Discrete Dynamics in Nature and Society
Volume 2010 (2010), Article ID 791610, 10 pages
doi:10.1155/2010/791610
Research Article

Asymptotic Stability for a Class of Nonlinear Difference Equations

Chang-you Wang,1,2,3 Shu Wang,3 Zhi-wei Wang,4 Fei Gong,2,5 and Rui-fang Wang2,5

1College of Mathematics and Physics, Chongqing University of Posts and Telecommunications, Chongqing 400065, China
2Key Laboratory of Network Control & Intelligent Instrument, Chongqing University of Posts and Telecommunications, Ministry of Education, Chongqing 400065, China
3College of Applied Sciences, Beijing University of Technology, Beijing 100124, China
4School of Communication and Information Engineering, Chongqing University of Posts and Telecommunications, Chongqing 400065, China
5College of Automation, Chongqing University of Posts and Telecommunications, Chongqing 400065, China

Received 9 January 2010; Accepted 5 February 2010

Academic Editor: Guang Zhang

Copyright © 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.

Abstract

We study the global asymptotic stability of the equilibrium point for the fractional difference equation 𝑥 𝑛 + 1 = ( 𝑎 𝑥 𝑛 − 𝑙 𝑥 𝑛 − 𝑘 ) / ( 𝛼 + 𝑏 𝑥 𝑛 − 𝑠 + 𝑐 𝑥 𝑛 − 𝑡 ) , 𝑛 = 0 , 1 , … , where the initial conditions 𝑥 − 𝑟 , 𝑥 − 𝑟 + 1 , … , 𝑥 1 , 𝑥 0 are arbitrary positive real numbers of the interval ( 0 , 𝛼 / 2 𝑎 ) , 𝑙 , 𝑘 , 𝑠 , 𝑡 are nonnegative integers, 𝑟 = m a x { 𝑙 , 𝑘 , 𝑠 , 𝑡 } and 𝛼 , 𝑎 , 𝑏 , 𝑐 are positive constants. Moreover, some numerical simulations are given to illustrate our results.

1. Introduction

Difference equations appear naturally as discrete analogues and as numerical solutions of differential and delay differential equations having applications in biology, ecology, physics, and so forth [1]. The study of nonlinear difference equations is of paramount importance not only in their own field but in understanding the behavior of their differential counterparts. There has been a lot of work concerning the global asymptotic behavior of solutions of rational difference equations [2–6]. In particular, Ladas [7] put forward the idea of investigating the global asymptotic stability of the following difference equation: 𝑥 𝑛 + 1 = 𝑥 𝑛 + 𝑥 𝑛 − 1 𝑥 𝑛 − 2 𝑥 𝑛 𝑥 𝑛 − 1 + 𝑥 𝑛 − 2 , 𝑛 = 0 , 1 , … , ( 1 . 1 ) where the initial values 𝑥 − 2 , 𝑥 − 1 , 𝑥 0 ∈ ( 0 , + ∞ ) .

In [8], Nesemann utilized the strong negative feedback property of [2] to study the following difference equation: 𝑥 𝑛 + 1 = 𝑥 𝑛 − 1 + 𝑥 𝑛 𝑥 𝑛 − 2 𝑥 𝑛 − 1 𝑥 𝑛 + 𝑥 𝑛 − 2 , 𝑛 = 0 , 1 , … , ( 1 . 2 ) where the initial values 𝑥 − 2 , 𝑥 − 1 , 𝑥 0 ∈ ( 0 , + ∞ ) .

By using semicycle analysis methods, the authors of [9] got a sufficient condition which guarantees the global asymptotic stability of the following difference equation: 𝑥 𝑛 + 1 = 𝑥 𝑏 𝑛 − 1 + 𝑥 𝑛 𝑥 𝑏 𝑛 − 2 + 𝑎 𝑥 𝑏 𝑛 − 1 𝑥 𝑛 + 𝑥 𝑏 𝑛 − 2 + 𝑎 , 𝑛 = 0 , 1 , … , ( 1 . 3 ) where 𝑎 , 𝑏 ∈ [ 0 , + ∞ ) and the initial values 𝑥 − 2 , 𝑥 − 1 , 𝑥 0 ∈ ( 0 , + ∞ ) .

