Hindawi Publishing Corporation
Discrete Dynamics in Nature and Society
Volume 2011, Article ID 153082, 14 pages
doi:10.1155/2011/153082
Research Article
Nontrivial Periodic Solutions for
Nonlinear Second-Order Difference Equations
Tieshan He,1 Yimin Lu,2 Youfa Lei,1 and Fengjian Yang1
1
School of Computational Science, Zhongkai University of Agriculture and Engineering, Guangdong,
Guangzhou 510225, China
2
School of Chemistry and Chemical Engineering, Zhongkai University of Agriculture and Engineering,
Guangdong, Guangzhou 510225, China
Correspondence should be addressed to Yimin Lu, luyimin68@163.com
Received 4 May 2011; Accepted 29 August 2011
Academic Editor: Zengji Du
Copyright q 2011 Tieshan He et al. 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.
This paper is concerned with the existence of nontrivial periodic solutions and positive periodic
solutions to a nonlinear second-order difference equation. Under some conditions concerning
the first positive eigenvalue of the linear equation corresponding to the nonlinear second-order
equation, we establish the existence results by using the topological degree and fixed point index
theories.
1. Introduction
Let R, Z, N be the sets of real numbers, integers, and natural numbers, respectively. For a, b ∈
Z, define Za, b {a, a 1, . . . , b} when a ≤ b.
In this paper, we deal with the existence of nontrivial periodic solutions and positive
periodic solutions for a nonlinear second-order difference equation
Δ2 ut − 1 qtut ft, ut,
ut T ut, t ∈ Z,
1.1
where T is a positive integer, q : Z → R and qt T qt for any t ∈ Z, f : Z × R → R
is continuous in the second variable and ft T, x ft, x for any t, x ∈ Z × R, and
Δut ut 1 − ut, Δ2 ut ΔΔut.
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Discrete Dynamics in Nature and Society
From the T -periodicity of q and f, it is easy to verify that the T -periodic solution to
1.1 is equivalent to the solution to the following periodic boundary value problem PBVP
for short:
Δ2 ut − 1 qtut ft, ut,
u0 uT ,
t ∈ Z1, T ,
Δu0 ΔuT .
1.2
The theory of nonlinear difference equations has been widely used to study discrete
models appearing in many fields such as computer science, economics, neural network,
ecology, and cybernetics. In recent years, there are many papers to study the existence of
periodic solutions for second-order difference equations. By using various methods and
techniques, for example, fixed point theorems, the method of upper and lower solutions,
coincidence degree theory, critical point theory, a series of existence results of periodic
solutions have been obtained. We refer the reader to 1–16 and references therein.
In 2, by using the method of upper and lower solutions, Atici and Cabada
investigated the existence and uniqueness of periodic solutions for PBVP 1.2 provided that
qt ≤ 0, qt /
≡ 0. Of course the natural question is what would happen if qt ≥ 0. In this
paper, we will assume that
0 ≤ qt < 4sin2
π
≡ 0.
, qt /
2T
H
And we will use the topological degree and fixed point index theories to establish the
existence of nontrivial periodic solutions and positive periodic solutions for 1.1. We note
that some ideas of this paper are from 17–19.
This paper is organized as follows. In Section 2, we give Green’s function associated
with PBVP 1.2 and then present some preliminary lemmas that will be used to prove our
main results. In Sections 4 and 5, by computing the topological degree and fixed point index,
we establish some existence results of nontrivial periodic solutions and positive periodic
solutions to 1.1. The final section of the paper contains some examples to illustrate our
results, and we also remark that the results obtained in previous papers and ours are mutually
independent.
