Available online at www.tjnsa.com
J. Nonlinear Sci. Appl. 9 (2016), 1505–1514
Research Article
Existence of periodic solutions for second-order
nonlinear difference equations
Zhiguo Rena , Jie Lib , Haiping Shic,∗
a
Department of Information Engineering, Jieyang Vocational and Technical College, Jieyang 522000, China.
b
Quality Control Office, Zhongshan Torch College, Zhongshan 528436, China.
c
Modern Business and Management Department, Guangdong Construction Vocational Technology Institute, Guangzhou 510440,
China.
Communicated by C. Park
Abstract
By using the critical point method, the existence of periodic solutions for second-order nonlinear difference equations is obtained. The proof is based on the Saddle Point Theorem in combination with variational
technique. The problem is to solve the existence of periodic solutions of second-order nonlinear difference
c
equations. One of our results obtained complements the result in the literature. 2016
All rights reserved.
Keywords: Existence, periodic solutions, second-order, nonlinear difference equations, discrete variational
theory.
2010 MSC: 39A23.
1. Introduction
Recently, the theory of nonlinear difference equations has been widely used to study discrete models
appearing in many fields such as computer science, economics, neural networks, ecology, cybernetics, etc.
For the general background of difference equations, one can refer to the monographs [1, 2, 3]. For the past
twenty years, there has been much progress on the qualitative properties of difference equations, which
included result in stability and attractive [13, 15] and result in oscillation and other topics, see [1, 2, 3, 8,
9, 10, 12, 21, 22, 23, 24, 25]. Therefore, it is worthwhile to explore this topic.
Let N, Z and R denote the sets of all natural numbers, integers and real numbers respectively. For any
a, b ∈ Z, define Z(a) = {a, a + 1, · · · }, Z(a, b) = {a, a + 1, · · · , b} when a ≤ b. Let the symbol * denote the
transpose of a vector.
∗
Corresponding author
Email addresses: rzhg-1225@163.com (Zhiguo Ren), 53457698@qq.com (Jie Li), shp7971@163.com (Haiping Shi)
Received 2015-09-27
Z. Ren, J. Li, H. Shi, J. Nonlinear Sci. Appl. 9 (2016), 1505–1514
The present paper considers the following second-order nonlinear difference equation
∆ pn (∆un−1 )δ + qn uδn + f (n, un+1 , un , un−1 ) = 0, n ∈ Z,
1506
(1.1)
where ∆ is the forward difference operator ∆un = un+1 − un , ∆2 un = ∆(∆un ), δ > 0 is the ratio of
odd positive integers, {pn } and {qn } are real sequences, f ∈ C(Z × R3 , R), T is a given positive integer,
pn+T = pn > 0, qn+T = qn < 0, f (n + T, v1 , v2 , v3 ) = f (n, v1 , v2 , v3 ).
Eq. (1.1) can be considered as a discrete analogue of a special case of the following second-order nonlinear
functional differential equation
0
p(t)ϕ(u0 ) + f (t, u(t + 1), u(t), u(t − 1)) = 0, t ∈ R.
(1.2)
Eq. (1.2) includes the following equation
0
p(t)ϕ(u0 ) + f (t, u(t)) = 0, t ∈ R,
which has arisen in the study of fluid dynamics, combustion theory, gas diffusion through porous media,
thermal self-ignition of a chemically active mixture of gases in a vessel, catalysis theory, chemically reacting
systems, and adiabatic reactor [5, 7, 11]. Equations similar in structure to (1.2) arise in the study of the
existence of solitary waves of lattice differential equations, see Smets and Willem [14].
When δ = 1, and f (n, un+1 , un , un−1 ) = 0, (1.1) becomes
∆ (pn ∆un−1 ) + qn un = 0,
(1.3)
