Abstract and Applied Analysis

Volume 2014, Article ID 705893, 5 pages

http://dx.doi.org/10.1155/2014/705893

## Global Behavior of the Difference Equation

^{1}College of Mathematics and Information Science, Guangxi University, Nanning, Guangxi 530004, China^{2}College of Information and Statistics, Guangxi University of Finance and Economics, Nanning, Guangxi 530003, China

Received 17 November 2013; Accepted 23 December 2013; Published 3 February 2014

Academic Editor: Tongxing Li

Copyright © 2014 Hongjian Xi 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 consider the following difference equation , where initial values and is a strictly decreasing continuous surjective function. We show the following. (1) Every positive solution of this equation converges to or for some . (2) Assume . Then the set of initial conditions such that the positive solutions of this equation converge to or is a unique strictly increasing continuous function or an empty set.

#### 1. Introduction

Recently there have been published quite a lot of works concerning global behavior of the difference equations [1–8]. These results are not only valuable in their own right, but they can provide insight into their differential counterparts.

In [9], Kulenović and Ladas considered the positive solutions for difference equation with . They gave some partial results on the convergence of this equation.

Kalikow et al. [10] studied the following difference equation: where initial values and is in a certain class of increasing continuous functions. They showed that the set of initial conditions of in the first quadrant that converge to any given boundary point of the first quadrant forms a unique strictly increasing continuous function.

Motivated by the above studies, in this paper, we consider the following difference equation: where initial values and is a strictly decreasing continuous surjective function. Our main result is the following theorem.

Theorem 1. * Every positive solution of (2) converges to
**
for some .** Assume . Then the set of initial conditions such that the positive solutions of (2) converge to
**
is a unique strictly increasing continuous function or an empty set.*

#### 2. The Main Result

*Proof of Theorem 1. *Let be a positive solution of (2). Then and are decreasing sequences since . Let and . Then we have
which implies or . If , then since is a strictly decreasing continuous surjective function with . This completes proof of Theorem 1.

Write and define by for all . It is easy to see that if is a solution of (2), then for any . In the following, let for some .

Lemma 2. *The following statements are true:*(i)*is a homeomorphism;*(ii)*;*(iii)* and .*

*Proof. *(i) Since for any with and is continuous for any , is a homeomorphism.

(ii) Let and . Then , , and
which implies .

Let . Then and . Choose . Then . Thus .

The proof of (iii) is similar to that of (ii). This completes the proof of Lemma 2.

*In order to show Theorem 1, we will construct two families of strictly increasing functions and () as follows. Set
Then is a strictly increasing function which maps onto . Set
Then is a strictly increasing function which maps onto .*

*Assume that, for some positive integer , we already define strictly increasing functions and such that both and map onto . Set
Then both and are strictly increasing functions which map onto . In such a way, we construct two families of strictly increasing functions and ().*

*Set and . For any , write
*

*Let . Since and , it follows that
Thus and .*

*Let . Since and , it follows that
Thus and . Using induction, one can easily show that, for any ,
Since is a homeomorphism and with is the boundary of , we have that, for any ,
In a similar fashion, we may show that
Since , , and is a homeomorphism, we have that and , which implies that, for any ,
It follows from (12) and (18) that, for ,
and for ,
Noting (19) and (20), we may assume that, for every ,
and for every ,
*

*Set
*

*Lemma 3. The following statements are true:(i) and ;(ii)both and are increasing continuous functions which map onto .*

*Proof. *(i) Let . Then we have , which follows that
Since , we have
Let . It follows from (24) and (25) that
so we have
It follows from (25) and (27) that
Thus we have .

Let . Then we have , which follows that
Since , we have
Let . It follows from (29) and (30) that
so we have
It follows from (31) and (32) that
Thus we have . In a similar fashion, we can show that .

(ii) Since are strictly increasing functions, we have that is an increasing function. For any , let
then .

Now we claim that . Indeed, if , then it follows from (6) that
So we have that
It follows from (34) and (36) that there exist such that
It follows from Lemma 3(i) and (37) that
and this is a contradiction. The claim is proven.

