Hindawi Publishing Corporation
Advances in Difference Equations
Volume 2010, Article ID 505906, 10 pages
doi:10.1155/2010/505906
Research Article
Asymptotic Behavior of Equilibrium Point for
a Family of Rational Difference Equations
Chang-you Wang,1, 2, 3 Qi-hong Shi,4 and Shu Wang3
1
College of Mathematics and Physics, Chongqing University of Posts and Telecommunications,
Chongqing 400065, China
2
Key Laboratory of Network Control and Intelligent Instrument, Chongqing University of
Posts and Telecommunications, Ministry of Education, Chongqing 400065, China
3
College of Applied Sciences, Beijing University of Technology, Beijing 100124, China
4
Fundamental Department, Hebei College of Finance, Baoding 071051, China
Correspondence should be addressed to Chang-you Wang, wangcy@cqupt.edu.cn
Received 7 August 2010; Accepted 19 October 2010
Academic Editor: Rigoberto Medina
Copyright q 2010 Chang-you Wang 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 following nonlinear difference equation xn1 li1 Asi xn−si /B k
C j1 xn−tj Dxn , n 0, 1, . . . , where the initial data x−m , x−m1 , . . . , x−1 , x0 ∈ R , m max{s1 , . . . , sl , t1 , . . . , tk }, s1 , . . . , sl , t1 , . . . , tk are nonnegative integers, and Asi , B, C, and D are
arbitrary positive real numbers. We give sufficient conditions under which the unique equilibrium
x 0 of this equation is globally asymptotically stable, which extends and includes corresponding
results obtained in the work of Çinar 2004, Yang et al. 2005, and Berenhaut et al. 2007. In
addition, some numerical simulations are also shown to support our analytic results.
1. Introduction
Difference equations appear naturally as discrete analogues and in the numerical solutions of
differential and delay differential equations, and they have applications in biology, ecology,
physics, and so forth 1. The study of properties of nonlinear difference equations has been
an area of intense interest in recent years. There has been a lot of work concerning the
globally asymptotic behavior of solutions of rational difference equations e.g., see 2–5.
In particular, Çinar 6 studied the properties of positive solution to
xn1 xn−1
,
1 xn xn−1
n 0, 1, . . . .
1.1
2
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Yang et al. 7 investigated the qualitative behavior of the recursive sequence
axn−1 bxn−2
,
c dxn−1 xn−2
xn1 n 0, 1, . . . .
1.2
More recently, Berenhaut et al. 8 generalized the result reported in 7 to
xn yn−k yn−m
,
1 yn−k yn−m
n 0, 1, . . . .
1.3
For more similar work, one can refer to 9–14 and references therein.
The main theorem in this paper is motivated by the above studies. The essential
problem we consider in this paper is the asymptotic behavior of the solutions for
l
xn1 Asi xn−si
Dxn ,
k
B C j1 xn−tj
i1
n 0, 1, . . . ,
1.4
with initial data x−m , x−m1 , . . . , x−1 , x0 ∈ R , m max{s1 , . . . , sl , t1 , . . . , tk }, s1 , . . . , sl , t1 , . . . , tk
are nonnegative integers, and Asi , B, C, and D are arbitrary positive real numbers. In addition,
some numerical simulations of the behavior are shown to illustrate our analytic results.
This paper proceeds as follows. In Section 2, we introduce some definitions and
preliminary results. The main results and their proofs are given in Section 3. Finally, some
numerical simulations are shown to support theoretical analysis.
2. Preliminaries and Notations
In this section, we prepare some materials used throughout this paper, namely, notations, the basic definitions, and preliminary results. We refer to the monographs of
Kocić and Ladas 2, and Kulenović and Ladas 3.
Let σ and κ be two nonnegative integers such that σ κ n. We usually write a vector
x x1 , x2 , . . . , xn with n components into x xσ , xκ , where xσ denotes a vector with
σ-components of x.
Lemma 2.1. Let I be some interval of real numbers and
f : I k1 −→ I
2.1
be a continuously differentiable function. Then, for every set of initial conditions x−k , x−k1 , . . . , x0 ∈
I,
xn1 fxn , xn−1 , . . . , xn−k ,
has a unique solution {xn }∞
n−k .
n 0, 1, . . .
2.2
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Definition 2.2. Function f : Rm → R is called mixed monotone in subset I m of Rm
if fxσ , xκ is monotone nondecreasing in each component of xσ and is monotone
nonincreasing in every component of xκ for x ∈ I m .
