Hindawi Publishing Corporation
Discrete Dynamics in Nature and Society
Volume 2009, Article ID 760328, 9 pages
doi:10.1155/2009/760328
Research Article
Periodic Solutions for a System of
Difference Equations
Shugui Kang1, 2 and Bao Shi2
1
2
College of Mathematics and Computer Science, Shanxi Datong University, Datong, Shanxi 037009, China
Department of Basic Sciences, Naval Aeronautical Engineering Institute, Yantai, Shandong 264001, China
Correspondence should be addressed to Shugui Kang, dtkangshugui@126.com
Received 9 January 2009; Accepted 8 March 2009
Recommended by Guang Zhang
This paper deals with the second-order nonlinear systems of difference equations, we obtain the
existence theorems of periodic solutions. The theorems are proved by using critical point theory.
Copyright q 2009 S. Kang and B. Shi. 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
Let N, Z, R be the set of all natural numbers, integers, and real numbers, respectively. For
a, b ∈ Z, note that Za, b {a, a 1, . . . , b}, where a ≤ b.
In this paper, we consider the existence of periodic solutions for the system of
difference equations of the form
δ δ
Δ pn1 Δxn−11
qn1 xn1 f1 n, Xn ,
δ δ
qn2 xn2 f2 n, Xn ,
Δ pn2 Δxn−12
..
.
δ δ
qnk xnk fk n, Xn ,
Δ pnk Δxn−1k
1.1
which can be recorded as
δ δ
T
Δ P n ΔXn−1
Qn XnT f n, Xn ,
n ∈ Z,
1.2
2
Discrete Dynamics in Nature and Society
where k is a positive integer,
⎛
pn1
0
···
0
⎞
⎟
⎜
⎜ 0 pn2 · · · 0 ⎟
⎟,
⎜
Pn ⎜
⎟
⎠
⎝···
0 0 · · · pnk
⎛
qn1 0 · · ·
⎜
⎜ 0 qn2 · · ·
Qn ⎜
⎜· · ·
⎝
0
0
0
⎞
⎟
0 ⎟
⎟,
⎟
⎠
· · · qnk
1.3
and P nω P n > 0 i.e., pn1 > 0, pn2 > 0, . . . , pnk > 0, Qnω Qn , f f1 , f2 , . . . , fk T ,
fi fi n, Xn fi n, xn1 , xn2 , . . . , xnk , fn ω, U fn, U for any n, U ∈ Z × Rk ,
ω > 0 is a positive integer, −1δ −1, δ is the ratio of odd positive integers, ΔXnT T
T
T
T
−XnT xn11 − xn1 , xn12 − xn2 , . . . , xn1k − xnk T , Δ2 Xn−1
ΔΔXn−1
ΔXnT −ΔXn−1
.
Xn1
δ
δ
δ
δ
k
δ
For U u1 , u2 , . . . , uk ∈ R , define U u1 , u2 , . . . , uk . |U| |u1 |, |u2 |, . . . , |uk |, |U| |u1 |δ , |u2 |δ , . . . , |uk |δ . A sequence X {Xn }n∈Z is a ω-periodic solution of 1.2 if substitution
of it into 1.2 yields an identity for all n ∈ Z.
In 1, 2, the qualitative behavior of linear difference equations
Δ pn Δxn qn xn 0
1.4
has been investigated. In 3, the nonlinear difference equation
Δ pn Δxn−1 qn xn f n, xn
1.5
has been considered. In 4, by critical point method, the existence of periodic and
subharmonic solutions of equation
Δ2 xn−1 f n, xn 0,
n∈Z
1.6
has been studied. Other interesting results can been found in 5–8. In 9, the authors
consider the existence of periodic solutions for second-order nonlinear difference equation
δ qn xnδ f n, xn ,
Δ pn Δxn−1
n ∈ Z,
1.7
using critical point theory, obtaining some new results. It is a discrete analogues of differential
equation
ptφ u ft, u 0.