Yang et al. [10] investigated the invariant intervals, the global attractivity of equilibrium points, and the asymptotic behavior of the solutions of the recursive sequence 𝑥 𝑛 + 1 = 𝑎 𝑥 𝑛 − 1 + 𝑏 𝑥 𝑛 − 2 𝑐 + 𝑑 𝑥 𝑛 − 1 𝑥 𝑛 − 2 , 𝑛 = 0 , 1 , … . ( 1 . 4 ) Berenhaut et al. [11] generalized the result reported in [12] to the following rational equation 𝑥 𝑛 = 𝑦 𝑛 − 𝑘 + 𝑦 𝑛 − 𝑚 1 + 𝑦 𝑛 − 𝑘 𝑦 𝑛 − 𝑚 , 𝑛 = 0 , 1 , … . ( 1 . 5 ) This work is motivated from [13–15]. For more similar work, one can refer to [12, 16–20] and references therein.

The purpose of this paper is to investigate the global attractivity of the equilibrium point, and the asymptotic behavior of the solutions of the difference equation: 𝑥 𝑛 + 1 = 𝑎 𝑥 𝑛 − 𝑙 𝑥 𝑛 − 𝑘 𝛼 + 𝑏 𝑥 𝑛 − 𝑠 + 𝑐 𝑥 𝑛 − 𝑡 , 𝑛 = 0 , 1 , … , ( 1 . 6 ) where the initial conditions 𝑥 − 𝑟 , 𝑥 − 𝑟 + 1 , … , 𝑥 1 , 𝑥 0 are arbitrary positive real numbers of the interval ( 0 , 𝛼 / 2 𝑎 ) , 𝑙 , 𝑘 , 𝑠 , 𝑡 is nonnegative integer, and 𝑟 = m a x { 𝑙 , 𝑘 , 𝑠 , 𝑡 } and 𝛼 , 𝑎 , 𝑏 , 𝑐 are positive constants. Moreover, some numerical simulations to the special case of (1.6) are given to illustrate our results.

This paper is arranged as follows: in Section 2, we give some definitions and preliminary results. The main results and their proofs are given in Section 3. Finally, some numerical simulations are given to illustrate our theoretical analysis.

2. Some Preliminary Results

To prove the main results in this paper, we first give some definitions and preliminary results [21, 22] which are basically used throughout this paper.

Lemma 2.1. Let 𝐼 be some interval of real numbers and let 𝑓 ∶ 𝐼 𝑘 + 1 ⟶ 𝐼 ( 2 . 1 ) be a continuously differentiable function. Then for every set of initial conditions 𝑥 − 𝑘 , 𝑥 − 𝑘 + 1 , … , 𝑥 0 ∈ 𝐼 , the difference equation 𝑥 𝑛 + 1 𝑥 = 𝑓 𝑛 , 𝑥 𝑛 − 1 , … , 𝑥 𝑛 − 𝑘 , 𝑛 = 0 , 1 , … , ( 2 . 2 ) has a unique solution { 𝑥 𝑛 } + ∞ 𝑛 = − 𝑘 .

Definition 2.2. A point 𝑥 ∈ 𝐼 is called an equilibrium point of (2.2), if 𝑥 = 𝑓 𝑥 , 𝑥 , … , 𝑥 . ( 2 . 3 ) That is, 𝑥 𝑛 = 𝑥 for 𝑛 ≥ 0 is a solution of (2.2), or equivalently, 𝑥 is a fixed point of 𝑓 .