which has been extensively investigated by many authors [1, 3, 6], for results on oscillation, asymptotic
behavior, boundary value problems, disconjugacy and disfocality.
In [21], the periodic solutions of second-order self-adjoint difference equation
∆ (pn ∆un−1 ) + qn un = f (n, un )
has been considered.
When f (n, un+1 , un , un−1 ) = 0, n ∈ Z(0), (1.1) reduces to the following equation
∆ pn (∆un−1 )δ + qn uδn = 0,
(1.4)
(1.5)
which has been studied in [1, 6, 22] for results on oscillation, asymptotic behavior and the existence of
positive solutions.
Moreover, if qn uδn + f (n, un+1 , un , un−1 ) = qn g(un ) + rn , (1.1) has been considered in [16] for oscillatory
properties of its all solutions.
When β > δ +1, in Theorem 3.2, Cai and Yu [4] have obtained some sufficient conditions for the existence
of periodic solutions of the following nonlinear difference equation
∆ pn (∆un−1 )δ + qn uδn = f (n, un ), n ∈ Z.
(1.6)
Furthermore, [4] is the only paper we found which deals with the problem of periodic solutions to secondorder difference equation (1.6). When β < δ + 1, can we still find the periodic solutions of (1.6)?
By using various methods and techniques, such as Schauder fixed point theorem, the cone theoretic fixed
point theorem, the method of upper and lower solutions, coincidence degree theory, a series of existence results of nontrivial solutions for differential equations have been obtained in [14, 16, 19]. Critical point theory
is also an important tool to deal with problems on differential equations [14, 19]. Because of applications in
many areas of difference equations [1, 2, 3], recently, a few authors have gradually paid attention to applying
critical point theory to deal with periodic solutions of discrete systems, see [8, 9, 10, 17, 21, 23]. Particularly,
Z. Ren, J. Li, H. Shi, J. Nonlinear Sci. Appl. 9 (2016), 1505–1514
1507
Guo and Yu [8, 9, 10] and Shi et al. [17] studied the existence of periodic solutions of second-order nonlinear
difference equations by using the critical point theory. However, to the best of our knowledge, when δ 6= 1
the results on periodic solutions of second-order nonlinear difference equation (1.1) are very scarce in the
literature (see[4]), because there are few known methods for considering the existence of periodic solutions
of discrete systems. Furthermore, since f in (1.1) depends on un+1 and un−1 , the traditional ways of establishing the functional in [8, 9, 10, 21, 23] are inapplicable to our case. The main purpose of this paper is
to give some sufficient conditions for the existence of periodic solutions to second-order nonlinear difference
equations. The main approach used in our paper are variational techniques and the Saddle Point Theorem.
In particular, one of our results obtained complements the result in the literature [4]. In fact, one can see
the Remark 1.4 for details. The motivation for the present work stems from the recent papers in [4, 23].
For basic knowledge on variational methods, we refer the reader to [14].
Let
p = min {pn }, p̄ = max {pn }, q = min {qn }, q̄ = max {qn }.
n∈Z(1,T )
n∈Z(1,T )
n∈Z(1,T )
n∈Z(1,T )
Now we state the main results of this paper.
Theorem 1.1. Assume that the following hypotheses are satisfied:
(F1 ) there exists a functional F (n, v1 , v2 ) ∈ C 1 (Z × R2 , R) such that
F (n + T, v1 , v2 ) = F (n, v1 , v2 ),
∂F (n − 1, v2 , v3 ) ∂F (n, v1 , v2 )
+
= f (n, v1 , v2 , v3 );
∂v2
∂v2
p
(F2 ) there exist constants R1 > 0 and 1 < α < 2 such that for n ∈ Z and v12 + v22 ≥ R1 ,
