δ1/22 − qmax

1/δ 3.31 Set

w0  − δ

δ  1

T

n1 h2

n

δ1/2δ

1/c1δ1/δ



pminλ δ1/22 − qmax

1/δ , 3.32 then byG2, we have

This implies that the assumption of saddle point theorem is satisfied Thus there exists at least

one critical point of J on E T , and the proof is complete When q n > 0, we have the following

result

Theorem 3.5 Assume that the following conditions are satisfied:

G3 2δ1 p n  p n1 < q n , q n > 0 for all n ∈ Z1, T;

G4 T

n1 h2

nδ1/2δT

n1 q n1/δ C δ1

1 < −A0T

n1 h nδ1/δ , where A0 maxn∈ Z1,T2δ1 p n  p n1  − q n

Then3.21 possesses at least one T-periodic solution.

Before provingTheorem 3.5, first, we prove the following result

Lemma 3.6 Assume that G3 holds, then Jx defined by 3.22 satisfies P-S condition.

Proof For any sequence {x l } ∈ E T with Jx l  being bounded and Jx l → 0 as

n → ∞, there exists a positive constant M such that |Jx l | ≤ M.

Thus

−M ≤ Jx l

≤ 1

δ  1

T



n1

p n



Δx l n−1δ1− 1

δ  1

T



n1

q n



x l n

δ1

T

n1

h n x l n

≤ 2δ1

δ  1

T



n1



p n  p n1



x l n δ1

− 1

δ  1

T



n1

q n



x l n δ1

T

n1

h n x l

n 

≤ 1

δ  1

T



n1



2δ1

p n  p n1− q nx l n

δ1



T

n1

h2n

1/2

x l

≤ 1

δ  1 A0x lδ1

δ1

T

n1

h2n

1/2

x l

≤ A0

δ  1 c

δ1

2 x lδ1



T



n1

h2n

1/2

x l.

3.34

That is,

−c δ1

2

A0

δ  1x lδ1−



T



n1

h2n

1/2

x l  ≤ M, ∀n ∈ N. 3.35

By δ  1 > 1, the above inequality implies that {x l } is a bounded sequence in E T Thus {x l} possesses convergent subsequences, and the proof is complete

Proof of Theorem 3.5 For any w  z, z, , z T ∈ W, we have

Jω  − 1

δ  1

T



n1

q n z δ1T

n1

Take z   T

n1 h n / T

n1 q n , ρ  w  T 1/2|T

n1 h n / T

n1|1/δ , then

Jw  δ

δ  1

T

n1 h n

δ1/δ



T n1 q n1/δ , ∀w ∈ ∂B ρW. 3.37 Set

σ  δ

δ  1

T

n1 h n

δ1/δ

|T n1 q n|1/δ , 3.38

then Jw  σ for all w ∈ ∂B ρ



W On the other hand, for any x ∈ Y, we have

Jx ≤ 1

δ  1

T



n1

2δ1 p n  p n1  − q n x δ1

T

n1

h2n

1/2

x

≤ A0

δ  1 c

δ1



T



n1

h2n

1/2

x

≤ − δ

δ  1

1

A0

!1/δ 1

c2

!δ1/δT

n1

h2n

δ1/2δ

.

3.39

Set w0  −δ/δ  11/A01/δ 1/c2δ1/δT

n1 h2

satisfies the assumption of saddle point theorem, that is, there exists at least one critical point

of J on E T This completes the proof ofTheorem 3.5

Acknowledgment

This project is supported by specialized research fund for the doctoral program of higher edu-cation, Grant no 20020532014

Proof of Theorem 3.3 For any w  z,... a2T. 3.6

In view of< /p>

T



n1...

δ1

, 3.19

where pmin  minn∈ Z1,T