Asymptotic Stability for a Class of Nonlinear Difference Equations
We study the global asymptotic stability of the equilibrium point for the fractional difference equation , , where the initial conditions are arbitrary positive real numbers of the interval are nonnegative integers, and are positive constants. Moreover, some numerical simulations are given to illustrate our results.
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 . 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  put forward the idea of investigating the global asymptotic stability of the following difference equation: where the initial values .
By using semicycle analysis methods, the authors of  got a sufficient condition which guarantees the global asymptotic stability of the following difference equation: where and the initial values .
Yang et al.  investigated the invariant intervals, the global attractivity of equilibrium points, and the asymptotic behavior of the solutions of the recursive sequenceBerenhaut et al.  generalized the result reported in  to the following rational equation 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: where the initial conditions are arbitrary positive real numbers of the interval is nonnegative integer, and and 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
Lemma 2.1. Let be some interval of real numbers and let be a continuously differentiable function. Then for every set of initial conditions , the difference equation has a unique solution .
Definition 2.3. Let be two nonnegative integers such that . Splitting into , where denotes a vector with -components of , we say that the function possesses a mixed monotone property in subsets of if is monotone nondecreasing in each component of and is monotone nonincreasing in each component of for . In particular, if , then it is said to be monotone nondecreasing in .
Definition 2.4. Let be an equilibrium point of (2.2).(i) is stable if for every , there exists such that for any initial conditions with , holds for(ii) is a local attractor if there exists such that holds for any initial conditions with .(iii) is locally asymptotically stable if it is stable and is a local attractor.(iv) is a global attractor if holds for any initial conditions .(v) is globally asymptotically stable if it is stable and is a global attractor.(vi) is unstable if it is not locally stable.
The linearized equation of (2.2) about the equilibrium is the linear difference equation Now assume that the characteristic equation associated with (2.4) is where .
Lemma 2.5. Assume that and . Then is a sufficient condition for the local asymptotically stability of the difference equation
Lemma 2.6. Assume that is a function and let 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 , then the equilibrium point 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 of (2.2) is unstable.
3. The Main Results and Their Proofs
In this section, we investigate the global asymptotic stability of the equilibrium point of (1.6).
Let be a function defined by then it follows that Let be the equilibrium points of (1.6), then we have where . If , then is a unique equilibrium point.
Moreover, Thus, the linearized equations of (1.6) about equilibrium points and are, respectively, where are nonnegative different integers.
The characteristic equation associated with (3.6) is where .
Theorem 3.1. The equilibrium point 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 , then the equilibrium point 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 of (1.6) is unstable.
Theorem 3.2. Let be an interval of real numbers and assume that is a continuous function satisfying the mixed monotone property. If there exists such that then there exist satisfying Moreover, if , then (2.2) has a unique equilibrium point and every solution of (2.2) converges to .
Proof. Using and as a couple of initial iteration, we construct two sequences and () from the equation It is obvious from the mixed monotone property of that the sequences and possess the following monotone property where , and Set then By the continuity of , we have Moreover, if , then , and then the proof is complete.
Theorem 3.3. The equilibrium point of (1.6) is a global attractor for any initial conditions
Proof. Let be a function defined by
We can easily see that the function is increasing in and decreasing in .
Let we have Then from (1.6) and Theorem 3.2, there exist satisfying thus In view of , we have Then It follows by Theorem 3.2 that the equilibrium point of (1.6) is a global attractor. The proof is therefore complete.
Theorem 3.4. The equilibrium point of (1.6) is a global asymptotic stability for any initial conditions
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 where the initial conditions . Let ; 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 , and the relations that when , and Figure 2 shows the numerical solutions of (4.2) with , and the relations that when
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.
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.
G. Ladas, “Open problem and conjectures,” Journal of Difference Equations and Applications, vol. 3, pp. 317–321, 1997.View at: Google Scholar
K. S. Berenhaut and S. Stevic, “The difference equation has solutions converging to zero,” Journal of Mathematical Analysis and Applications, vol. 326, pp. 1466–1471, 2007.View at: Google Scholar
M. R. S. Kulenovic and G. Ladas, Dynamics of Second Order Rational Difference Equations with Open Problems and Conjectures, Chapman & Hall/CRC Press, Boca Raton, Fla, USA, 2001.
V. L. Kocic and G. Ladas, Global Behavior of Nonlinear Difference Equations of Higher Order with Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1993.