2. Preliminaries
In this section, we are going to construct Green’s function associated with PBVP 1.2 and
then present some preliminary lemmas. Consider T -dimensional Banach space
E u {ut}T1 : ut ∈ R, t ∈ Z1, T 2.1
equipped with the norm u max{|ut|, t ∈ Z1, T } for all u ∈ E and the cone P {u ∈ E :
ut ≥ 0, t ∈ Z1, T }. Then the cone P is normal and has nonempty interiors int P . It is clear
that P is also a total cone of E, that is, E P − P , which means the set P − P {u − v : u, v ∈ P }
is dense in E. For each u, v ∈ E, we write u ≤ v if v − u ∈ P . For r > 0, let Br {u ∈ E : u < r}
and ∂Br {u ∈ E : u r}. Put Q maxt∈Z1,T qt.
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Lemma 2.1. If 0 < Q < 4 sin2 π/2T , then, for each v ∈ E, the problem
Δ2 ut − 1 Qut vt,
u0 uT ,
t ∈ Z1, T ,
Δu0 ΔuT .
2.2
has a unique solution
ut T
Gt, kvk,
t ∈ Z0, T 1,
2.3
k1
where Gt, k is given by
Gt, k ⎧
⎨ρt − k,
0 ≤ k ≤ t ≤ T 1,
⎩ρT t − k, 0 ≤ t ≤ k ≤ T 1
2.4
with ρt 1/2 sin ϕ sinϕT/2 cos ϕT/2 − t and ϕ : arctan Q4 − Q/2 − Q.
2
Proof. i Taking
into account that Q ∈ 0, 4 sin π/2T , an easy computation ensures that
ϕ : arctan Q4 − Q/2 − Q ∈ 0, π/T . Hence, ρt > 0, t ∈ Z0, T . It is easy to verify
that
Δ2 ρt − 1 Qρt 0,
Let ut T
k1
ρ0 ρT ,
2.5
Gt, kvk, t ∈ Z0, T 1. Then
ut t
T
ρt − kvk ρT t − kvk,
k1
t ∈ Z1, T ,
2.6
kt1
where, and in what follows, we denote
Δut ρ1 ρT 1 1.
l
ks
xk 0 when l < s. We have
t
T
Δρt − kvk ρ0vt 1 ΔρT t − kvk − ρT vt 1.
k1
2.7
kt1
Then,
Δ2 ut − 1 Qut t−1 Δ2 ρt − k − 1 Qρt − k vk Qρ0vt Δρ0vt
k1
T Δ2 ρT t − k − 1 QρT t − k vk − QρT vt − ΔρT vt
kt
vt,
t ∈ Z1, T .
2.8
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From the definitions of u and G, it is easy to see that u0 uT and Δu0 ΔuT . This
completes the proof of the lemma.
From the expression of G, we see that Gt, k > 0 and Gt, k Gk, t for all t, k ∈
Z1, T . Define operators K, L : E → E, respectively, by
Kut T
Gt, kuk,
k1
Lut Q − qt ut,
u ∈ E, t ∈ Z1, T ,
2.9
u ∈ E, t ∈ Z1, T .
Obviously, KP ⊆ P and LP ⊆ P . It is clear that K is strongly positive, that is, Ku ∈ int P
for u ∈ P \ {θ}.
Lemma 2.2. Assume that H holds. Then, KL : E → E is a linear completely continuous operator
with KL < 1, and I − KL−1 , the inverse mapping of I − KL, exists and is bounded.
Proof. It is obvious that KL : E → E is a linear completely continuous operator. Since u0 t ≡
1/Q is a solution of PBVP 2.2 with v0 t ≡ 1, we have
T
Gt, k k1
1
,
Q
t ∈ Z1, T .
2.10
Then by H and the fact that K is strongly positive, one has
|KLut| T
T
Gt, k Q − qk |uk| ≤ u Gt, k Q − qk u 1 − Kq t < u,
k1
k1
2.11
where u ∈ E, t ∈ Z1, T . Hence KL < 1, and I − KL−1 , the inverse mapping of I − KL,
exists and is bounded. The proof of Lemma 2.2 is completed.
Let
S : I − KL−1 K I KL · · · KLn · · · K K KLK · · · KLn K · · · .
2.12
The complete continuity of K together with the continuity of I − KL−1 implies that the
operator S : E → E is completely continuous.