0<
∂F (n, v1 , v2 )
∂F (n, v1 , v2 )
α
v1 +
v2 ≤ (δ + 1)F (n, v1 , v2 );
∂v1
∂v2
2
(F3 ) there exist constants a1 > 0, a2 > 0 and 1 < γ ≤ α such that
q
γ (δ+1)
2
2
2
v1 + v2
F (n, v1 , v2 ) ≥ a1
− a2 , ∀(n, v1 , v2 ) ∈ Z × R2 .
Then for any given positive integer m > 0, (1.1) has at least one mT -periodic solution.
Remark 1.2. Assumption (F2 ) implies that for each n ∈ Z there exist constants a3 > 0 and a4 > 0 such that
p
α (δ+1)
2
v12 + v22
(F20 ) F (n, v1 , v2 ) ≤ a3
+ a4 , ∀(n, v1 , v2 ) ∈ Z × R2 .
In fact, let v = (v1 , v2 ) and ∇v F (n, v) be the gradient of F (n, v) in v. From (F2 ), we have
α
(δ+1)
∇v F (n,v)
v
2
·
≤
|v|
F (n,v)
|v| , for n ∈ Z and |v| ≥ R1 .
Thus,
α
(δ + 1)
d ln F (n, v)
≤ 2
,
d|v|
|v|
which implies
d
α
(ln F (n, v) − (δ + 1) ln |v|) ≤ 0,
d|v|
2
for n ∈ Z and |v| ≥ R1 .
Denote G = max{ln F (n, v) − α2 (δ + 1) ln |v| : n ∈ Z, |v| = R1 }. By (1.7),
ln F (n, v) − α2 (δ + 1) ln |v| ≤ G, for n ∈ Z and |v| ≥ R1 .
That is,
α
F (n, v) ≤ a3 |v| 2 (δ+1) , for n ∈ Z and |v| ≥ R1 ,
(1.7)
where a3 = eG .
Let a4 = max{|F (n, v)| : n ∈ Z, |v| ≤ R1 }. Then (F20 ) holds. If f (n, un+1 , un , un−1 ) = −f (n, un ), (1.1)
reduces to (1.6). Then, we have the following results.
Z. Ren, J. Li, H. Shi, J. Nonlinear Sci. Appl. 9 (2016), 1505–1514
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Theorem 1.3. Assume that the following hypotheses are satisfied:
(F4 ) there exists a functional F (n, v) ∈ C 1 (Z × R, R), F (n + T, v) = F (n, v) such that
∂F (n, v)
= f (n, v);
∂v
(F5 ) there exist constants R2 > 0 and 1 < α < 2 such that for n ∈ Z and
p
v12 + v22 ≥ R2 ,
α
(δ + 1)F (n, v) ≤ vf (n, v) < 0;
2
(F6 ) there exist constants a5 > 0, a6 > 0 and 1 < γ ≤ α such that
γ
F (n, v) ≤ −a5 |v| 2 (δ+1) + a6 , ∀(n, v) ∈ Z × R.
Then for any given positive integer m > 0, (1.6) has at least one mT -periodic solution.
Remark 1.4. When β > δ + 1, in Theorem 3.2, Cai and Yu [4] have obtained some criteria for the existence
of periodic solutions of (1.6). When β < δ + 1, we can still find the periodic solutions of (1.6). Hence,
Theorem 1.3 complements the existing one.
The rest of the paper is organized as follows. First, in Section 2, we shall establish the variational
framework associated with (1.1) and transfer the problem of the existence of periodic solutions of (1.1) into
that of the existence of critical points of the corresponding functional. Some related fundamental results
will also be recalled. Then, in Section 3, we shall complete the proof of the results by using the critical point
method. Finally, in Section 4, we shall give an example to illustrate the main result.
2. Variational structure and some lemmas
In order to apply the critical point theory, we shall establish the corresponding variational framework
for (1.1) and give some lemmas which will be of fundamental importance in proving our main results. We
start by some basic notations.
Let S be the set of sequences u = (· · · , u−n , · · · , u−1 , u0 , u1 , · · · , un , · · · ) = {un }+∞
n=−∞ , that is
S = {{un }|un ∈ R, n ∈ Z}.
For any u, v ∈ S, a, b ∈ R, au + bv is defined by
au + bv = {aun + bvn }+∞
n=−∞ .
Then S is a vector space.
For any given positive integers m and T , EmT is defined as a subspace of S by
EmT = {u ∈ S|un+mT = un , ∀n ∈ Z}.
Clearly, EmT is isomorphic to RmT . EmT can be equipped with the inner product
hu, vi =
mT
X
uj vj , ∀u, v ∈ EmT ,
(2.1)
j=1
by which the norm k · k can be induced by

