# Periodic Solutions with Minimal Period for Fourth-Order Nonlinear Difference Equations

Authors: Xia Liu; Tao Zhou; Haiping Shi; Yuhua Long; Zongliang Wen
Journal: Discrete Dynamics in Nature and Society
 (2018)

---

## Abstract

A fourth-order nonlinear difference equation is considered. By making use of critical point theory, some new criteria are obtained for the existence of periodic solutions with minimal period. The main methods used are a variational technique and the Linking Theorem.


---

## Body

## 1. Introduction

LetN, Z, and R denote the sets of all natural numbers, integers, and real numbers, respectively. [
·
] denotes the greatest-integer function. For any a
,
b
∈
Z, define Z
(
a
,
b
)
=
{
a
,
a
+
1
,
…
,
b
} when a
≤
b. x
∗ denotes the transpose of a vector x.Consider the following fourth-order nonlinear difference equation:(1)
Δ
4
u
n
-
2
=
f
n
,
u
n
,
n
∈
Z
,where Δ is the forward difference operator Δ
u
n
=
u
n
+
1
-
u
n; Δ
i
u
n
=
Δ
(
Δ
i
-
1
u
n
) for i
≥
2, and f
∈
C
1
(
R
2
,
R
) and f
(
n
,
u
) are T-periodic in n for a given positive integer T.Equation (1) can be considered as a discrete analogue of continuous versions of problem like(2)
u
4
t
=
f
t
,
u
t
,
t
∈
R
,which is used to describe the stationary states of the deflection of an elastic beam [1]. Equations similar to (2) arise in the study of the existence of solutions to differential equations; we refer the reader to [2–7] and the references therein.The theory of nonlinear difference equations has been widely used to study discrete models in many fields such as finance insurance, computing, electrical circuit analysis, dynamical systems, physical field, and biology. Because of their importance, many literature and monographs deal with their existence and uniqueness problems; see [8–27].Using the critical point theory and monotone operator theory, He and Su [13] studied the following discrete nonlinear fourth-order boundary value problems:(3)
Δ
4
u
n
-
2
+
η
Δ
2
u
n
-
1
-
ξ
u
n
=
λ
f
n
,
u
n
,
n
∈
Z
a
+
1
,
b
+
1
,with three parameters. Some existence, multiplicity, and nonexistence results of nontrivial solutions are obtained.Chen and Tang [12] in 2011 were concerned with the existence of infinitely many homoclinic orbits from 0 of the fourth-order difference system(4)
Δ
4
u
n
-
2
+
q
n
u
n
=
f
n
,
u
n
+
1
,
u
n
,
u
n
-
1
,
n
∈
Zby using the symmetric Mountain Pass Lemma and established some existence criteria to guarantee that (4) has infinitely many homoclinic orbits.In 2012, Ma and Lu [19] showed the existence and multiplicity of positive solutions of the nonlinear discrete fourth-order boundary value problem for(5)
Δ
4
u
n
-
2
=
λ
h
n
f
u
n
,
n
∈
2,3
,
…
,
T