Definition 2.3. If there exists a point x ∈ I such that x fx, x, . . . , x, x, x is called an
equilibrium point of 2.2.
Definition 2.4. Let x be an equilibrium point of 2.2.
1 The equilibrium x of 2.2 is locally stable if for every ε > 0, there exists δ > 0
such that for any initial data x−m , x−m1 , . . . , x−1 , x0 ∈ I m1 satisfying max{|x−m −
x|, |x−m1 − x|, . . . , |x0 − x|} < δ, |xn − x| < ε holds for all n ≥ −m.
2 The equilibrium x of 2.2 is a local attractor if there exists δ > 0 such that
limn → ∞ xn x for any data x−m , x−m1 , . . . , x−1 , x0 ∈ I m1 satisfying max{|x−m −
x|, |x−m1 − x|, . . . , |x0 − x|} < δ.
3 The equilibrium x of 2.2 is locally asymptotically stable if it is stable and is a local
attractor.
4 The equilibrium x of 2.2 is a global attractor if for all x−m , x−m1 , . . . , x−1 , x0 ∈ I,
limn → ∞ xn x holds.
5 x is globally asymptotically stable if it is stable and is a global attractor.
6 x is unstable if it is not locally stable.
Lemma 2.5. Assume that s1 , s2 , . . . , sk ∈ R and k ∈ N. Then,
|s1 | |s2 | · · · |sk | < 1
2.3
is a sufficient condition for the local stability of the difference equation
xnk s1 xnk−1 · · · sk xn 0,
n 0, 1, . . . .
2.4
3. The Main Results and Their Proofs
In this section, we investigate the globally asymptotic stability of the equilibrium point of
1.4.
It is obvious that x 0 is a unique equilibrium point of 1.4 provided either 0 < D <
1, li1 Asi < B1 − D or D ≥ 1.
Let f : R m → R be a multivariate continuous function defined by
l
fxn−s1 , . . . , xn−sl , xn−t1 , . . . , xn−tk , xn Asi xn−si
Dxn .
k
B C j1 xn−tj
i1
3.1
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Advances in Difference Equations
··· /
sl /
t1 /
··· /
tk , we have
If s1 /
fxn−si xn−s1 , . . . , xn−sl , xn−t1 , . . . , xn−tk , xn fxn−si xn−s1 , . . . , xn−sl , xn−t1 , . . . , xn−tk , xn BC
BC
As i
k
,
As i
k
D,
j1 xn−tj
j1 xn−tj
si / 0, i 1, 2, . . . , l,
for some si 0,
i ∈ {1, 2, . . . , l},
l
k
i1 Asi xn−si
fxn−tj xn−s1 , . . . , xn−sl , xn−t1 , . . . , xn−tk , xn xn−tr ,
2
B C kr1 xn−tr r1, r / j
−C
tj /
0,
3.2
j 1, 2, . . . , k,
k
−C li1 Asi xn−si fxn−tj xn−s1 , ..., xn−sl , xn−t1 , ..., xn−tk , xn xn−tr D,
2
B C kr1 xn−tr r1, r / j
for some tj 0,
0,
si , tj /
fxn xn−s1 , . . . , xn−sl , xn−t1 , . . . , xn−tk , xn D,
j ∈ {1, 2, . . . , k},
i 1, 2, . . . , l,
j 1, 2, . . . k.
By constructing 1.4 and applying Lemma 2.5, we have the following affirmation.
··· /
sl /
t1 /
··· /
tk , and li1 Asi < B1 − D with 0 < D < 1, then the
Theorem 3.1. If s1 /
unique equilibrium point x 0 of 1.4 is locally stable. If D ≥ 1, x 0 is unstable.
Proof. Considering the linearized equation of 1.4 with respect to equilibrium point x 0,
l
zn1 Dzn l
zn1 Dzn l
i1
i1
Asi zn−si
0,
B
Asi zn−si
0,
B
for si /
0, i 1, 2, . . . , l,
for some tj 0, j ∈ {1, 2, . . . , k},
Asi zn−si
zn1 Dzn 0, for si , tj /
0, i 1, 2, . . . , l, j 1, 2, . . . k,
B
l
As p
i1, i /
p Asi zn−si
zn 0, for some sp 0, p ∈ {1, 2, . . . , l}.
zn1 D B
B
3.3
i1
By Lemma 2.5, 1.4 is stable if the following inequality holds
l
D
which implies our claim.
i1
B
As i
< 1,
3.4
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When D ≥ 1, note that the solution of 1.4 is xn > 0, hence xn1 ≥ Dxn ≥ xn , which
implies that {xn }∞
n1 is a dispersed sequence.