kuk = 
mT
X
j=1
1
2
u2j 
, ∀u ∈ EmT .
(2.2)
Z. Ren, J. Li, H. Shi, J. Nonlinear Sci. Appl. 9 (2016), 1505–1514
1509
It is obvious that EmT with the inner product (2.1) is a finite dimensional Hilbert space and linearly
homeomorphic to RmT .
On the other hand, we define the norm k · ks on EmT as follows:

1
s
mT
X
s

,
(2.3)
kuks =
|uj |
j=1
for all u ∈ EmT and s > 1.
Since kuks and kuk2 are equivalent, there exist constants c1 , c2 such that c2 ≥ c1 > 0, and
c1 kuk2 ≤ kuks ≤ c2 kuk2 , ∀u ∈ EmT .
(2.4)
Clearly, kuk = kuk2 . For all u ∈ EmT , define the functional J on EmT as follows:
J(u) = −
mT
mT
mT
n=1
n=1
n=1
X
1 X
1 X
pn+1 (∆un )δ+1 +
qn uδ+1
+
F (n, un+1 , un ),
n
δ+1
δ+1
mT
mT
n=1
n=1
X
1 X
J(u) := −H(u) +
qn uδ+1
+
F (n, un+1 , un ),
n
δ+1
where
(2.5)
mT
H(u) =
∂F (n − 1, v2 , v3 ) ∂F (n, v1 , v2 )
1 X
pn+1 (∆un )δ+1 ,
+
= f (n, v1 , v2 , v3 ).
δ+1
∂v2
∂v2
n=1
C 1 (EmT , R)
Clearly, J ∈
and for any u = {un }n∈Z ∈ EmT , by using u0 = umT , u1 = umT +1 , we can
compute the partial derivative as
∂J
= ∆ pn (∆un−1 )δ + qn uδn + f (n, un+1 , un , un−1 ).
∂un
Thus, u is a critical point of J on EmT if and only if
∆ pn (∆un−1 )δ + qn uδn + f (n, un+1 , un , un−1 ) = 0, ∀n ∈ Z(1, mT ).
Due to the periodicity of u = {un }n∈Z ∈ EmT and f (n, v1 , v2 , v3 ) in the first variable n, we reduce the
existence of periodic solutions of (1.1) to the existence of critical points of J on EmT . That is, the functional
J is just the variational framework of (1.1).
Let


2 −1 0 · · · 0 −1
 −1 2 −1 · · · 0
0 


 0 −1 2 · · · 0
0 


P =

 ··· ··· ··· ··· ··· ··· 
 0
0
0 · · · 2 −1 
−1
0
0
···
−1
2
be a mT × mT matrix. By matrix theory, we see that the eigenvalues of P are
2k
λk = 2 1 − cos
π , k = 0, 1, 2, · · · , mT − 1.
mT
Thus, λ0 = 0, λ1 > 0, λ2 > 0, · · · , λmT −1 > 0. Therefore,

2
λmin = min{λ1 , λ2 , · · · , λmT −1 } = 2 1 − cos mT
π ,



4,


when mT is even,
 λmax = max{λ1 , λ2 , · · · , λmT −1 } =
1
π , when mT is odd.
2 1 + cos mT
(2.6)
(2.7)
Z. Ren, J. Li, H. Shi, J. Nonlinear Sci. Appl. 9 (2016), 1505–1514
1510
Let
W = ker P = {u ∈ EmT |P u = 0 ∈ RmT }.
Then
W = {u ∈ EmT |u = {c}, c ∈ R}.
Let V be the direct orthogonal complement of EmT to W , i.e., EmT = V ⊕ W . For convenience, we
identify u ∈ EmT with u = (u1 , u2 , · · · , umT )∗ .
Let E be a real Banach space, J ∈ C 1 (E, R), i.e., J is a continuously Fréchet-differentiable functional
defined
the Palais-Smale condition
(P.S. condition for short) if any sequence
(k) on E. J is said
to(k)satisfy
0
(k)
u
⊂ E for which J u
is bounded and J u
→ 0(k → ∞) possesses a convergent subsequence
in E.
Let Bρ denote the open ball in E about 0 of radius ρ and let ∂Bρ denote its boundary.
Lemma 2.1 (Saddle Point Theorem [14]). Let E be a real Banach space, E = E1 ⊕ E2 , where E1 6= {0}
and is finite dimensional. Suppose that J ∈ C 1 (E, R) satisfies the P.S. condition and
(J1 ) there exist constants σ, ρ > 0 such that J|∂Bρ ∩E1 ≤ σ;
(J2 ) there exists e ∈ Bρ ∩ E1 and a constant ω ≥ σ such that Je+E2 ≥ ω.
Then J possesses a critical value c ≥ ω, where
c = inf
max J(h(u)), Γ = {h ∈ C(B̄ρ ∩ E1 , E) | h|∂Bρ ∩E1 = id}
h∈Γ u∈Bρ ∩E1
and id denotes the identity operator.
Lemma 2.2. Assume that (F1 ) − (F3 ) are satisfied. Then J satisfies the P.S. condition.
Proof. Let u(k) ⊂ EmT be such that J u(k)
is bounded and J 0 u(k) → 0 as k → ∞. Then there
exists a positive constant M1 such that J u(k) ≤ M1 .
For k large enough, we have
D E (k) 0 (k)
(k) ,u
≤ u .
J u
2
So
1 (k) u δ+1
2
E
D
1
≥ J u(k) −
J 0 u(k) , u(k)
δ+1