1.8
They do have physical applications in the study of nuclear physics, gas aerodynamics, and so
on see 10, 11. In this paper, we obtain some new results of existence of periodic solution
for the second-order nonlinear system of difference equations by using critical point theory.
We remark, however, the result in 9 is only good for 1.7 which is much less general than
our results in what follows.
Discrete Dynamics in Nature and Society
3
2. Some Basic Lemmas
Let E be a real Hilbert space, I ∈ C1 E, R mean that I is continuously Fréchet differentiable
functional defined on E. I is said to be satisfying Palais-Smale condition P-S condition if
any bounded sequence {Iun } and I un → 0 n → ∞ possess a convergent subsequence
in E. Let Bρ be the open ball in E with radius ρ and centered at θ, and let ∂Bρ denote its
boundary, θ is null element of E.
Lemma 2.1 see 12. Let E be a real Hilbert space, and assume that I ∈ C1 E, 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 ∈ ∂Bρ , where Bρ {x ∈
E : x < ρ};
∈Bρ such that Ix0 ≤ 0.
I2 I0 ≤ 0 and there exists x0 /
Then c infh∈Γ sups∈0,1 Ihs is a positive critical value of I, where
Γ {h ∈ C0, 1, X : h0 θ, h1 x0 }.
2.1
Let Ω∗ be the set of sequences
X Xn n∈Z . . . , X−n , . . . , X−1 , X0 , X1 , . . . , Xn , . . . ,
2.2
where Xn xn1 , xn2 , . . . , xnk ∈ Rk , that is,
Ω∗ X Xn n∈Z : Xn ∈ Rk , n ∈ Z .
2.3
For any X, Y ∈ Ω∗ , a, b ∈ R, aX bY is defined by
∞
aX bY aXn bYn n−∞ ,
2.4
then Ω∗ is a vector space. For given positive integer ω, Eω is defined as a subspace of Ω∗ by
Eω X Xn ∈ Ω∗ : Xnω Xn , n ∈ Z .
2.5
Obviously, Eω is isomorphic to Rkω , for any X, Y ∈ Eω , defined inner product
X, Y ω
Xi , Y i ,
2.6
i1
by which the norm · can be induced by
X ω Xi 2
i1
1/2
,
X ∈ Eω .
2.7
4
Discrete Dynamics in Nature and Society
1/2
where Xi kj1 |xij |2 . It is obvious that Eω with the inner product defined by 2.6 is a
finite-dimensional Hilbert space and linearly homeomorphic to Rkω . Define the functional J
on Eω as follows:
JX ω
ω
ω
δ1 1 1 Pn , ΔXn−1
−
Qn , Xnδ1 F n, Xn ,
δ 1 n1
δ 1 n1
n1
X ∈ Eω ,
2.8
where Fn, Xn such that ∇U Fn, U fn, U, that is,
∂ F n, u1 , u2 , . . . , uk
fi n, U fi n, u1 , u2 , . . . , uk ∂ui
2.9
for any n, U ∈ Z1, ω × Rk , Pn pn1 , pn2 , . . . , pnk , Qn qn1 , qn2 , . . . , qnk . Clearly J ∈
C1 Eω , R, and for any X {Xn }n∈Z ∈ Eω , by X0 Xω and X1 Xω1 , we have
δ δ
∂JX
−Δ pnl Δxn−1l
− qnl xnl fl n, Xn ,
∂xnl
l ∈ Z1, k, n ∈ Z1, ω.
2.10
Thus X {Xn }n∈Z is a critical point of J on Eω J X 0 if and only if
δ δ
Δ pnl Δxn−1l
qnl xnl fl n, Xn ,
l ∈ Z1, k, n ∈ Z1, ω.
2.11
That is,
δ δ
T
Δ P n ΔXn−1
Qn XnT f n, Xn ,
n ∈ Z.
2.12
By the periodicity of Xn and fn, Xn in the first variable n, we know that if X {Xn }n∈Z ∈ Eω
is a critical point of the real functional J defined by 2.8, then it is a periodic solution of 1.2.