Theorem 3.2. Let f : Rm1 → R defined by 2.2 be a continuous function satisfying the mixed
monotone property. If there exists two real numbers Ω0 , Θ0 satisfying
Ω0 ≤ min{x−m , x−m1 , . . . , x−1 , x0 } ≤ max{x−m , x−m1 , . . . , x−1 , x0 } ≤ Θ0 ,
3.5
such that
Ω0 ≤ f
≤ f Θ0 , Ω0
≤ Θ0 ,
Ω0 , Θ0
σ
κ
σ
3.6
κ
2
then there exists Ω, Θ ∈ Ω0 , Θ0 satisfying
Ω fΩσ , Θκ ,
Θ fΘσ , Ωκ .
3.7
Moreover, if Ω Θ, then 2.2 has a unique equilibrium point x ∈ Ω0 , Θ0 , and every solution of
2.2 converges to x.
Proof. Let Ω0 and Θ0 be a couple of initial iteration data, then we construct two sequences
∞
∞
{Ωi }i1 and {Θi }i1 in the form
Ωi f
,
Ωi−1 , Θi−1
σ
κ
Θi f
.
Θi−1 , Ωi−1
σ
3.8
κ
∞
Note that mixed monotone property of f and initial assumption, the sequences {Ωi }i1 and
∞
{Θi }i1 thus possess
Ω0 ≤ Ω1 ≤ · · · ≤ Ωi ≤ Θi ≤ · · · ≤ Θ1 ≤ Θ0 ,
i 0, 1, 2, . . . .
3.9
i
i
It guarantees one can choose a new sequence {xl }∞
l1 satisfying Ω ≤ xl ≤ Θ for l ≥ m 1i 1.
Denote
Ω lim Ωi ,
i→∞
Θ lim Θi ,
i→∞
3.10
then
Ω ≤ lim inf xi ≤ lim sup xi ≤ Θ.
i→∞
i→∞
3.11
By the continuity of f, we have
Ω fΩσ , Θκ ,
Θ fΘσ , Ωκ .
Moreover, if Ω Θ, then Ω Θ limi → ∞ xi x, and then the proof is complete.
3.12
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0j 1, 2, . . . , k,
Theorem 3.3. If 0 < D < 1, li1 Asi < B2 1 − D/k 1C B, and tj /
then the unique equilibrium point x 0 of 1.4 is global attractor for any initial conditions
x−m , x−m1 , . . . , x1 , x0 ∈ 0, 1m1 .
Proof. For any nonnegative initial data x−i , i ∈ {0, 1, 2, . . . , m}, it is obvious that the function
fxn−s1 , . . . , xn−sl , xn−t1 , . . . , xn−tk , xn defined by 3.1 is nondecreasing in xn−s1 , . . . , xn−sl , xn and nonincreasing in xn−t1 , . . . , xn−tk .
Let
Ω0 0,
by view of the assumption
l
i1
Θ0 max{x−m , . . . , x−1 , x0 },
3.13
Asi < B2 1 − D/k 1C B, we have
l As Ω0
i1 i k DΩ0
Ω ≤ f Ω0 , Θ0
σ
k
B CΘ0 l As Θ0
i1 i k DΘ0
≤ f Θ0 , Ω0
σ
κ
B CΩ0 Θ0 li1 Asi
DΘ0
≤
B
Θ0 k 1C B li1 Asi
≤
DΘ0 ≤ Θ0 .
B2
0
3.14
2
From 1.4 and Theorem 3.2, there exists Ω, Θ ∈ Ω0 , Θ0 satisfying
l
As i Ω
Ω
DΩ,
B CΘk
l
As i Θ
DΘ.
