Asymptotic Stability for a Class of Nonlinear Difference Equations

Abstract

We study the global asymptotic stability of the equilibrium point for the fractional difference equation 𝑥_𝑛+1=(𝑎𝑥_𝑛−𝑙𝑥_𝑛−𝑘)/(𝛼+𝑏𝑥_𝑛−𝑠+𝑐𝑥_𝑛−𝑡), 𝑛=0,1,…, where the initial conditions 𝑥_−𝑟,𝑥_−𝑟+1,…,𝑥₁,𝑥₀ are arbitrary positive real numbers of the interval (0,𝛼/2𝑎),𝑙,𝑘,𝑠,𝑡 are nonnegative integers, 𝑟=max{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−1𝑥_𝑛−2𝑥_𝑛𝑥_𝑛−1+𝑥_𝑛−2,𝑛=0,1,…,(1.1) where the initial values 𝑥₋₂,𝑥₋₁,𝑥₀∈(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_𝑛𝑥_𝑛−2𝑥_𝑛−1𝑥_𝑛+𝑥_𝑛−2,𝑛=0,1,…,(1.2) where the initial values 𝑥₋₂,𝑥₋₁,𝑥₀∈(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^𝑏𝑛−1+𝑥_𝑛𝑥^𝑏𝑛−2+𝑎𝑥^𝑏𝑛−1𝑥_𝑛+𝑥^𝑏𝑛−2+𝑎,𝑛=0,1,…,(1.3) where 𝑎,𝑏∈[0,+∞) and the initial values 𝑥₋₂,𝑥₋₁,𝑥₀∈(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 sequence𝑥_𝑛+1=𝑎𝑥^𝑛−1+𝑏𝑥_𝑛−2𝑐+𝑑𝑥_𝑛−1𝑥_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−𝑚1+𝑦_𝑛−𝑘𝑦_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=𝑎𝑥^𝑛−𝑙𝑥_𝑛−𝑘𝛼+𝑏𝑥_𝑛−𝑠+𝑐𝑥_𝑛−𝑡,𝑛=0,1,…,(1.6) where the initial conditions 𝑥_−𝑟,𝑥_−𝑟+1,…,𝑥₁,𝑥₀ are arbitrary positive real numbers of the interval (0,𝛼/2𝑎),𝑙,𝑘,𝑠,𝑡 is nonnegative integer, and 𝑟=max{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,…,𝑥₀∈𝐼, the difference equation 𝑥_𝑛+1=𝑓𝑥_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=(𝑥₁,𝑥₂,…,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(𝑥₁,𝑥₂,…,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,…,𝑥₀)∈𝐼^𝑘+1 with |𝑥_−𝑘−𝑥|+|𝑥_−𝑘+1−𝑥|+⋯+|𝑥₀−𝑥|<𝛿, |𝑥_𝑛−𝑥|<𝜀holds for𝑛=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,…,𝑥₀)∈𝐼^𝑘+1 with |𝑥_−𝑘−𝑥|+|𝑥_−𝑘+1−𝑥|+⋯+|𝑥₀−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,…,𝑥₀)∈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−𝑖𝑦_𝑛−𝑖.(2.4) Now assume that the characteristic equation associated with (2.4) is 𝑃(𝜆)=𝑃₀𝜆^𝑘+𝑃₁𝜆^𝑘−1+⋯+𝑃_𝑘−1𝜆+𝑃_𝑘=0,(2.5) where 𝑃_𝑖=𝜕𝑓(𝑥,𝑥,…,𝑥)/𝜕𝑥_𝑛−𝑖.