,by using Dancer’s global bifurcation theorem.Liu et al. [18] studied the existence and multiplicity of periodic and subharmonic solutions to the following nonlinear difference equation:(6)
Δ
2
r
n
-
2
Δ
2
u
n
-
2
=
f
n
,
u
n
+
1
,
u
n
,
u
n
-
1
,
n
∈
Z
,using variational technique and the Linking Theorem.By employing the variational methods, Yu et al. [22] got some new criteria for the existence of subharmonic solutions with prescribed minimal period of second-order nonlinear difference equation(7)
Δ
2
u
n
-
1
+
A
sin
⁡
u
n
=
f
n
,
n
∈
Z
.Applying the direct method of the calculus of variations and the mountain pass technique, Leszczyński [17] in 2015 proved the existence of at least one and at least two solutions to the fourth-order discrete anisotropic boundary value problem with both advance and retardation of form(8)
Δ
2
γ
n
-
1
ϕ
p
n
Δ
2
u
n
-
2
=
f
n
,
u
n
+
1
,
u
n
,
u
n
-
1
,
n
∈
Z
1
,
k
.Nonexistence of nontrivial solutions was also obtained.Motivated by the recent papers [12, 16], our purpose in this work is to apply Linking Theorem in critical point theory to establish some conditions for the nonlinear function f which are able to guarantee the existence of at least two nontrivial periodic solutions with minimal period m
T for the above problem.Throughout this paper, we suppose thatm is a given integer and m
>
1. Let (9)
ω
=
2
π
T
.To wit, we get the following.Theorem 1.
Assume that the following hypotheses are satisfied:(
F
1
)
there exists a functionF
∈
C
2
(
R
2
,
R
) with F
(
n
+
T
,
u
)
=
F
(
n
,
u
), F
(
-
n
,
-
u
)
=
F
(
n
,
u
), and F
(
n
,
u
)
≥
0 and it satisfies (10)
∂
F
n
,
u
∂
u
=
f
n
,
u
,
∀
n
∈
Z
;
(
F
2
)
there exist positive constantsϵ
1, α
∈
0,8
s
i
n
4
π
/
m
T such that (11)
F
n
,
u
≤
α
u
2
,
∀
n
∈
Z
,
u
≤
ϵ
1
;
(
F
3
)
there exist positive constantsζ, β
∈
8
,
+
∞ such that (12)
F
n
,
u
≥
β
u
2
-
ζ
,
∀
n
,
u
∈
Z
×
R
;
(
F
4
)
there exist positive constantsρ and μ
>
ν such that (13)
∂
2
F
n
,
u
∂
u
2
η
,
η
≤
μ
η
2
,
∀
n
,
u
∈
Z
×
R
,
η
∈
R
,
∂
2
F
n
,
u
∂
u
2
η
,
η
≥
ν
η
2
,
∀
u
≤
ρ
,
n
∈
Z
,
η
∈
R
;
(
F
5
)
ifu is a solution of (1) with minimal period l
T, l is a rational number, and f
(
n
,
u
) also has a minimal period l
T, then l must be an integer.
(
F
6
)
letτ
m be the least prime factor of m, (14)
ν
>
16
s
i
n
4
ω
2
m
,
16
s
i
n
4
ω
τ
m
2
m
>
μ
,
∑
n
=
1
m
T
f
2
n
,
0
<
π
ρ
2
32
s
i
n
4
ω
τ
m
/
2
m
-
2
μ
2
ν
-
32
s
i
n
4
ω
/
2
m
ω
.
Then (1) has at least two nontrivial periodic solutions with minimal period m
T.Theorem 2.
Assume that(
F
1
), (
F
4
)
–
(
F
6
), and the following conditions are satisfied: (
F
7
)
l
i
m
|
u
|
→
0
F
(
n
,
u
)
/
u
2
=
0, ∀
(
n
,
u
)
∈
Z
×
R;
(
F
8
)
there exist positive constantsϵ
2 and γ
>
2 such that for n
∈
Z and |
u
|
≥
ϵ
2, (15)
0
<
γ
F
n
,
u
≤
u
f
n
,
u
.
Then (1) has at least two nontrivial periodic solutions with minimal period m
T.