Lemma 2.3. Assume that H holds. Then, for each v ∈ E, the following linear periodic boundary
value problem
Δ2 ut − 1 qtut vt,
u0 uT ,
t ∈ Z1, T ,
Δu0 ΔuT 2.13
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5
1
, where ut Svt, t ∈ Z1, T , and u0 uT , u1 has a unique solution {ut}Tt0
uT 1.
Proof. It is easy to see that PBVP 2.13 is equivalent to the operator equation u KLu 1
, where ut Svt I −
Kv. Therefore, PBVP 2.13 has a unique solution {ut}Tt0
−1
KL Kvt, t ∈ Z1, T , and u0 uT , uT 1 u1.
Lemma 2.4. Assume that H holds. Then, for the operator S defined by 2.12, the spectral radius
rS > 0 and there exists ξ ∈ E with ξ > 0 on Z1, T such that Sξ rSξ and Tt1 ξt 1/rS.
Moreover, λ1 1/rS is the first positive eigenvalue of the linear PBVP corresponding to PBVP
1.2 and
T
T
1
utξt,
Sutξt λ1 t1
t1
∀u ∈ E.
2.14
Proof. An obvious modification of the proof of 13, Lemma 2.3 yields this result. We omit the
details here.
Lemma 2.5. Assume that H holds. Then, SP ⊆ P1 , where
P1 u∈P :
T
utξt ≥ δu ,
t1
2 sin ϕ sin ϕT/2 mint∈Z1,T ξt
,
δ
λ1 I − KL−1 2.15
and λ1 , ξ are given in Lemma 2.4, ϕ is given in Lemma 2.1.
Proof. By 2.14, we have, for any u ∈ P ,
T
Sutξt t1
T
T
mint∈Z1,T ξt 1
utξt ≥
ut.
λ1 t1
λ1
t1
2.16
On the other hand, one has
T
Gt, kuk
Su I − KL−1 Ku ≤ I − KL−1 Ku I − KL−1 max
t∈Z1,T k1
T
T
− KL−1 I
−1 ≤ I − KL max Gt, k uk uk.
t,k∈Z1,T 2 sin ϕ sin ϕT/2 k1
k1
2.17
Then,
T
t1
Sutξt ≥
2 sin ϕ sin ϕT/2 mint∈Z1,T ξt
Su δSu.
λ1 I − KL−1 Hence, SP ⊆ P1 . The proof is complete.
2.18
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Discrete Dynamics in Nature and Society
Define operators f, A : E → E, respectively, by
fut ft, ut,
u ∈ E, t ∈ Z1, T ,
A Sf.
2.19
It follows from the continuity of f together with the complete continuity of S that A : E → E
is completely continuous.
Remark 2.6. By Lemma 2.3, it is easy to see that u {ut}Tt1 ∈ E is a fixed point of the operator
1
is a solution of PBVP 1.2, where u0 uT , u1 uT 1.
A if and only if u {ut}Tt0
The proofs of the main theorems of this paper are based on the topological degree and
fixed point index theories. The following four well-known lemmas in 20–22 are needed in
our argument.
Lemma 2.7. Let Ω be a bounded open set in a real Banach space E with θ ∈ Ω, let and A : Ω → E be
μx0 for all x ∈ ∂Ω and μ ≥ 0,
completely continuous. If there exists x0 ∈ E \ {θ} such that x − Ax /
then the topological degree degI − A, Ω, θ 0.
Lemma 2.8. Let Ω be a bounded open set in a real Banach space E with θ ∈ Ω, and let A : Ω → E
be completely continuous. If Ax /
μx for all x ∈ ∂Ω and μ ≥ 1, then the topological degree degI −
A, Ω, θ 1.
Lemma 2.9. Let E be a Banach space and X ⊂ E a cone in E. Assume that Ω is a bounded open subset
of E. Suppose that A : X ∩ Ω → X is a completely continuous operator. If there exists x0 ∈ X \ {θ}
such that x − Ax /
μx0 for all x ∈ X ∩ ∂Ω and μ ≥ 0, then the fixed point index iA, X ∩ Ω, X 0.