Lemma 2.5. Assume that 𝑃₁,𝑃₂,…,𝑃_𝑘∈𝑅 and 𝑘∈{0,1,2,…}. Then ||𝑃₁||+||𝑃₂||+⋯+||𝑃_𝑘||<1(2.6) is a sufficient condition for the local asymptotically stability of the difference equation 𝑥_𝑛+𝑘+𝑃₁𝑥_{𝑛+𝑘−1}+⋯+𝑃_𝑘𝑥_𝑛=0,𝑛=0,1,….(2.7)

Lemma 2.6. Assume that 𝑓 is a 𝐶¹ 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 disk|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 disk|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 by𝑓(𝑢,𝑣,𝑤,𝑠)=𝑎𝑢𝑣𝛼+𝑏𝑤+𝑐𝑠,(3.1) then it follows that𝑓_𝑢(𝑢,𝑣,𝑤,𝑠)=𝑎𝑣𝛼+𝑏𝑤+𝑐𝑠,𝑓^𝑣(𝑢,𝑣,𝑤,𝑠)=𝑎𝑢𝑓𝛼+𝑏𝑤+𝑐𝑠_𝑤,(𝑢,𝑣,𝑤,𝑠)=−𝑎𝑏𝑢𝑣(𝛼+𝑏𝑤+𝑐𝑠)²,𝑓_𝑠(𝑢,𝑣,𝑤,𝑠)=−𝑎𝑐𝑢𝑣(𝛼+𝑏𝑤+𝑐𝑠)².(3.2) Let 𝑥,𝑥 be the equilibrium points of (1.6), then we have𝑥=0,𝑥=𝛼𝑎−(𝑏+𝑐),(3.3) where 𝑎≠𝑏+𝑐. If 𝑎=𝑏+𝑐, then 𝑥=0 is a unique equilibrium point.

Moreover, 𝑓𝑓_𝑢𝑥,𝑥,𝑥,𝑥=𝑓_𝑣𝑥,𝑥,𝑥,𝑥=𝑓_𝑤𝑥,𝑥,𝑥,𝑥=𝑓_𝑠𝑥,𝑥,𝑥,𝑥=0,_𝑢𝑥,𝑥,𝑥,𝑥=𝑓_𝑣𝑥,𝑥,𝑥,𝑥=1,𝑓_𝑤𝑥,𝑥,𝑥,𝑥=−𝑎𝑏,𝑓^𝑠𝑥,𝑥,𝑥,𝑥=−𝑎𝑐.(3.4) Thus, the linearized equations of (1.6) about equilibrium points 𝑥 and 𝑥 are, respectively, 𝑧𝑧_𝑛+1=0,(3.5)_𝑛+1=𝑧_𝑛−𝑘+𝑧_𝑛−𝑙−𝑏𝑎𝑧^𝑛−𝑠−𝑐𝑎𝑧^𝑛−𝑡,(3.6) where 𝑙,𝑘,𝑠,𝑡 are nonnegative different integers.

The characteristic equation associated with (3.6) is𝑃(𝜆)=𝜆^𝑟−𝑘+𝜆^𝑟−𝑙−𝑏𝑎𝜆^𝑟−𝑠−𝑐𝑎𝜆^𝑟−𝑡=0,(3.7) where 𝑟=max{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 disk|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 𝑚₀≤min𝑥_−𝑘,𝑥_−𝑘+1,…,𝑥₋₁,𝑥₀≤max𝑥_−𝑘,𝑥_−𝑘+1,…,𝑥₋₁,𝑥₀≤𝑀₀(3.8) such that 𝑚₀≤𝑓𝑚₀_𝑝,𝑀₀_𝑞≤𝑓𝑀⁰_𝑝,𝑚₀_𝑞≤𝑀⁰,(3.9) then there exist (𝑚,𝑀)∈[𝑚₀,𝑀₀]² satisfying 𝑀=𝑓[𝑀]_𝑝,[𝑚]_𝑞,𝑚=𝑓[𝑚]_𝑝,[𝑀]_𝑞.(3.10) Moreover, if 𝑚=𝑀, then (2.2) has a unique equilibrium point 𝑥∈[𝑚₀,𝑀₀] and every solution of (2.2) converges to 𝑥.