## 2. Variational Structure and Some Lemmas

Define the functionalJ as follows: (16)
J
u
≔
1
2
∑
n
=
1
m
T
Δ
2
u
n
-
1
2
-
∑
n
=
1
m
T
F
n
,
u
n
,on the finite-dimensional Hilbert space (17)
E
m
T
=
u
∣
u
n
+
m
T
=
u
n
,
∀
n
∈
Z
,where (18)
u
=
…
,
u
-
n
,
…
,
u
-
1
,
u
0
,
u
1
,
…
,
u
n
,
…
.Form
T
>
2, define (19)
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
m
T
×
m
T
.Form
T
=
2, define (20)
M
≔
2
-
2
-
2
2
.The argument need not be changed and we omit it.J
(
u
) can be rewritten as (21)
J
u
=
1
2
∑
n
=
1
m
T
Δ
2
u
n
,
Δ
2
u
n
-
∑
n
=
1
m
T
F
n
,
u
n
=
1
2
x
∗
M
x
-
∑
n
=
1
m
T
F
n
,
u
n
,where x
=
(
Δ
u
1
,
Δ
u
2
,
…
,
Δ
u
m
T
)
∗.Clearly,J
∈
C
1
(
E
m
T
,
R
) and for any u
∈
E
m
T, by using u
0
=
u
m
T, u
1
=
u
m
T
+
1, we can compute the partial derivative as (22)
∂
J
∂
u
n
=
Δ
4
u
n
-
2
-
f
n
,
u
n
,
∀
n
∈
Z
1
,
m
T
.Therefore, u
∈
E
m
T is a critical point of J on E
m
T if and only if (23)
Δ
4
u
n
-
2
=
f
n
,
u
n
,
∀
n
∈
Z
1
,
m
T
.Since u
=
{
u
n
}
n
∈
Z
∈
E
m
T and f
(
n
,
u
) in the first variable n are m
T-periodic in n, we can reduce the existence of periodic solutions of (1) to the existence of critical points of J on E
m
T.By matrix theory, the eigenvalues ofM can be given by(24)
λ
j
=
4
s
i
n
2
j
π
m
T
,
j
=
0,1
,
2
,
…
,
m
T
-
1
.It is easy to see that 0 is an eigenvalue of M and λ
j
>
0, j
=
1,2
,
…
,
m
T
-
1, and (25)
min
⁡
λ
1
,
λ
2
,
…
,
λ
m
T
-
1
=
4
s
i
n
2
π
m
T
,
max
⁡
λ
1
,
λ
2
,
…
,
λ
m
T
-
1
≤
4
.Denote(26)
C
=
k
e
r
M
=
u
∈
E
m
T
∣
M
u
=
0
∈
R
m
T
.Therefore, (27)
C
=
u
∈
E
m
T
∣
u
=
ξ
,
ξ
∈
R
.LetD be the direct orthogonal complement of E
m
T to C; that is, E
m
T
=
C
⊕
D.For1
≤
j
≤
(
m
T
-
1
)
/
2, the eigenvectors of M corresponding to λ
j are (28)
ξ
j
=
cos
⁡
2
j
π
m
T
,
cos
⁡
2
j
π
·
2
m
T
,
…
,
cos
⁡
2
j
π
·
m
T
m
T
∗
,
ζ
j
=
sin
⁡
2
j
π
m
T
,
sin
⁡
2
j
π
·
2
m
T
,
…
,
sin
⁡
2
j
π
·
m
T
m
T
∗
.Whenm
T is even, the eigenvector corresponding to 4 is ξ
=
(
-
1,1
,
-
1,1
,
…
,
-
1,1
)
∗. Let P
=
s
p
a
n
{
ξ
}, Q
=
s
p
a
n
ξ
j
,
j
=
1,2
,
…
,
(
m
T
-
1
)
/
2, and S
=
s
p
a
n
ζ
j
,
j
=
1,2
,
…
,
(
m
T
-
1
)
/
2. Then E
m
T
=
C
⊕
P
⊕
Q
⊕
S. For any u
∈
E
m
T, (29)
u
n
=
a
+
∑
j
=
1
m
T
-
1
/
2
a
j
cos
⁡
ω
j
m
n
+
b
j
sin
⁡
ω
j
m
n
,
∀
n
∈
Z
,where a, a
j, and b
j are constants.Whenm
T is odd, E
m
T
=
C
⊕
Q
⊕
S. For any u
∈
E
m
T, (30)
u
n
=
a
+
-
1
n
b
+
∑
j
=
1
m
T
-
1
/
2
a
j
cos
⁡
ω
j
m
n
+
b
j
sin
⁡
ω
j
m
n
,
∀
n
∈
Z
,where a, b, a
j, and b
j are constants.Denote(31)
E
~
m
T
=
u
∈
E
m
T
∣
u
-
n
=
u
n
,
∀
n
∈
Z
.Then E
~
m
T
=
S and (32)
u
n
=
∑
j
=
1
m
T
-
1
/
2
b
j
sin
⁡
ω
j
m
n
,
∀
n
∈
Z
.LetB
r denote the open ball in E about 0 of radius r and let ∂
B
r denote its boundary.Lemma 3 (Linking Theorem [1]).