Lemma 2.10. Let E be a Banach space and X ⊂ E a cone in E. Assume that Ω is a bounded open
subset of E with θ ∈ Ω. Suppose that A : X ∩ Ω → X is a completely continuous operator. If
Ax /
μx for all x ∈ X ∩ ∂Ω and μ ≥ 1, then the fixed point index iA, X ∩ Ω, X 1.
3. Existence of Nontrivial Periodic Solutions
Theorem 3.1. Assume that H holds. If the following conditions are satisfied
ft, x < λ1 ,
lim sup max x x → ∞ t∈Z1,T ft, x
> λ1 ,
t∈Z1,T x
3.2
ft, x
< λ1 ,
x
t∈Z1,T 3.3
lim inf
min
x→0
lim sup max
x → 0−
3.1
where λ1 is the first positive eigenvalue of the linear operator S given in Lemma 2.4, then 1.1 has at
least one nontrivial periodic solution.
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Proof. In view of Remark 2.6, it suffices to prove that the operator A has at least fixed point in
E \ {θ}. It follows from 3.2 and 3.3 that there exist r > 0 and σ ∈ 0, 1 such that
ft, x ≥ λ1 1 σx ≥ λ1 1 − σx,
∀x ∈ 0, r, t ∈ Z1, T ,
ft, x ≥ λ1 1 − σx ≥ λ1 1 σx,
∀x ∈ −r, 0, t ∈ Z1, T .
3.4
By the above two inequalities, we have
ft, x ≥ λ1 1 σx,
∀|x| ≤ r, t ∈ Z1, T ,
3.5
ft, x ≥ λ1 1 − σx,
∀|x| ≤ r, t ∈ Z1, T .
3.6
We may suppose that A has no fixed point on ∂Br . Otherwise, the proof is finished. Now we
will prove
u
/ Au μξ,
∀u ∈ ∂Br , μ ≥ 0,
3.7
where ξ is given in Lemma 2.4. Suppose the contrary; then there exist u0 ∈ ∂Br and μ0 ≥ 0
such that u0 Au0 μ0 ξ. Then μ0 > 0. Multiplying the equality u0 Au0 μ0 ξ by ξ on its both
sides, summing from 1 to T , and using 2.14 and 3.5, it follows that
T
t1
u0 tξt T
T
T
T
1
ft, u0 tξt μ0 ξ2 t
Au0 tξt μ0 ξ2 t λ1 t1
t1
t1
t1
T
T
≥ 1 σ u0 tξt μ0 ξ2 t.
t1
3.8
t1
Similarly, by 3.6, we know also that
T
t1
u0 tξt ≥ 1 − σ
T
T
u0 tξt μ0 ξ2 t.
t1
3.9
t1
If Tt1 u0 tξt ≥ 0, then 3.8 implies that Tt1 ξ2 t ≤ 0, which contradicts ξ > 0 on Z1, T .
T
If t1 u0 tξt < 0, then 3.9 also implies that Tt1 ξ2 t < 0, which is a contradiction. Thus,
3.7 holds. On the basis of Lemma 2.7, we have
degI − A, Br , θ 0.
3.10
From 3.1 it follows that there exist G > 0 and ε ∈ 0, 1 such that |ft, x| < λ1 1 − ε|x|
for |x| > G, t ∈ Z1, T . Let C supt∈Z1,T ,|x|≤G ft, x. Obviously,
ft, x ≤ λ1 1 − ε|x| C,
∀x ∈ R, t ∈ Z1, T .