In a similar fashion, we may show that Thus is an increasing continuous function. In a similar fashion, we may show that is an increasing continuous function. Lemma 3 is proven.

*Let
where and . It follows from Lemma 2(iii) and Lemma 3(ii) that
*

*Proof of Theorem 1. *Noting (40), we consider the following two cases.*Case 1 *. It follows from (40) that
Let and be a solution of (2) with initial value ; it follows from Lemma 3(i) that
which implies that . It follows from (42) and Theorem 1 that
Next we claim that is a strictly increasing function. Indeed, if there exists , such that and , then there exist such that . Set
Then we have
Using induction, one can show that, for any ,
It follows from (44) and (47) that
This is a contradiction. The claim is proven.

Now let with and be a solution of (2) with initial value .

If , then it follows from Theorem 1 and (2) that which implies .

If and , then there exists such that
from which it follows that
Then we have , which implies .

If and , then let and , and there exists such that . We can show that, for any ,
which implies
From all abovementioned, the set of initial conditions such that the positive solutions of (2) converge to
is .

In a similar fashion, we also may show that the set of initial conditions such that the positive solutions of (2) converge to
is .*Case 2* (). It follows from (41) and Case 1 that the set of initial conditions such that the positive solutions of (2) converge to
is an empty set. This completes the proof of Theorem 1.

*Conflict of Interests*

*Conflict of Interests*

*The authors declare that there is no conflict of interests regarding the publication of this paper.*

*Acknowledgments*

*Acknowledgments*

*This project was supported by NNSF of China (11261005), NSF of Guangxi (2012GXNSFDA276040), and SF of ED of Guangxi (2013ZD061).*

*References*

*References*

- A. Janssen and D. Tjaden, “Solution to problem 86-2,”
*Mathematical Intelligencer*, vol. 9, pp. 40–43, 1987. View at Google Scholar - C. Kent, “Convergence of solutions in a nonhyperbolic case,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 47, no. 7, pp. 4651–4665, 2001. View at Google Scholar - Q. He, T. Sun, and H. Xi, “Dynamics of a family of nonlinear delay difference equations,”
*Abstract and Applied Analysis*, vol. 2013, Article ID 456530, 4 pages, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - W. Liu and S. Stević, “Global attractivity of a family of nonautonomous max-type difference equations,”
*Applied Mathematics and Computation*, vol. 218, no. 11, pp. 6297–6303, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - F. Sun, “On the asymptotic behavior of a difference equation with maximum,”
*Discrete Dynamics in Nature and Society*, vol. 2008, Article ID 243291, 6 pages, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Stević, “Global stability of a max-type difference equation,”
*Applied Mathematics and Computation*, vol. 216, no. 1, pp. 354–356, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - T. Sun, H. Xi, C. Han, and B. Qin, “Dynamics of the max-type difference equation ${x}_{n}=max$$\{$$1/{x}_{n-m}$, ${A}_{n}/{x}_{n-r}$$\}$,”
*Journal of Applied Mathematics and Computing*, vol. 38, no. 1-2, pp. 173–180, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - T. Sun, X. Wu, Q. He, and H. Xi, “On boundedness of solutions of the difference equation ${x}_{n+1}={p+x}_{n-1}/{x}_{n}$ for $p<1$,”
*Journal of Applied Mathematics and Computing*, vol. 44, no. 1-2, pp. 61–68, 2014. View at Publisher · View at Google Scholar - M. R. S. Kulenović and G. Ladas,
*Dynamics of Second Order Rational Difference Equations: With Open Problems and Conjectures*, Chapman & Hall/CRC Press, Boca Raton, Fla, USA, 2002. View at MathSciNet - S. Kalikow, P. M. Knopf, Y. S. Huang, and G. Nyerges, “Convergence properties in the nonhyperbolic case ${x}_{n+1}={x}_{n-1}/\left(1+f\left({x}_{n}\right)\right)$,”
*Journal of Mathematical Analysis and Applications*, vol. 326, no. 1, pp. 456–467, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet

*
*