Definition 2.3. Let 𝑝 , 𝑞 be two nonnegative integers such that 𝑝 + 𝑞 = 𝑛 . Splitting 𝑥 = ( 𝑥 1 , 𝑥 2 , … , 𝑥 𝑛 ) into 𝑥 = ( [ 𝑥 ] 𝑝 , [ 𝑥 ] 𝑞 ) , where [ 𝑥 ] 𝜎 denotes a vector with 𝜎 -components of 𝑥 , we say that the function 𝑓 ( 𝑥 1 , 𝑥 2 , … , 𝑥 𝑛 ) possesses a mixed monotone property in subsets 𝐼 𝑛 of 𝑅 𝑛 if 𝑓 ( [ 𝑥 ] 𝑝 , [ 𝑥 ] 𝑞 ) is monotone nondecreasing in each component of [ 𝑥 ] 𝑝 and is monotone nonincreasing in each component of [ 𝑥 ] 𝑞 for 𝑥 ∈ 𝐼 𝑛 . In particular, if 𝑞 = 0 , then it is said to be monotone nondecreasing in 𝐼 𝑛 .

Definition 2.4. Let 𝑥 be an equilibrium point of (2.2).(i) 𝑥 is stable if for every 𝜀 > 0 , there exists 𝛿 > 0 such that for any initial conditions ( 𝑥 − 𝑘 , 𝑥 − 𝑘 + 1 , … , 𝑥 0 ) ∈ 𝐼 𝑘 + 1 with | 𝑥 − 𝑘 − 𝑥 | + | 𝑥 − 𝑘 + 1 − 𝑥 | + ⋯ + | 𝑥 0 − 𝑥 | < 𝛿 , | 𝑥 𝑛 − 𝑥 | < 𝜀 holds for 𝑛 = 1 , 2 , … . (ii) 𝑥 is a local attractor if there exists 𝛾 > 0 such that 𝑥 𝑛 → 𝑥 holds for any initial conditions ( 𝑥 − 𝑘 , 𝑥 − 𝑘 + 1 , … , 𝑥 0 ) ∈ 𝐼 𝑘 + 1 with | 𝑥 − 𝑘 − 𝑥 | + | 𝑥 − 𝑘 + 1 − 𝑥 | + ⋯ + | 𝑥 0 − 𝑥 | < 𝛾 .(iii) 𝑥 is locally asymptotically stable if it is stable and is a local attractor.(iv) 𝑥 is a global attractor if 𝑥 𝑛 → 𝑥 holds for any initial conditions ( 𝑥 − 𝑘 , 𝑥 − 𝑘 + 1 , … , 𝑥 0 ) ∈ 𝐼 𝑘 + 1 .(v) 𝑥 is globally asymptotically stable if it is stable and is a global attractor.(vi) 𝑥 is unstable if it is not locally stable.
The linearized equation of (2.2) about the equilibrium 𝑥 is the linear difference equation 𝑦 𝑛 + 1 = 𝑘 𝑖 = 1 𝜕 𝑓 𝑥 , 𝑥 , … , 𝑥 𝜕 𝑥 𝑛 − 𝑖 𝑦 𝑛 − 𝑖 . ( 2 . 4 ) Now assume that the characteristic equation associated with (2.4) is 𝑃 ( 𝜆 ) = 𝑃 0 𝜆 𝑘 + 𝑃 1 𝜆 𝑘 − 1 + ⋯ + 𝑃 𝑘 − 1 𝜆 + 𝑃 𝑘 = 0 , ( 2 . 5 ) where 𝑃 𝑖 = 𝜕 𝑓 ( 𝑥 , 𝑥 , … , 𝑥 ) / 𝜕 𝑥 𝑛 − 𝑖 .