3.11
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μu,
Choose R such that R > max{r, aε−1 C}, where a mint∈Z1,T ξt. We next show Au /
for all u ∈ ∂BR , μ ≥ 1. In fact, if there exist u1 ∈ ∂BR and μ1 ≥ 1 such that Au1 μ1 u1 , then,
by the definition of A and 3.11, we obtain
T
Gt, k fk, u1 k
|u1 t| ≤ |Au1 t| ≤ I − KL
−1
k1
T
T
−1
≤ λ1 1 − εI − KL
Gt, k|u1 k| CI − KL−1 Gt, k.
k1
3.12
k1
Set u2 t |u1 t|. Then u2 ∈ P \ {θ}, and, for any t ∈ Z1, T , λ1 1 − εSu2 t CSv0 t ≥
u2 t, where v0 t ≡ 1. Then, by 2.14, we have
T t1
T
T
T
C
u2 tξt.
1 − εu2 t v0 t ξt λ1 1 − ε Su2 tξt C Sv0 tξt ≥
λ1
t1
t1
t1
3.13
Using the above inequality and noticing that Tt1 ξt λ1 see Lemma 2.4, we have that
ε−1 C ≥ Tt1 u2 tξt ≥ a Tt1 u2 t. This implies that R u2 ≤ Tt1 u2 t ≤ aε−1 C, which
contradicts the choice of R. It follows from Lemma 2.8 that
degI − A, BR , θ 1.
3.14
According to the additivity of Leray-Schauder degree, by 3.14 and 3.10, we get
deg I − A, BR \ Br , θ degI − A, BR , θ − degI − A, Br , θ 1,
3.15
which implies that the nonlinear operator A has at least one fixed point in BR \ Br . Thus, 1.1
has at least one nontrivial periodic solution. The proof is complete.
Theorem 3.2. Assume that H holds. If the following conditions are satisfied
ft, x < λ1 ,
lim sup max x t∈Z1,T x→0
ft, x
> λ1 ,
t∈Z1,T x
3.17
ft, x
< λ1 ,
x
t∈Z1,T 3.18
lim inf min
x → ∞
lim sup max
x → −∞
3.16
where λ1 is the first positive eigenvalue of the linear operator S given in Lemma 2.4, then 1.1 has at
least one nontrivial periodic solution.
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Proof. It suffices to prove that the operator A has at least fixed point in E \ {θ}. From 3.16,
we find that there exist ε ∈ 0, 1 and r > 0 such that
ft, x ≤ λ1 1 − ε|x|,
∀|x| ≤ r, t ∈ Z1, T ,
3.19
Now we prove
Au /
μu,
∀u ∈ ∂Br , μ ≥ 1.
3.20
If 3.20 does hold, there exist μ0 ≥ 1 and u0 ∈ ∂Br such that Au0 μ0 u0 . Then, by 3.19, we
have
|u0 t| ≤ |Au0 t| ≤ I − KL
−1
T
Gt, kfk, u0 k
k1
T
−1
≤ λ1 1 − εI − KL
Gt, k|u0 k| ,
3.21
t ∈ Z1, T .
k1
Set u1 t |u0 t|. Then u1 ∈ P \ {θ} and λ1 1 − εSu1 ≥ u1 . Multiplying this inequality by ξ
and summing from 1 to T , it follows from 2.14 that
T
T
T
u1 tξt.
1 − ε u1 tξt λ1 1 − ε Su1 tξt ≥
t1
t1
3.22
t1
This together with Tt1 u1 tξt > 0 implies that 1 − ε ≥ 1, which contradicts the choice of ε,
and so 3.20 holds. It follows from Lemma 2.8 that
degI − A, Br , θ 1.
3.23
By 3.17, 3.18, and the continuity of ft, x with respect to x, we know that there
exist σ ∈ 0, 1 and C > 0 such that
ft, x ≥ λ1 1 σx − C,
∀x ≥ 0, t ∈ Z1, T ,
ft, x ≥ λ1 1 − σx − C,
∀x ≤ 0, t ∈ Z1, T .
3.24
Then,
ft, x ≥ λ1 1 σx − C ≥ λ1 1 − σx − C,
∀x ≥ 0, t ∈ Z1, T ,
ft, x ≥ λ1 1 − σx − C ≥ λ1 1 σx − C,
∀x ≤ 0, t ∈ Z1, T .