(k) (k)
(k)
(k)
mT
∂F
n
−
1,
u
,
u
∂F
n,
u
,
u
X
n
n
n−1
n+1
1 
(k)
F n, u(k)

=
−
· u(k)
· u(k)
n +
n
n+1 , un
δ+1
∂v2
∂v2
M1 +
n=1
mT
X
=


(k)
F n, u(k)
−
n+1 , un
n=1
1 
δ+1
(k)
(k)
∂F n, un+1 , un
(k)
· un+1 +
∂v1
(k)
(k)
∂F n, un+1 , un
∂v2

 .
· u(k)
n
Take
(
I1 =
r
n ∈ Z(1, mT )|
By (F2 ), we have
M1 +
1 (k) u δ+1
2
(k) 2
un+1
+
(k) 2
un
)
≥ R1
(
, I2 =
)
r
(k) 2
(k) 2
n ∈ Z(1, mT )|
un+1 + un
< R1 .
Z. Ren, J. Li, H. Shi, J. Nonlinear Sci. Appl. 9 (2016), 1505–1514
≥
mT
X

(k)
F n, un+1 , u(k)
−
n
n=1

−
≥
1
δ+1
X
1
δ+1
(k)
X

n∈I1
(k)
∂F n, un+1 , un

∂v1
n∈I2
(k)
(k)
∂F n, un+1 , un
(k)
· un+1 +
(k)
· un+1 +
∂v1
(k)
(k)
∂F n, un+1 , un
∂v2
1511
(k)
(k)
∂F n, un+1 , un
∂v2


· u(k)
n


· u(k)
n
mT
X
α X (k)
(k)
(k)
F n, un+1 , u(k)
−
F
n,
u
,
u
n
n+1 n
2
n=1
n∈I1


(k)
(k)
(k)
(k)
∂F
n,
u
,
u
∂F
n,
u
,
u
X
n
n
n+1
n+1
1
(k)


−
· un+1 +
· u(k)
n
δ+1
∂v1
∂v2
n∈I2
mT
α X (k)
= 1−
F n, un+1 , u(k)
n
2
n=1


(k)
(k)
(k)
(k)
∂F
n,
u
,
u
∂F
n,
u
,
u
X
n
n
n+1
n+1
1
(k)
(k)
 α (δ + 1)F n, u(k)
+
· un+1 −
· un(k)  .
−
n+1 , un
δ+1
2
∂v1
∂v2
n∈I2
1 ,v2 )
1 ,v2 )
v1 − ∂F (n,v
v2 with respect to the second and third
The continuity of α2 (δ + 1)F (n, v1 , v2 ) − ∂F (n,v
∂v1
∂v2
variables implies that there exists a constant M2 > 0 such that
∂F (n, v1 , v2 )
α
∂F (n, v1 , v2 )
(δ + 1)F (n, v1 , v2 ) −
v1 −
v2 ≥ −M2 ,
2
∂v1
∂v2
p
for n ∈ Z(1, mT ) and v12 + v22 ≤ R1 . Therefore,
M1 +
mT
1 α X 1
(k) (k)
mT M2 .
F n, un+1 , un(k) −
u ≥ 1 −
δ+1
2
δ+1
2
n=1
By (F3 ), we get
"
#γ
mT r
2 (δ+1) α
1 α X
1
(k) (k) 2
(k) 2
M1 +
a1
un+1 + un
− 1−
a2 mT −
mT M2
u ≥ 1 −
δ+1
2
2
δ+1
2
≥ 1−
where M3 = 1 −
α
2
α
2
a2 mT +
M1 +
Thus,
1−
a1
n=1
mT X
γ
(k) 2 (δ+1)
− M3 ,
un n=1
1
δ+1 mT M2 .
Combining with (2.4), we have
γ
γ
1 α
(δ+1) (k) 2 (δ+1)
(k) − M3 .
a1 c12
u ≥ 1 −
u δ+1
2
2
2
γ
γ
α
1 (δ+1) (k) 2 (δ+1)
(k) a1 c12
−
u u ≤ M1 + M3 .
2
δ+1
2
2
This implies that u(k) 2 is bounded on the finite dimensional space EmT . As a consequence, it has a
convergent subsequence.
Z. Ren, J. Li, H. Shi, J. Nonlinear Sci. Appl. 9 (2016), 1505–1514
1512
3. Proof of the main results
In this Section, we shall prove our main results by using the critical point theory.
Proof. By Lemma 2.2, J satisfies the P.S. condition. To apply the Saddle Point Theorem, it suffices to prove
that J satisfies the conditions (J1 ) and (J2 ).
For any w ∈ W , since H(w) = 0, we have
J(w) =
mT
mT
n=1
n=1
X
1 X
qn wnδ+1 +
F (n, wn+1 , wn ).
δ+1
By (F3 ),
γ
(δ+1)
mT q
X
2
2
2
J(w) ≥ a1
wn+1 + wn
− a2 mT ≥ −a2 mT.
n=1
Since