B CΩk
3.15
l
As i Ω
i1 Asi Θ
−
DΩ − Θ,
B CΘk
B CΩk
3.16
i1
Θ
i1
Taking the difference
l
Ω−Θ
i1
we deduce that
l
1 − DΩ − Θ −
i1
Asi Ω B CΩk − li1 Asi Θ B CΘk
0,
B CΘk B CΩk
3.17
which implies
⎡
⎣1 − D −
where
pqk
l
i1
⎤
Asi B C li1 Asi pqk Ωp Θq
⎦Ω − Θ 0,
B CΘk B CΩk
Ωp Θq Ωk Ωk−1 Θ · · · ΩΘk−1 Θk .
3.18
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By view of 0 < D < 1, li1 Asi < B2 1 − D/k 1C B and initial conditions
x−m , x−m1 , . . . , x1 , x0 ∈ 0, 1m1 , therefore we have
l
1 − D −
i1
Asi B C li1 Asi pqk Ωp Θq
B Ck 1 li1 Asi
> 0,
> 1 − D −
B2
B CΘk B CΩk
3.19
thus, we have
Ω Θ.
3.20
By Theorem 3.2, then every solution of 1.4 converges to the unique equilibrium point x 0.
Theorem 3.4. If 0 < D < 1, li1 Asi < B2 1 − D/k 1C B, s1 /
··· /
sl /
t1 /
··· /
tk , and
0j ∈ {1, 2, . . . , k}, then the unique equilibrium point x 0 of 1.4 is globally asymptotically
tj /
stable for any initial conditions x−m , x−m1 , . . . , x1 , x0 ∈ 0, 1m1 .
Proof. The result holds from Theorems 3.1 and 3.3.
Theorem 3.5. If 0 < D < 1, li1 Asi < B2 1 − D/k 1μk C B, and tj / 0j ∈ {1, 2, . . . , k},
then the unique equilibrium point x 0 of 1.4 is global attractor for any initial conditions
x−m , x−m1 , . . . , x1 , x0 ∈ 0, μm1 .
Proof. The proof of this Theorem is similar to that of Theorem 3.3. We omit the details.
Theorem 3.6. If 0 < D < 1, li1 Asi < B2 1−D/k 1μk C B, s1 /
··· /
sl /
t1 /
··· /
tk , and
0j ∈ {1, 2, . . . , k}, then the unique equilibrium point x 0 of 1.4 is globally asymptotically
tj /
stable for any initial conditions x−m , x−m1 , . . . , x1 , x0 ∈ 0, μm1 .
Proof. The result holds from Theorems 3.1 and 3.5.
4. Numerical Simulation
In this section, we give some numerical simulations supporting our theoretical analysis via
the software package Matlab 7.1. As examples, we consider
xn1 0.05xn−2 0.02xn−4 0.03xn 1
xn ,
8 xn−1 xn−3
2
4.1
0.3xn−4 0.2xn−3 1
xn ,
10 3xn−1 xn−2
2
4.2
xn−4 xn−3
2xn .
8 3xn−1 xn−2
4.3
xn1 xn1 8
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3.5
3
2.5
y
2
y xn
1.5
1
0.5
0
−10
0
10
20
30
40
50
60
30
40
50
60
n
Figure 1
3
2.5
2
y
1.5
y xn
1
0.5
0
−10
0
10
20
n
Figure 2
Let μ 4, it is obvious that 4.1 and 4.2 satisfy the conditions of Theorem 3.6 when
the initial datas x−4 , x−3 , x−2 , x−1 , x0 ∈ 0, 45 . Equation 4.3 satisfies the conditions of
Theorem 3.1 for the initial data x−4 , x−3 , x−2 , x−1 , x0 ∈ R .
By employing the software package Matlab 7.1, we can solve the numerical solutions
of 4.1, 4.2, and 4.3 which are shown, respectively, in Figures 1, 2, and 3. More precisely,
Figure 1 shows the asymptotic behavior of the solution to 4.1 with initial data x−4 2, x−3 0.2, x−2 1.1, x−1 3.7, and x0 0.5, Figure 2 shows the asymptotic behavior of the solution
to 4.2 with initial data x−4 1, x−3 2.2, x−2 2, x−1 2.7, and x0 0.5, and Figure 3
shows the asymptotic behavior of the solution to 4.3 with initial data x−4 0.1, x−3 2.7, x−2 1.1, x−1 3.7, and x0 0.5.