3.25
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Discrete Dynamics in Nature and Society
By the above two inequalities, we have
ft, x ≥ λ1 1 σx − C,
∀x ∈ R, t ∈ Z1, T ,
3.26
ft, x ≥ λ1 1 − σx − C,
∀x ∈ R, t ∈ Z1, T .
3.27
Set
Ω {u ∈ E : u Au τξ for some τ ≥ 0},
3.28
where ξ is given in Lemma 2.4. We claim that Ω is bounded in E. In fact, for any u ∈ Ω, there
exists τ ≥ 0 such that u Au τξ ≥ Au. Then, by 3.26, we have
ut ≥ λ1 1 σSut − CSv0 t,
t ∈ Z1, T ,
3.29
where v0 t ≡ 1. Multiplying the above inequality by ξt on both sides and summing from 1
to T , it follows from 2.14 that
T
T
T
T
T
C
utξt ≥ λ1 1 σ Sutξt − C Sv0 tξt 1 σ utξt −
ξt.
λ1 t1
t1
t1
t1
t1
3.30
Then, noticing that
T
t1
ξt λ1 , we have
σ
T
utξt ≤ C.
3.31
t1
On the other hand, bearing in mind that ξ λ1 Sξ, we obtain that, for u ∈ Ω,
u − λ1 1 − σSu CSv0 Sfu − λ1 1 − σSu CSv0 τξ Sfu − λ1 1 − σu Cv0 τλ1 ξ.
3.32
By 3.27, we obtain that fu − λ1 1 − σu Cv0 τλ1 ξ ∈ P . Lemma 2.5 yields that u − λ1 1 −
σSu CSv0 ∈ P1 . Then by 2.14 and 3.31, we obtain that
u − λ1 1 − σSu CSv0 ≤
T
1
ut − λ1 1 − σSut CSv0 tξt
δ t1
T 1
2C
C
.
utξt − 1 − σutξt ξt ≤
δ t1
λ1
δ
3.33
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This gives
u − λ1 1 − σSu ≤
2C
CSv0 ,
δ
∀u ∈ Ω.
3.34
Hence, I−λ1 1−σSΩ ⊂ BR1 , where R1 2C/δCSv0 > 0. It follows from λ1 1−σrS <
1 that I − λ1 1 − σS has a linear bounded inverse I − λ1 1 − σS−1 . Therefore, there exists
R2 > 0 such that
Ω ⊂ I − λ1 1 − σS−1 BR1 ⊂ BR2 .
3.35
Then, we can conclude that Ω is bounded in E, proving our claim. Thus, there exists R >
max{r, R2 } such that
u
/ Au τξ,
∀u ∈ ∂BR , τ ≥ 0.
3.36
This and Lemma 2.7 give degI − A, BR , θ 0. Taking 3.23 into account, we have degI −
A, BR \ Br , θ −1. Then, A has at least one fixed point in BR \ Br , which means that 1.1 has
at least one nontrivial periodic solution. The proof is completed.
4. Existence of Positive Periodic Solutions
Theorem 4.1. Assume that H holds. If the following conditions are satisfied
xft, x ≥ 0,
∀x ∈ R, t ∈ Z1, T ,
ft, x
< λ1 ,
x
x → ∞ t∈Z1,T ft, x
lim inf min
> λ1 ,
x → 0 t∈Z1,T x
lim sup max
4.1
4.2
4.3
where λ1 is the first positive eigenvalue of the linear operator S given in Lemma 2.4, then 1.1 has at
least one positive periodic solution and one negative periodic solution.
Proof. From 4.1, we know that AP ⊂ P . Similar to the proof of Theorem 3.1, it follows from
4.1–4.3 and Lemmas 2.9 and 2.10 that there exist 0 < r < R such that
iA, Br ∩ P, P 0,
iA, BR ∩ P, P 1.
4.4
Hence, by the additivity of the fixed point index, we have
i A, BR \ Br ∩ P, P iA, BR ∩ P, P − iA, Br ∩ P, P 1.