Lemma 2.5. Assume that 𝑃 1 , 𝑃 2 , … , 𝑃 𝑘 ∈ 𝑅 and 𝑘 ∈ { 0 , 1 , 2 , … } . Then | | 𝑃 1 | | + | | 𝑃 2 | | | | 𝑃 + ⋯ + 𝑘 | | < 1 ( 2 . 6 ) is a sufficient condition for the local asymptotically stability of the difference equation 𝑥 𝑛 + 𝑘 + 𝑃 1 𝑥 𝑛 + 𝑘 − 1 + ⋯ + 𝑃 𝑘 𝑥 𝑛 = 0 , 𝑛 = 0 , 1 , … . ( 2 . 7 )

Lemma 2.6. Assume that 𝑓 is a 𝐶 1 function and let 𝑥 be an equilibrium of (2.2). Then the following statements are true. (a)If all roots of the polynomial equation (2.5) lie in the open unite d i s k | 𝜆 | < 1 , then the equilibrium point 𝑥 of (2.2) is locally asymptotically stable.(b) If at least one root of (2.2) has absolute value greater than one, then the equilibrium point 𝑥 of (2.2) is unstable.

Remark 2.7. The condition (2.6) implies that all the roots of the polynomial equation (2.5) lie in the open unite d i s k | 𝜆 | < 1 .

3. The Main Results and Their Proofs

In this section, we investigate the global asymptotic stability of the equilibrium point of (1.6).

Let 𝑓 ∶ ( 0 , ∞ ) 4 → ( 0 , ∞ ) be a function defined by 𝑓 ( 𝑢 , 𝑣 , 𝑤 , 𝑠 ) = 𝑎 𝑢 𝑣 𝛼 + 𝑏 𝑤 + 𝑐 𝑠 , ( 3 . 1 ) then it follows that 𝑓 𝑢 ( 𝑢 , 𝑣 , 𝑤 , 𝑠 ) = 𝑎 𝑣 𝛼 + 𝑏 𝑤 + 𝑐 𝑠 , 𝑓 𝑣 ( 𝑢 , 𝑣 , 𝑤 , 𝑠 ) = 𝑎 𝑢 , 𝑓 𝛼 + 𝑏 𝑤 + 𝑐 𝑠 𝑤 ( 𝑢 , 𝑣 , 𝑤 , 𝑠 ) = − 𝑎 𝑏 𝑢 𝑣 ( 𝛼 + 𝑏 𝑤 + 𝑐 𝑠 ) 2 , 𝑓 𝑠 ( 𝑢 , 𝑣 , 𝑤 , 𝑠 ) = − 𝑎 𝑐 𝑢 𝑣 ( 𝛼 + 𝑏 𝑤 + 𝑐 𝑠 ) 2 . ( 3 . 2 ) Let 𝑥 , 𝑥 be the equilibrium points of (1.6), then we have 𝑥 = 0 , 𝛼 𝑥 = 𝑎 − ( 𝑏 + 𝑐 ) , ( 3 . 3 ) where 𝑎 ≠ 𝑏 + 𝑐 . If 𝑎 = 𝑏 + 𝑐 , then 𝑥 = 0 is a unique equilibrium point.

Moreover, 𝑓 𝑢 𝑥 , 𝑥 , 𝑥 , 𝑥 = 𝑓 𝑣 𝑥 , 𝑥 , 𝑥 , 𝑥 = 𝑓 𝑤 𝑥 , 𝑥 , 𝑥 , 𝑥 = 𝑓 𝑠 𝑥 , 𝑥 , 𝑥 , 𝑥 𝑓 = 0 , 𝑢 𝑥 , 𝑥 , 𝑥 , 𝑥 = 𝑓 𝑣 𝑥 , 𝑥 , 𝑥 , 𝑥 = 1 , 𝑓 𝑤 𝑥 , 𝑥 , 𝑥 , 𝑥 𝑏 = − 𝑎 , 𝑓 𝑠 𝑥 , 𝑥 , 𝑥 , 𝑥 𝑐 = − 𝑎 . ( 3 . 4 ) Thus, the linearized equations of (1.6) about equilibrium points 𝑥 and 𝑥 are, respectively, 𝑧 𝑛 + 1 𝑧 = 0 , ( 3 . 5 ) 𝑛 + 1 = 𝑧 𝑛 − 𝑘 + 𝑧 𝑛 − 𝑙 − 𝑏 𝑎 𝑧 𝑛 − 𝑠 − 𝑐 𝑎 𝑧 𝑛 − 𝑡 , ( 3 . 6 ) where 𝑙 , 𝑘 , 𝑠 , 𝑡 are nonnegative different integers.