LetG be a real Banach space, G
=
G
1
⊕
G
2, where G
1 is finite dimensional. Assume that J
∈
C
1
(
G
,
R
) satisfies the PS condition and(
J
1
)
there exist constantsb
>
0 and ϱ
>
0 such that J
|
∂
B
ϱ
∩
G
2
≥
b;
(
J
2
)
there exists ane
∈
∂
B
1
∩
G
2 and a constant ς
≥
ϱ such that J
|
∂
Φ
≤
0, where Φ
=
(
B
¯
ς
∩
G
1
)
⊕
{
s
e
∣
0
<
s
<
ς
}.
ThenJ possesses a critical value c
≥
b, where (33)
c
=
i
n
f
h
∈
Γ
s
u
p
u
∈
Φ
J
h
u
,and Γ
=
{
h
∈
C
Φ
¯
,
G
∣
h
|
∂
Φ
=
I
}, where I denotes the identity operator.Lemma 4.
Assume that the hypotheses(
F
1
)
–
(
F
3
) are satisfied. Then the functional J is bounded from above in E
m
T.Proof.
According to(
F
3
) and (21), for any u
∈
E
m
T, (34)
J
u
=
1
2
x
∗
M
x
-
∑
n
=
1
m
T
F
n
,
u
n
≤
1
2
x
∗
M
x
-
∑
n
=
1
m
T
β
u
n
2
-
ζ
≤
2
x
2
-
β
u
2
+
m
T
ζ
,where x
=
(
Δ
u
1
,
Δ
u
2
,
…
,
Δ
u
m
T
)
∗. Since (35)
x
2
=
∑
n
=
1
m
T
u
n
+
1
-
u
n
,
u
n
+
1
-
u
n
=
u
∗
M
u
≤
4
u
2
,we get(36)
J
u
≤
8
-
β
u
2
+
m
T
ζ
≤
m
T
ζ
.The desired results are obtained.Lemma 5.
Assume that the hypotheses(
F
1
)
–
(
F
3
) are satisfied. Then the functional J satisfies the PS condition.Proof.
LetJ
u
(
k
) be a bounded sequence from below, that is, there exists a constant b
1
>
0 such that (37)
-
b
1
≤
J
u
k
,
∀
k
∈
N
.Due to the proof of Lemma 4, it is obvious that (38)
-
b
1
≤
J
u
k
≤
8
-
β
u
k
2
+
m
T
ζ
,
∀
k
∈
N
.Thus, (39)
u
k
2
≤
b
1
+
m
T
ζ
β
-
8
.That is, u
(
k
) is a bounded sequence in the finite-dimensional space E
m
T. Consequently, it has a convergent subsequence. The proof is finished.Lemma 6.
Ifu is a critical point of J
(
u
) on E
~
m
T, then u is a critical point of J
(
u
) on E
m
T.In a similar fashion to the proof of Lemma4 and the process in [22], we can prove Lemma 6. The detailed proof is omitted.Set(40)
Ω
l
=
-
m
32
s
i
n
4
ω
l
/
2
m
-
2
μ
∑
n
=
1
m
T
f
2
n
,
0
.Lemma 7.
Assume that the hypotheses(
F
4
)
–
(
F
6
) are satisfied. If u is a critical point of J
(
u
) on E
~
m
T, J
(
u
)
<
Ω
τ
m, then u has minimal period m
T.Proof.
If not, there is a positive integerl such that u has minimal period m
T
/
l. By (
F
5
), l
≥
τ
m. For any u
∈
E
~
m
T, (41)
u
n
=
∑
j
=
1
m
T
-
l
/
2
l
b
j
sin
⁡
ω
l
j
m
n
,and then (42)
J
u
=
1
2
x
∗
M
x
-
∑
n
=
1
m
T
F
n
,
u
n
≥
2
s
i
n
2
ω
l
2
m
x
2
-
∑
n
=
1
m
T
F
n
,
u
n
,where x
=
(
Δ
u
1
,
Δ
u
2
,
…
,
Δ
u
m
T
)
∗. Since (43)
x
2
=
∑
n
=
1
m
T
u
n
+
1
-
u
n
,
u
n
+
1
-
u
n
=
u
∗
M
u
≥
4
s
i
n
2
ω
l
2
m
u
2
,we get (44)
J
u
≥
8
s
i
n
4
ω
l
2
m
u
2
-
∑
n
=
1
m
T
f
2
n
,
0
1
/
2
u
-
μ
2
u
2
≥
-
1
32
s
i
n
4
ω
l
/
2
m
-
2
μ
∑
n
=
1
m
T
f
2
n
,
0
≥
-
m
32
s
i
n
4
ω
l
/
2
m
-
2
μ
∑
n
=
1
m
T
f
2
n
,
0
.Then (45)
J
u
≥
Ω
l
≥
Ω
τ
m
,which contradicts J
(
u
)
<
Ω
τ
m. The proof is complete.