4.5
Then, the nonlinear operator A has at least one fixed point on BR \ Br ∩ P . So 1.1 has at
least one positive periodic solution.
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Put f1 t, x −ft, −x, for all t, x ∈ Z1, T × R. Define operators f1 , A1 : E → E,
respectively, by
f1 ut f1 t, ut,
u ∈ E, t ∈ Z1, T ,
A1 Sf1 .
4.6
Obviously, A1 P ⊂ P . Following almost the same procedure as above, from lim supx → ∞
maxt∈Z1,T f1 t, x/x < λ1 and lim infx → 0 mint∈Z1,T f1 t, x/x > λ1 , we know also that the
nonlinear operator A1 has at least one fixed point ζ ∈ P \ {θ}. Then A1 ζ ζ. This means that
A−ζ Sf−ζ S−f1 ζ −A1 ζ −ζ. Hence, 1.1 has at least one negative periodic
solution −ζ, and the conclusion is achieved.
Theorem 4.2. Assume that H and 4.1 hold. If the following conditions are satisfied
ft, x
< λ1 ,
x
t∈Z1,T lim sup max
x→0
ft, x
> λ1 ,
lim inf min
x → ∞ t∈Z1,T x
4.7
where λ1 is the first positive eigenvalue of the linear operator S given in Lemma 2.4, then 1.1 has at
least one positive periodic solution and one negative periodic solution.
The proof is similar to that of Theorem 4.1 and so we omit it here.
5. Examples
Example 5.1. Let ft, x |x|. It is easy to see that lim supx → ∞ maxt∈Z1,T |ft, x/x| 0 < λ1 ,
lim infx → 0 mint∈Z1,T ft, x/x ∞ > λ1 , and lim supx → 0− maxt∈Z1,T ft, x/x −∞ <
λ1 . Then, it follows from Theorem 3.1 that 1.1 has at least one nontrivial periodic solution.
Example 5.2. Let ft, x 2x4 x3 . It is not difficult to see that lim supx → 0 maxt∈Z1,T |ft, x/x| 0 < λ1 , lim infx → ∞ mint∈Z1,T ft, x/x ∞ > λ1 , and lim supx → −∞
maxt∈Z1,T ft, x/x −∞ < λ1 . Then, it follows from Theorem 3.2 that 1.1 has at least
one nontrivial periodic solution.
Example 5.3. Let ft, x 3x5 ex . Obviously, xft, x ≥ 0 for all x ∈ R and t ∈ Z1, T .
Moreover, lim supx → 0 maxt∈Z1,T ft, x/x 0 < λ1 and lim infx → ∞ mint∈Z1,T ft, x/x ∞ > λ1 . Then it follows from Theorem 4.2 that 1.1 has at least one positive periodic
solution and one negative periodic solution.
6
Remark 5.4. It is easy to see that the existence of nontrivial periodic solutions in Examples
5.1–5.3 could not be obtained by any theorems in 1–16, 19.
Discrete Dynamics in Nature and Society
13
Ackowledments
This project is supported by the Natural Science Foundation of Guangdong Province no.
S2011010001900 and by the Scientific Research Plan Item of Fujian Provincial Department of
Education no. JA06035.
References
1 F. M. Atici and G. S. Guseinov, “Positive periodic solutions for nonlinear difference equations with
periodic coefficients,” Journal of Mathematical Analysis and Applications, vol. 232, no. 1, pp. 166–182,
1999.
2 F. M. Atici and A. Cabada, “Existence and uniqueness results for discrete second-order periodic
boundary value problems,” Computers & Mathematics with Applications, vol. 45, no. 6-9, pp. 1417–1427,
2003.
3 J. Yu, Z. Guo, and X. Zou, “Periodic solutions of second order self-adjoint difference equations,”
Journal of the London Mathematical Society. Second Series, vol. 71, no. 1, pp. 146–160, 2005.