The characteristic equation associated with (3.6) is 𝑃 ( 𝜆 ) = 𝜆 𝑟 − 𝑘 + 𝜆 𝑟 − 𝑙 − 𝑏 𝑎 𝜆 𝑟 − 𝑠 − 𝑐 𝑎 𝜆 𝑟 − 𝑡 = 0 , ( 3 . 7 ) where 𝑟 = m a x { 𝑙 , 𝑘 , 𝑠 , 𝑡 } .

By Lemmas 2.5 and 2.6, we have the following result.

Theorem 3.1. The equilibrium point 𝑥 = 0 of (1.6) is locally asymptotically stable. Moreover, we have the following. (a)If all roots of the characteristic equation (3.7) lie in the open unite d i s k | 𝜆 | < 1 , then the equilibrium point 𝑥 of (1.6) is locally asymptotically stable.(b) If at least one root of (3.7) has absolute value greater than one, then the equilibrium point 𝑥 of (1.6) is unstable.

Theorem 3.2. Let [ 𝛾 , 𝛿 ] be an interval of real numbers and assume that 𝑓 ∶ [ 𝛾 , 𝛿 ] 𝑘 + 1 → 𝑅 is a continuous function satisfying the mixed monotone property. If there exists 𝑚 0 𝑥 ≤ m i n − 𝑘 , 𝑥 − 𝑘 + 1 , … , 𝑥 − 1 , 𝑥 0 𝑥 ≤ m a x − 𝑘 , 𝑥 − 𝑘 + 1 , … , 𝑥 − 1 , 𝑥 0 ≤ 𝑀 0 ( 3 . 8 ) such that 𝑚 0 𝑚 ≤ 𝑓 0 𝑝 , 𝑀 0 𝑞 𝑀 ≤ 𝑓 0 𝑝 , 𝑚 0 𝑞 ≤ 𝑀 0 , ( 3 . 9 ) then there exist ( 𝑚 , 𝑀 ) ∈ [ 𝑚 0 , 𝑀 0 ] 2 satisfying [ 𝑀 ] 𝑀 = 𝑓 𝑝 , [ 𝑚 ] 𝑞 [ 𝑚 ] , 𝑚 = 𝑓 𝑝 , [ 𝑀 ] 𝑞 . ( 3 . 1 0 ) Moreover, if 𝑚 = 𝑀 , then (2.2) has a unique equilibrium point 𝑥 ∈ [ 𝑚 0 , 𝑀 0 ] and every solution of (2.2) converges to 𝑥 .