## 3. Proof of Results

Proof of Theorem1.
It comes from Lemma4 that the functional J is bounded from above on E
m
T.
Take(46)
k
0
=
s
u
p
u
∈
E
m
T
J
u
.
On one hand, there exists a sequenceu
(
k
) on E
m
T such that (47)
k
0
=
l
i
m
k
→
∞
J
u
k
.
On the other hand, from (36), we get (48)
J
u
≤
8
-
β
u
2
+
m
T
ζ
≤
m
T
ζ
,
∀
u
∈
E
m
T
.Thus, l
i
m
u
→
+
∞
J
(
u
)
=
-
∞, which implies that u
(
k
) is bounded. Therefore, u
(
k
) has a convergent subsequence, denoted by u
(
k
j
). Set (49)
u
¯
=
l
i
m
j
→
+
∞
u
k
j
.Due to the continuity of J
(
u
), it is obvious that J
(
u
¯
)
=
k
0. That is, u
¯ is a critical point of J on E
m
T.
Assumption(
F
2
) implies that, for any u
∈
D, u
≤
ϵ
1, (50)
J
u
=
1
2
x
∗
M
x
-
∑
n
=
1
m
T
F
n
,
u
n
≥
1
2
x
∗
M
x
-
α
∑
n
=
1
m
T
u
n
2
≥
2
s
i
n
2
π
m
T
x
2
-
α
u
2
,where x
=
(
Δ
u
1
,
Δ
u
2
,
…
,
Δ
u
m
T
)
∗. Since (51)
x
2
=
∑
n
=
1
m
T
u
n
+
1
-
u
n
,
u
n
+
1
-
u
n
=
u
∗
M
u
≥
4
s
i
n
2
π
m
T
u
2
,then (52)
J
u
≥
8
s
i
n
4
π
m
T
-
α
u
2
.Let σ
=
8
s
i
n
4
π
/
m
T
-
α
ϵ
1
2. Therefore, the functional (53)
J
∂
B
ϵ
1
∩
D
≥
σ
.Thus, we have proved that k
0
=
s
u
p
u
∈
E
m
T
J
(
u
)
≥
σ
>
0. At the same time, we have also proved that there exist constants σ
>
0 and ϵ
1
>
0 such that J
|
∂
B
ϵ
1
∩
D
≥
σ.
By∑
n
=
1
m
T
‍
Δ
2
u
n
-
1
2
=
0, ∀
u
∈
C and assumption (
F
1
), (54)
J
u
=
1
2
∑
n
=
1
m
T
Δ
2
u
n
-
1
2
-
∑
n
=
1
m
T
F
n
,
u
n
=
-
∑
n
=
1
m
T
F
n
,
u
n
≤
0
.Then u
¯
∉
C and the critical point u
¯ of J corresponding to the critical value c
0 is a nontrivial periodic solution of (1).
Letv
∈
∂
B
1
∩
D. Then, for any ψ
∈
C and t
∈
R, let u
=
t
v
+
ψ. Then (55)
J
u
=
1
2
∑
n
=
1
m
T
Δ
2
u
n
,
Δ
2
u
n
-
∑
n
=
1
m
T
F
n
,
u
n
≤
t
2
2
∑
n
=
1
m
T
Δ
2
v
n
,
Δ
2
v
n
-
∑
n
=
1
m
T
F
n
,
t
v
n
