Asymptotic Behavior of Equilibrium Point for a Class of Nonlinear Difference Equation
© Chang-you Wang et al. 2009
Received: 17 February 2009
Accepted: 17 September 2009
Published: 9 November 2009
We study the asymptotic behavior of the solutions for the following nonlinear difference equation where the initial conditions are arbitrary nonnegative real numbers, are nonnegative integers, , and are positive constants. Moreover, some numerical simulations to the equation are given to illustrate our results.
Difference equations appear naturally as discrete analogues and in the 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 globally asymptotic behavior of solutions of rational difference equations [2–6]. In particular, Elabbasy et al.  investigated the global stability and periodicity of the solution for the following recursive sequence:
In  Elabbasy et al. investigated the global stability, boundedness, and the periodicity of solutions of the difference equation:
Yang et al.  investigated the global attractivity of equilibrium points and the asymptotic behavior of the solutions of the recursive sequence:
The purpose of this paper is to investigate the global attractivity of the equilibrium point, and the asymptotic behavior of the solutions of the following difference equation
where the initial conditions are arbitrary nonnegative real numbers, , are nonnegative integers, and , are positive constants. Moreover, some numerical simulations to the equation 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
has a unique solution .
That is, for is a solution of (2.2), or equivalently, is a fixed point of .
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 .
Let be an equilibrium point of (2.2).
(i) is stable if, for every , there exists such that for any initial conditions with , hold 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.
3. The Main Results and Their Proofs
In this section we investigate the globally asymptotic stability of the equilibrium point of (1.4).
Let be a function defined by
If , then it follows that
Let be the equilibrium points of (1.4), then we have
Moreover, we have that
Thus, the linearized equation of (1.4) about is
If and , then the equilibrium point of (1.4) is locally stable.
from which the result follows.
Moreover, if , then (2.2) has a unique equilibrium point and every solution of (2.2) converges to .
Moreover, if , then , and then the proof is complete.
then the equilibrium point of (1.4) is global attractor when .
It follows by Theorem 3.2 that the equilibrium point of (1.4) is global attractor. The proof is therefore complete.
4. Numerical Simulations
In this section, we give numerical simulations supporting our theoretical analysis. As examples, we consider the following difference equations:
Let . It is obvious that (4.1) and (4.2) satisfy the conditions of Theorem 3.3 when the initial conditions are .
The authors are grateful to the referees for their comments. This work is supported by the Science and Technology Project of Chongqing Municipal Education Commission (Grant no. KJ 080511) of China, Natural Science Foundation Project of CQ CSTC (Grant no. 2008BB 7415) of China, the NSFC (Grant no.10471009), and the BSFC (Grant no. 1052001) of China.
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