Proof. Using 𝑚 0 and 𝑀 0 as a couple of initial iteration, we construct two sequences { 𝑚 𝑖 } and { 𝑀 𝑖 } ( 𝑖 = 1 , 2 , … ) from the equation 𝑚 𝑖 𝑚 = 𝑓 𝑖 − 1 𝑝 , 𝑀 𝑖 − 1 𝑞 , 𝑀 𝑖 𝑀 = 𝑓 𝑖 − 1 𝑝 , 𝑚 𝑖 − 1 𝑞 . ( 3 . 1 1 ) It is obvious from the mixed monotone property of 𝑓 that the sequences { 𝑚 𝑖 } and { 𝑀 𝑖 } possess the following monotone property 𝑚 0 ≤ 𝑚 1 ≤ ⋯ ≤ 𝑚 𝑖 ≤ ⋯ ≤ 𝑀 𝑖 ≤ ⋯ ≤ 𝑀 1 ≤ 𝑀 0 , ( 3 . 1 2 ) where 𝑖 = 0 , 1 , 2 , … , and 𝑚 𝑖 ≤ 𝑥 𝑙 ≤ 𝑀 𝑖 f o r 𝑙 ≥ ( 𝑘 + 1 ) 𝑖 + 1 . ( 3 . 1 3 ) Set 𝑚 = l i m 𝑖 → ∞ 𝑚 𝑖 , 𝑀 = l i m 𝑖 → ∞ 𝑀 𝑖 , ( 3 . 1 4 ) then 𝑚 ≤ l i m 𝑖 → ∞ i n f 𝑥 𝑖 ≤ l i m 𝑖 → ∞ s u p 𝑥 𝑖 ≤ 𝑀 . ( 3 . 1 5 ) By the continuity of 𝑓 , we have [ 𝑀 ] 𝑀 = 𝑓 𝑝 , [ 𝑚 ] 𝑞 [ 𝑚 ] , 𝑚 = 𝑓 𝑝 , [ 𝑀 ] 𝑞 . ( 3 . 1 6 ) Moreover, if 𝑚 = 𝑀 , then 𝑚 = 𝑀 = l i m 𝑖 → ∞ 𝑥 𝑖 = 𝑥 , and then the proof is complete.

Theorem 3.3. The equilibrium point 𝑥 = 0 of (1.6) is a global attractor for any initial conditions 𝑥 − 𝑟 , 𝑥 − 𝑟 + 1 , … , 𝑥 1 , 𝑥 0 ∈ 𝛼 0 , 2 𝑎 𝑟 + 1 . ( 3 . 1 7 )

Proof. Let 𝑓 ∶ ( 0 , ∞ ) 4 → ( 0 , ∞ ) be a function defined by 𝑓 ( 𝑢 , 𝑣 , 𝑤 , 𝑠 ) = 𝑎 𝑢 𝑣 𝛼 + 𝑏 𝑤 + 𝑐 𝑠 . ( 3 . 1 8 ) We can easily see that the function 𝑓 ( 𝑢 , 𝑣 , 𝑤 , 𝑠 ) is increasing in 𝑢 , 𝑣 and decreasing in 𝑤 , 𝑠 .
Let 𝑀 0 𝑥 = m a x − 𝑟 , 𝑥 − 𝑟 + 1 , … , 𝑥 − 1 , 𝑥 0 , 𝑎 𝑀 0 − 𝛼 𝑏 + 𝑐 < 𝑚 0 < 0 ; ( 3 . 1 9 ) we have 𝑚 0 ≤ 𝑎 𝑚 2 0 𝛼 + 𝑏 𝑀 0 + 𝑐 𝑀 0 ≤ 𝑎 𝑀 2 0 𝛼 + 𝑏 𝑚 0 + 𝑐 𝑚 0 ≤ 𝑀 0 . ( 3 . 2 0 ) Then from (1.6) and Theorem 3.2, there exist 𝑚 , 𝑀 ∈ [ 𝑚 0 , 𝑀 0 ] satisfying 𝑚 = 𝑎 𝑚 2 𝛼 + 𝑏 𝑀 + 𝑐 𝑀 , 𝑀 = 𝑎 𝑀 2 𝛼 + 𝑏 𝑚 + 𝑐 𝑚 , ( 3 . 2 1 ) thus [ ] 𝛼 − 𝑎 ( 𝑚 + 𝑀 ) ( 𝑚 − 𝑀 ) = 0 . ( 3 . 2 2 ) In view of 2 𝑎 𝑀 0 < 𝛼 , we have 𝛼 − 𝑎 ( 𝑚 + 𝑀 ) > 0 . ( 3 . 2 3 ) Then 𝑀 = 𝑚 . ( 3 . 2 4 ) It follows by Theorem 3.2 that the equilibrium point 𝑥 = 0 of (1.6) is a global attractor. The proof is therefore complete.