+
ψ
n
≤
t
2
2
Υ
∗
M
Υ
-
∑
n
=
1
m
T
β
t
v
n
+
ψ
n
2
-
ζ
≤
2
t
2
Υ
2
-
β
∑
n
=
1
m
T
t
v
n
+
ψ
n
2
+
m
T
ζ
=
2
t
2
Υ
2
-
β
t
2
-
β
ψ
2
+
m
T
ζ
,where Υ
=
(
Δ
v
1
,
Δ
v
2
,
…
,
Δ
v
m
T
)
∗. Since (56)
Υ
2
=
∑
n
=
1
m
T
v
n
+
1
-
v
n
,
v
n
+
1
-
v
n
=
v
∗
M
v
≤
4
,then (57)
J
u
≤
8
-
β
t
2
-
β
ψ
2
+
m
T
ζ
≤
-
β
ψ
2
+
m
T
ζ
.Therefore, there is a constant χ
>
ϵ
1
>
0 such that, for any u
∈
∂
Φ, J
(
u
)
≤
0, where Φ
=
(
B
¯
χ
∩
C
)
⊕
{
t
v
∣
0
<
t
<
χ
}. By the Linking Theorem, J possesses a critical value c
≥
σ
>
0, where (58)
c
=
i
n
f
h
∈
Γ
s
u
p
u
∈
Φ
J
h
u
,and Γ
=
{
h
∈
C
Φ
¯
,
E
m
T
∣
h
|
∂
Φ
=
I
}.
Similar to the proof of [18], we can prove that (1) has at least two m
T-periodic nontrivial solutions. For simplicity, its proof is omitted.
By(
F
4
), for |
u
|
≤
ρ, we have (59)
F
n
,
u
=
f
n
,
0
u
+
1
2
∂
2
F
n
,
θ
u
∂
u
2
u
2
≥
f
n
,
0
u
+
ν
2
u
2
.Then, for |
u
|
≤
ρ, (60)
J
u
=
1
2
∑
n
=
1
m
T
Δ
2
u
n
,
Δ
2
u
n
-
∑
n
=
1
m
T
F
n
,
u
n
≤
1
2
∑
n
=
1
m
T
Δ
2
u
n
,
Δ
2
u
n
-
ν
2
∑
n
=
1
m
T
u
n
2
-
∑
n
=
1
m
T
f
n
,
0
u
n
.
Takeu
n
=
ρ
sin
⁡
ω
/
m
n. From (
F
1
), we have (61)
f
n
,
0
=
∑
j
=
1
T
-
1
/
2
a
j
sin
⁡
2
j
π
T
n
=
∑
j
=
1
T
-
1
/
2
a
j
sin
⁡
2
j
π
m
T
m
n
,where a
j is a constant. Note that m
>
1; we get (62)
∑
n
=
1
m
T
f
n
,
0
u
n
=
∑
j
=
1
T
-
1
/
2
ρ
a
j
∑
n
=
1
m
T
sin
⁡
2
j
π
m
T
m
n
·
sin
⁡
2
π
m
T
n
=
0
.Therefore, we have (63)
J
u
≤
32
s
i
n
4
ω
2
m
-
2
ν
u
2
.Since (64)
u
=
ρ
m
π
ω
1
/
2
,then (65)
J
u
=
32
s
i
n
4
ω
/
2
m
-
2
ν
ρ
2
m
π
ω
<
Ω
τ
m
.It comes from Lemma 7 that the desired result is obtained.Remark 8.
Similarly to the above argument, we can also prove Theorem2. For simplicity, we omit its proof.



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*Source: 4376156-2018-05-10.xml*