For X {Xn }n∈Z ∈ Eω , Xn xn1 , xn2 , . . . , xnk ∈ Rk , r > 1, denote
1/r
ω r
Xi
Xr ,
i1
Xn r
1/r
k xni r
.
2.13
i1
Clearly, X2 X, Xn 2 Xn . Because of ·r1 and ·r2 being equivalent when r1 , r2 > 1,
so there exist constants c1 , c2 , c3 , c4 , 1 , 2 , 3 , and 4 such that c2 ≥ c1 > 0, c4 ≥ c3 > 0, 2 ≥
1 > 0, and 4 ≥ 3 > 0,
c1 X ≤ Xδ1 ≤ c2 X,
c3 X ≤ Xβ ≤ c4 X,
1 Xn ≤ Xn δ1 ≤ 2 Xn ,
3 Xn ≤ Xn β ≤ 4 Xn ,
for all X ∈ Eω , δ > 0, and β > 1.
2.14
Discrete Dynamics in Nature and Society
5
Lemma 2.2. Suppose that
F1 there exist constants a1 > 0, a2 > 0, β > δ 1 such that
Fn, U ≤ −a1 Uβ a2
2.15
for any n, U ∈ Z1, ω × Rk ;
F2 qni ≤ 0,
n ∈ Z, i ∈ Z1, k.
2.16
Then
JX ω
ω
ω
δ1 1 1 Pn , ΔXn−1
−
Qn , Xnδ1 F n, Xn
δ 1 n1
δ 1 n1
n1
2.17
satisfies P-S condition.
l
l
l
l
l
Proof. For any sequence {X l } {. . . , X−n , . . . , X−1 , X0 , X1 , . . . , Xn , . . .} ∈ Eω , JX l is
bounded and J X l → 0 l → ∞. Then there exists a positive constant M > 0, such
that |JX l | ≤ M. From F1 , we have
−M ≤ JX l ω l
l δ1 1 l δ1
Pn , Xn − Xn−1
− Qn , Xn
δ 1 n1
ω
l F n, Xn
n1
ω
l δ1 1 l δ1
2δ1 Pn , Xn Xn−1 δ 1 n1
ω ω
l δ1 1 l −
Qn , Xn F n, Xn
δ 1 n1
n1
ω ω δ1 l δ1 l δ1 1 2
Pn Pn1 , Xn −
Qn , Xn ≤
δ 1 n1
δ 1 n1
≤
ω
l F n, Xn
n1
≤
ω ω l δ1 l δ1 1 2
Pn Pn1 , Xn −
Qn , Xn δ 1 n1
δ 1 n1
δ1
− a1
ω β
Xnl a2 ω
n1
ω l δ1 β
1 2δ1 Pn Pn1 − Qn , Xn − a1 X l β a2 ω.
δ 1 n1
2.18
6
Discrete Dynamics in Nature and Society
Set
A0 max
δ1 2
pni pn1i − qni .
n∈Z1,ω,i∈Z1,k
2.19
Then A0 > 0, and
−M ≤ J X l
≤
ω δ1
A0 Xnl − a1 X l β a2 ω
β
δ1
δ 1 n1
≤
ω δ1
A0 δ1
2
Xnl − a1 X l β a2 ω
β
δ 1 n1
β
A0 δ1
2 l δ1
X δ1 − a1 X l β a2 ω.
δ1
2.20
Because of β > δ 1, and β − δ − 1/β δ 1/β 1, in view of Hölder inequality, we have
δ1/β
ω ω δ1
β
l β−δ−1/β
Xnl Xn
≤ω
.
2.21
l β
X ≥ ωδ1−β/δ1 X l β .
β
δ1
2.22
n1
n1
Thus
Then we have
−M ≤ JX l ≤
β
A0 δ1
2 l δ1
X δ1 − a1 X l β a2 ω
δ1
≤
β
A0 δ1
2 l δ1
X δ1 − a1 ωδ1−β/δ1 X l δ1 a2 ω.