Proof. Using 𝑚₀ and 𝑀₀ as a couple of initial iteration, we construct two sequences {𝑚_𝑖} and {𝑀_𝑖} (𝑖=1,2,…) from the equation 𝑚_𝑖=𝑓𝑚_𝑖−1_𝑝,𝑀_𝑖−1_𝑞,𝑀^𝑖=𝑓𝑀_𝑖−1_𝑝,𝑚_𝑖−1_𝑞.(3.11) It is obvious from the mixed monotone property of 𝑓 that the sequences {𝑚_𝑖} and {𝑀_𝑖} possess the following monotone property 𝑚₀≤𝑚₁≤⋯≤𝑚_𝑖≤⋯≤𝑀_𝑖≤⋯≤𝑀₁≤𝑀₀,(3.12) where 𝑖=0,1,2,…, and 𝑚_𝑖≤𝑥_𝑙≤𝑀_𝑖for𝑙≥(𝑘+1)𝑖+1.(3.13) Set 𝑚=lim_𝑖→∞𝑚_𝑖,𝑀=lim_𝑖→∞𝑀_𝑖,(3.14) then 𝑚≤lim_𝑖→∞inf𝑥_𝑖≤lim_𝑖→∞sup𝑥_𝑖≤𝑀.(3.15) By the continuity of 𝑓, we have 𝑀=𝑓[𝑀]_𝑝,[𝑚]_𝑞,𝑚=𝑓[𝑚]_𝑝,[𝑀]_𝑞.(3.16) Moreover, if 𝑚=𝑀, then 𝑚=𝑀=lim_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,…,𝑥₁,𝑥₀∈0,2𝛼𝑎^𝑟+1.(3.17)

Proof. Let 𝑓∶(0,∞)⁴→(0,∞) be a function defined by 𝑓(𝑢,𝑣,𝑤,𝑠)=𝑎𝑢𝑣𝛼+𝑏𝑤+𝑐𝑠.(3.18) We can easily see that the function 𝑓(𝑢,𝑣,𝑤,𝑠) is increasing in 𝑢,𝑣 and decreasing in 𝑤,𝑠.
Let 𝑀₀=max𝑥_−𝑟,𝑥_−𝑟+1,…,𝑥₋₁,𝑥₀,𝑎𝑀⁰−𝛼𝑏+𝑐<𝑚⁰<0;(3.19) we have 𝑚₀≤𝑎𝑚²⁰𝛼+𝑏𝑀₀+𝑐𝑀₀≤𝑎𝑀²⁰𝛼+𝑏𝑚₀+𝑐𝑚₀≤𝑀₀.(3.20) Then from (1.6) and Theorem 3.2, there exist 𝑚,𝑀∈[𝑚₀,𝑀₀] satisfying 𝑚=𝑎𝑚²𝛼+𝑏𝑀+𝑐𝑀,𝑀=𝛼+𝑎𝑏𝑀𝑚+²𝑐𝑚,(3.21) thus [𝛼−𝑎(𝑚+𝑀)](𝑚−𝑀)=0.(3.22) In view of 2𝑎𝑀₀<𝛼, we have 𝛼−𝑎(𝑚+𝑀)>0.(3.23) Then 𝑀=𝑚.(3.24) 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,…,𝑥₁,𝑥₀∈0,2𝛼𝑎^𝑟+1.(3.25)

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 equations𝑥_𝑛+1=𝑥^𝑛𝑥_𝑛−15+2𝑥_𝑛+𝑥_𝑛−1𝑥,𝑛=0,1,…,(4.1)_𝑛+1=𝑥^𝑛𝑥_𝑛−15+2𝑥_𝑛−2+𝑥_𝑛−3,𝑛=0,1,…,(4.2) where the initial conditions 𝑥₋₃,𝑥₋₂,𝑥₋₁,𝑥₀∈(0,2.5). Let 𝑚₀=−0.5,𝑀₀=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,𝑥₀=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,𝑥₀=1.4, and the relations that 𝑚_𝑖≤𝑥_𝑙≤𝑀_𝑖 when 𝑙≥(𝑘+1)𝑖+1,𝑖=0,1,2,….

Figure 1 

Chart of (4.1) with 𝑥(−1)=1.2,𝑥(0)=1.8.

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.