p δ+1

c
δ+1 1
mT
X
! 21 δ+1
(∆vn )2

≤ H (v) ≤

n=1
and
λmin kvk22 ≤
mT
X
p̄ δ+1 
c
δ+1 2
mT
X
! 12 δ+1
(∆vn )2

,
n=1
(∆vn )2 = v ∗ P v ≤ λmax kvk22 ,
n=1
we get
p δ+1 δ+1
p̄ δ+1 δ+1
2
2
c1 λmin
kvkδ+1
≤ H (v) ≤
c λmax
kvkδ+1
.
2
2
δ+1
δ+1 2
(3.1)
Besides,
kvkδ+1
≤q
qcδ+1
2
2
mT
X
vnδ+1 ≤
n=1
mT
X
qn vnδ+1 ≤ q̄
n=1
mT
X
kvkδ+1
.
vnδ+1 ≤ q̄cδ+1
1
2
n=1
Combining with (F20 ), (2.4), (3.1) and (3.2), for any v ∈ V , we have
J(v) = −H(v) +
mT
mT
n=1
n=1
X
1 X
qn vnδ+1 +
F (n, vn+1 , vn ),
δ+1
α (δ+1)
mT q
X
2
p δ+1 δ+1
q̄ δ+1
δ+1
δ+1
2
2
2
≤−
c1 λmin kvk2 +
c1 kvk2 + a3
vn+1 + vn
+ a4 mT
δ+1
δ+1
n=1
" mT
# α4 (δ+1)
X
α
p δ+1 δ+1
q̄
(δ+1)
2
2
≤−
c λmin
kvkδ+1
+
cδ+1 kvkδ+1
+ a3 c22
vn+1
+ vn2
+ a4 mT
2
2
δ+1 1
δ+1 1
n=1
α
α
p δ+1 δ+1
α
q̄ δ+1
(δ+1)
(δ+1)
δ+1
(δ+1)
2
2
2
4
≤−
c1 λmin
kvkδ+1
+
c
kvk
+
2
a
c
kvk
+ a4 mT.
3
1
2
2
2
2
δ+1
δ+1
Let µ = −a2 mT , since 1 < α < 2, there exists a constant ρ > 0 large enough such that
J(v) ≤ µ − 1 < µ, ∀v ∈ V, kvk2 = ρ.
Thus, by Lemma 2.1, Eq. (1.1) has at least one mT -periodic solution.
Remark 3.1. Due to Theorem 1.1, the conclusion of Theorem 1.3 is obviously true.
(3.2)
Z. Ren, J. Li, H. Shi, J. Nonlinear Sci. Appl. 9 (2016), 1505–1514
1513
4. Example
As an application of the main theorem, we give an example to illustrate our result.
Example 4.1. For all n ∈ Z, assume that
πn h
πn 2
2 i
(∆un−1 )3 + cos2
u3n + 6un ψ(n) u2n+1 + u2n + ψ(n − 1) u2n + u2n−1
= 0,
∆ sin2
T
T
(4.1)
where ψ is continuously differentiable and ψ(n) > 0, T is a given positive integer, ψ(n + T ) = ψ(n). We
have
h
2
2 i
f (n, v1 , v2 , v3 ) = 6v2 ψ(n) v12 + v22 + ψ(n − 1) v22 + v32
and
F (n, v1 , v2 ) = ψ(n) v12 + v22
Then
3
.
h
i
∂F (n − 1, v2 , v3 ) ∂F (n, v1 , v2 )
2
2 2
2
2 2
.
+
= 6v2 ψ(n) v1 + v2 + ψ(n − 1) v2 + v3
∂v2
∂v2
It is easy to verify all the assumptions of Theorem 1.1 are satisfied. Consequently, for any given positive
integer m > 0, (4.1) has at least one mT -periodic solution.
Acknowledgement
This project is supported by the National Natural Science Foundation of China (No. 11401121) and Department of Education of Guangdong Province for Excellent Young College Teacher of Guangdong Province.
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