Theorem 3.4. The equilibrium point 𝑥 = 0 of (1.6) is a global asymptotic stability for any initial conditions 𝑥 − 𝑟 , 𝑥 − 𝑟 + 1 , … , 𝑥 1 , 𝑥 0 ∈ 𝛼 0 , 2 𝑎 𝑟 + 1 . ( 3 . 2 5 )

Proof. The result follows from Theorems 3.1 and 3.3.

4. Numerical Simulations

In this section, we give numerical simulations to support our theoretical analysis via the software package Matlab7.0. As an example, we consider the following difference equations 𝑥 𝑛 + 1 = 𝑥 𝑛 𝑥 𝑛 − 1 5 + 2 𝑥 𝑛 + 𝑥 𝑛 − 1 𝑥 , 𝑛 = 0 , 1 , … , ( 4 . 1 ) 𝑛 + 1 = 𝑥 𝑛 𝑥 𝑛 − 1 5 + 2 𝑥 𝑛 − 2 + 𝑥 𝑛 − 3 , 𝑛 = 0 , 1 , … , ( 4 . 2 ) where the initial conditions 𝑥 − 3 , 𝑥 − 2 , 𝑥 − 1 , 𝑥 0 ∈ ( 0 , 2 . 5 ) . Let 𝑚 0 = − 0 . 5 , 𝑀 0 = 2 . 5 ; it is obvious that (4.1) and (4.2) satisfy the conditions of Theorems 3.2 and 3.3.

By employing the software package MATLAB7.0, we can solve the numerical solutions of (4.1) and (4.2) which are shown, respectively, in Figures 1 and 2. More precisely, Figure 1 shows the numerical solution of (4.1) with 𝑥 − 1 = 1 . 2 , 𝑥 0 = 1 . 8 , and the relations that 𝑚 𝑖 ≤ 𝑥 𝑙 ≤ 𝑀 𝑖 when 𝑙 ≥ ( 𝑘 + 1 ) 𝑖 + 1 ,    𝑖 = 0 , 1 , 2 , … , and Figure 2 shows the numerical solutions of (4.2) with 𝑥 − 3 = 1 . 5 , 𝑥 − 2 = 1 . 8 , 𝑥 − 1 = 1 . 3 , 𝑥 0 = 1 . 4 , and the relations that 𝑚 𝑖 ≤ 𝑥 𝑙 ≤ 𝑀 𝑖 when 𝑙 ≥ ( 𝑘 + 1 ) 𝑖 + 1 , 𝑖 = 0 , 1 , 2 , … .

791610.fig.001
Figure 1: Chart of (4.1) with 𝑥 ( − 1 ) = 1 . 2 , 𝑥 ( 0 ) = 1 . 8 .
791610.fig.002
Figure 2: Chart of (4.2) with 𝑥 ( − 3 ) = 1 . 5 , 𝑥 ( − 2 ) = 1 . 8 , 𝑥 ( − 1 ) = 1 . 3 , 𝑥 ( 0 ) = 1 . 4 .

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 MATLAB7.0.

We have dealt with the problem of global asymptotic stability analysis for a class of nonlinear difference equations. 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, an illustrate example is 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 equations.

Acknowledgments

The authors are grateful to the referee for her/his comments. This work is supported by Science and Technology Study Project of Chongqing Municipal Education Commission (Grant no. KJ 080511) of China, Natural Science Foundation Project of CQ CSTC (Grant no. 2008BB7415) of China, Foundation Project of Doctor Graduate Student Innovation of Beijing University of Technology of China, the NSFC (Grant no. 10471009), and BSFC (Grant no. 1052001) of China.

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