δ1
2.23
Thus, for any l ∈ N,
β
A0 δ1
2 l δ1
a1 ωδ1−β/δ1 X l δ1 −
X δ1 ≤ M a2 ω.
δ1
2.24
Because of β > δ 1, it is easily seen that the inequality 2.24 implies that {X l } is a bounded
sequence in Eω . Thus {X l } possesses convergent subsequences. The proof is complete.
Discrete Dynamics in Nature and Society
7
3. Main Result
Theorem 3.1. Suppose that condition (F1 ) holds, and
F3 for each n ∈ Z,
lim
Fn, U
U → 0
Uδ1
0;
3.1
F4 for any i ∈ Z1, k, n ∈ Z1, ω,
qni < 0;
3.2
F5 Fn, θ 0.
Then 1.2 has at least two nontrivial ω-periodic solutions.
Proof. By Lemma 2.2, J satisfies P-S condition. Next, we will verify the conditions I1 and
I2 of Lemma 2.1. By F3 , there exists ρ > 0, such that
|Fn, U| ≤ −
qmax δ1
1
Uδ1
2δ 1
3.3
for any U < ρ and n ∈ Z1, ω, where qmax maxn∈Z1,ω, i∈Z1,k qni < 0. Thus
JX ≥ −
ω
ω
1 Qn , Xnδ1 F n, Xn
δ 1 n1
n1
≥−
ω ω q δ1 qmax Xn δ1 max 1
Xn δ1
δ1
δ 1 n1
2δ 1 n1
≥−
ω ω qmax δ1
q δ1 1
Xn δ1 max 1
Xn δ1
δ 1 n1
2δ 1 n1
−
qmax δ1
1 δ1
X δ1
2δ 1
≥−
δ1
qmax δ1
1 c1
Xδ1
2δ 1
3.4
δ1
δ1
for any X ∈ Eω with X ≤ ρ. We choose a −δ1
, then we have
1 c1 qmax /2δ 1ρ
JX|∂Bρ ≥ a > 0,
that is, the condition I1 of Lemma 2.1 holds.
3.5
8
Discrete Dynamics in Nature and Society
Obviously, J0 0. For any given V ∈ Eω with V 1 and constant α > 0,
JαV ω ω
ω
δ1
1 1 Pn , αVn − αVn−1
−
Qn , αVn δ1 Fn, αVn δ 1 n1
δ 1 n1
n1
ω δ1
δ1
1 pn1 αvn1 − αvn−11
pn2 αvn2 − αvn−12
δ 1 n1
δ1 · · · pnk αvnk − αvn−1k
−
ω
δ1
δ1
δ1 1 qn2 αvn2
· · · qnk αvnk
qn1 αvn1
δ 1 n1
ω
F n, αVn
n1
≤
ω
1 pn1 2αδ1 pn2 2αδ1 · · · pnk 2αδ1
δ 1 n1
ω
1 −
qn1 αδ1 qn2 αδ1 · · · qnk αδ1
δ 1 n1
− a1
3.6
ω αVn β a2 ω
n1
≤
ω
ω β
δ1 δ1
1 2 Pn 1 Qn 1 α − a1 αβ Vn a2 ω
δ 1 n1
n1
≤
ω
δ1 δ1
1 β
2 Pn 1 Qn 1 α − a1 αβ V β a2 ω
δ 1 n1
≤
ω
δ1 δ1
1 β
2 Pn 1 Qn 1 α − a1 c3 αβ a2 ω
δ 1 n1
−→ −∞,
α −→ ∞.
Thus we can choose a sufficiently large α such that α > ρ, and X αV ∈ Eω , JX < 0.
According to Lemma 2.1, there exists at least one critical value c ≥ a > 0. We suppose that X ∗
is a critical point corresponding to c, then JX ∗ c and J X ∗ 0.