4 H. Bin, J. Yu, and Z. Guo, “Nontrivial periodic solutions for asymptotically linear resonant difference
problem,” Journal of Mathematical Analysis and Applications, vol. 322, no. 1, pp. 477–488, 2006.
5 J. Yu and B. Zheng, “Multiplicity of periodic solutions for second-order discrete Hamiltonian systems
with a small forcing term,” Nonlinear Analysis. Theory, Methods & Applications, vol. 69, no. 9, pp. 3016–
3029, 2008.
6 Z. Guo and J. Yu, “The existence of periodic and subharmonic solutions of subquadratic second order
difference equations,” Journal of the London Mathematical Society. Second Series, vol. 68, no. 2, pp. 419–
430, 2003.
7 J. Yu, X. Deng, and Z. Guo, “Periodic solutions of a discrete Hamiltonian system with a change of sign
in the potential,” Journal of Mathematical Analysis and Applications, vol. 324, no. 2, pp. 1140–1151, 2006.
8 Z. Guo and J. Yu, “Existence of periodic and subharmonic solutions for second-order superlinear
difference equations,” Science in China. Series A. Mathematics, vol. 46, no. 4, pp. 506–515, 2003.
9 Z. Zhou, J. Yu, and Y. Chen, “Periodic solutions of a 2nth-order nonlinear difference equation,” Science
China. Mathematics, vol. 53, no. 1, pp. 41–50, 2010.
10 Y. Xue and C. Tang, “Multiple periodic solutions for superquadratic second-order discrete
Hamiltonian systems,” Applied Mathematics and Computation, vol. 196, no. 2, pp. 494–500, 2008.
11 Y. Xue and C. Tang, “Existence of a periodic solution for subquadratic second-order discrete
Hamiltonian system,” Nonlinear Analysis. Theory, Methods & Applications, vol. 67, no. 7, pp. 2072–2080,
2007.
12 T. He and W. Chen, “Periodic solutions of second order discrete convex systems involving the pLaplacian,” Applied Mathematics and Computation, vol. 206, no. 1, pp. 124–132, 2008.
13 T. He and Y. Xu, “Positive solutions for nonlinear discrete second-order boundary value problems
with parameter dependence,” Journal of Mathematical Analysis and Applications, vol. 379, no. 2, pp.
627–636, 2011.
14 F. Lian and Y. Xu, “Multiple solutions for boundary value problems of a discrete generalized EmdenFowler equation,” Applied Mathematics Letters, vol. 23, no. 1, pp. 8–12, 2010.
15 R. Ma and H. Ma, “Positive solutions for nonlinear discrete periodic boundary value problems,”
Computers & Mathematics with Applications, vol. 59, no. 1, pp. 136–141, 2010.
16 X. He and X. Wu, “Existence and multiplicity of solutions for nonlinear second order difference
boundary value problems,” Computers & Mathematics with Applications, vol. 57, no. 1, pp. 1–8, 2009.
17 F. Li and Z. Liang, “Existence of positive periodic solutions to nonlinear second order differential
equations,” Applied Mathematics Letters, vol. 18, no. 11, pp. 1256–1264, 2005.
18 Z. Yang, “Existence of nontrivial solutions for a nonlinear Sturm-Liouville problem with integral
boundary conditions,” Nonlinear Analysis, vol. 68, no. 1, pp. 216–225, 2008.
19 A. Cabada and V. Otero-Espinar, “Optimal existence results for nth order periodic boundary value
difference equations,” Journal of Mathematical Analysis and Applications, vol. 247, no. 1, pp. 67–86, 2000.
14
Discrete Dynamics in Nature and Society
20 K. Deimling, Nonlinear Functional Analysis, Springer, Berlin, Germany, 1985.
21 D. J. Guo and V. Lakshmikantham, Nonlinear Problems in Abstract Cones, vol. 5, Academic Press,
Boston, Mass, USA, 1988.
22 D. J. Guo, Nonlinear Functional Analysis, Shandong Science and Technology Press, Ji’nan, China, 2nd
edition, 2001.
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