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×104
2.5
2
1.5
y
y xn
1
0.5
0
−4
−2
0
2
4
6
8
10
12
14
16
n
Figure 3
5. Conclusions
This paper presents the use of a variational iteration method for systems of nonlinear
difference equations. This technique is a powerful tool for solving various difference
equations and can also be applied to other nonlinear differential equations in mathematical
physics. The numerical simulations show that this method is an effective and convenient
one. The variational iteration method provides an efficient method to handle the nonlinear
structure. Computations are performed using the software package Matlab 7.1.
We have dealt with the problem of global asymptotic stability analysis for a class of
nonlinear difference equation. The general sufficient conditions have been obtained to ensure
the existence, uniqueness, and global asymptotic stability of the equilibrium point for the
nonlinear difference equation. These criteria generalize and improve some known results.
In particular, some examples are given to show the effectiveness of the obtained results. In
addition, the sufficient conditions that we obtained are very simple, which provide flexibility
for the application and analysis of nonlinear difference equation.
Acknowledgments
The authors are grateful to the referee for giving us lots of precious comments. This work is
supported by Natural Science Foundation Project of CQ CSTC Grant no. 2008BB 7415 of
China, and National Science Foundation Grant no. 40801214 of China.
References
1 W.-T. Li and H.-R. Sun, “Global attractivity in a rational recursive sequence,” Dynamic Systems and
Applications, vol. 11, no. 3, pp. 339–345, 2002.
2 V. L. Kocić and G. Ladas, Global Behavior of Nonlinear Difference Equations of Higher Order with
Applications, vol. 256 of Mathematics and Its Applications, Kluwer Academic Publishers, Dordrecht, The
Netherlands, 1993.
10
Advances in Difference Equations
3 M. R. S. Kulenović and G. Ladas, Dynamics of Second Order Rational Difference Equations with Open
Problems and Conjectures, Chapman & Hall/CRC, Boca Raton, Fla, USA, 2002.
4 H. M. El-Owaidy, A. M. Ahmed, and M. S. Mousa, “On the recursive sequences xn1 −αxn−1 /β ±
xn ,” Applied Mathematics and Computation, vol. 145, no. 2-3, pp. 747–753, 2003.
5 X.-Y. Xianyi and D.-M. Zhu, “Global asymptotic stability in a rational equation,” Journal of Difference
Equations and Applications, vol. 9, no. 9, pp. 833–839, 2003.
6 C. Çinar, “On the positive solutions of the difference equation xn1 xn−1 /1 xn xn−1 ,” Applied
Mathematics and Computation, vol. 150, no. 1, pp. 21–24, 2004.
7 X. Yang, W. Su, B. Chen, G. M. Megson, and D. J. Evans, “On the recursive sequence xn axn−1 bxn−2 /c dxn−1 xn−2 ,” Applied Mathematics and Computation, vol. 162, no. 3, pp. 1485–1497, 2005.
8 K. S. Berenhaut, J. D. Foley, and S. Stević, “The global attractivity of the rational difference equation
yn yn−k yn−m /1 yn−k yn−m ,” Applied Mathematics Letters, vol. 20, no. 1, pp. 54–58, 2007.
9 K. S. Berenhaut and S. Stević, “The difference equation xn1 a xn−k / k−1
i0 ci xn−i has solutions
converging to zero,” Journal of Mathematical Analysis and Applications, vol. 326, no. 2, pp. 1466–1471,
2007.
10 R. Memarbashi, “Sufficient conditions for the exponential stability of nonautonomous difference
equations,” Applied Mathematics Letters, vol. 21, no. 3, pp. 232–235, 2008.
11 M. Aloqeili, “Global stability of a rational symmetric difference equation,” Applied Mathematics and
Computation, vol. 215, no. 3, pp. 950–953, 2009.
12 M. Aprahamian, D. Souroujon, and S. Tersian, “Decreasing and fast solutions for a second-order
difference equation related to Fisher-Kolmogorov’s equation,” Journal of Mathematical Analysis and
Applications, vol. 363, no. 1, pp. 97–110, 2010.
13 C.-y. Wang, F. Gong, S. Wang, L.-r. Li, and Q.-h. Shi, “Asymptotic behavior of equilibrium point for a
class of nonlinear difference equation,” Advances in Difference Equations, vol. 2009, Article ID 214309,
8 pages, 2009.
14 C.-y. Wang, S. Wang, Z.-w. Wang, F. Gong, and R.-f. Wang, “Asymptotic stability for a class of
nonlinear difference equations,” Discrete Dynamics in Nature and Society, vol. 2010, Article ID 791610,
10 pages, 2010.