By similar argument of Lemma 2.2, we know that JX is bounded from above, so
0. If
there exists X ∗∗ ∈ Eω such that JX ≤ JX ∗∗ cmax for any X ∈ Eω . Obviously, X ∗∗ /
X ∗ , then the proof is complete. Otherwise, X ∗∗ X ∗ , c cmax . In view of Lemma 2.1,
X ∗∗ /
c inf sup Jhs,
h∈Γ s∈0,1
3.7
where Γ {h ∈ C0, 1, Eω : h0 θ, h1 X}. Then cmax maxs∈0,1 Jhs for any h ∈ Γ
holds. In view of the continuity of Jhs in s, Jθ ≤ 0, and JX < 0, we know that there
Discrete Dynamics in Nature and Society
9
exists some s0 ∈ 0, 1 such that Jhs0 cmax . If we choose h1 , h2 ∈ Γ such that
{h1 s : s ∈ 0, 1} ∩ {h2 s : s ∈ 0, 1} φ,
3.8
then there exist s1 , s2 ∈ 0, 1 such that Jh1 s1 Jh2 s2 cmax . Then J possesses
two different critical points Y h1 s1 and Ž h2 s2 in Eω , hence, we obtain at least two
nontrivial critical points which correspond to the critical value cmax . Thus 1.2 possesses at
least two nontrivial ω-periodic solutions. The proof is complete.
Acknowledgments
This work is supported by Natural Science Foundation of Shanxi Province 2008011002-1
and Shanxi Datong University and by the Development Foundation of Higher Education
Department of Shanxi Province.
References
1 C. D. Ahlbrandt and A. C. Peterson, Discrete Hamiltonian Systems: Difference Equations, Continued
Fractions, and Riccati Equations, vol. 16 of Kluwer Texts in the Mathematical Sciences, Kluwer Academic
Publishers, Dordrecht, The Netherlands, 1996.
2 S.-S. Cheng, H.-J. Li, and W. T. Patula, “Bounded and zero convergent solutions of second-order
difference equations,” Journal of Mathematical Analysis and Applications, vol. 141, no. 2, pp. 463–483,
1989.
3 T. Peil and A. Peterson, “Criteria for C-disfocality of a selfadjoint vector difference equation,” Journal
of Mathematical Analysis and Applications, vol. 179, no. 2, pp. 512–524, 1993.
4 Z. Guo and J. Yu, “Existence of periodic and subharmonic solutions for second-order superlinear
difference equations,” Science in China Series A, vol. 46, no. 4, pp. 506–515, 2003.
5 R. P. Agarwal and P. J. Y. Wong, Advanced Topics in Difference Equations, vol. 404 of Mathematics and Its
Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1997.
6 M. Cecchi, Z. Došlá, and M. Marini, “Positive decreasing solutions of quasi-linear difference
equations,” Computers & Mathematics with Applications, vol. 42, no. 10-11, pp. 1401–1410, 2001.
7 P. J. Y. Wong and R. P. Agarwal, “Oscillations and nonoscillations of half-linear difference equations
generated by deviating arguments,” Computers & Mathematics with Applications, vol. 36, no. 10–12, pp.
11–26, 1998.
8 P. J. Y. Wong and R. P. Agarwal, “Oscillation and monotone solutions of second order quasilinear
difference equations,” Funkcialaj Ekvacioj, vol. 39, no. 3, pp. 491–517, 1996.
9 X. Cai and J. Yu, “Existence theorems of periodic solutions for second-order nonlinear difference
equations,” Advances in Difference Equations, vol. 2008, Article ID 247071, 11 pages, 2008.
10 R. P. Agarwal and P. Y. H. Pang, “On a generalized difference system,” Nonlinear Analysis: Theory,
Methods & Applications, vol. 30, no. 1, pp. 365–376, 1997.
11 M. Marini, “On nonoscillatory solutions of a second-order nonlinear differential equation,” Bollettino
della Unione Matemàtica Italiana, vol. 3, no. 1, pp. 189–202, 1984.
12 J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems, vol. 74 of Applied Mathematical
Sciences, Springer, New York, NY, USA, 1989.