Discrete Dynamics in Nature and Society

Volume 2013 (2013), Article ID 963757, 5 pages

http://dx.doi.org/10.1155/2013/963757

Research Article

Global Behavior of

Chenquan Gan, Xiaofan Yang, and Wanping Liu

College of Computer Science, Chongqing University, Chongqing 400044, China

Received 29 June 2013; Revised 9 October 2013; Accepted 16 October 2013

Academic Editor: M. De la Sen

Copyright © 2013 Chenquan Gan 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.

This paper aims to investigate the global stability of negative solutions of the difference equation , , where the initial conditions is a positive integer, and the parameters . By utilizing the invariant interval and periodic character of solutions, it is found that the unique negative equilibrium is globally asymptotically stable under some parameter conditions. Additionally, two examples are given to illustrate the main results in the end.

1. Introduction

The study of nonlinear difference equations has always attracted a considerable attention (see, e.g., [1–30] and the references cited therein). In particular, some references investigated the dynamical behavior of positive solutions of difference equations (see, e.g., [3–5, 11]), and some references examined the dynamical behavior of negative solutions of some difference equations (see, e.g., [6, 7]).

Gibbons et al. [4] studied the behavior of nonnegative solutions to the recursive sequence

with , and also presented an open problem, which had been solved by in [8]. and Ladas, in addition, considered (1) in their book [9].

[10] considered the behavior of nonnegative solutions of the following second-order difference equation

where is a nonnegative increasing mapping.

Douraki et al. [11] studied the qualitative behavior of positive solutions of the difference equation

where the initial values , is a positive integer, and . Moreover, (3) is a special case of the following open problem (see also in [11]), which was proposed by and Ladas in [9].

Open Problem (equation in [9]). Assume that . Investigate the global behavior of positive solutions of (3).

[12] considered the boundedness, oscillatory behavior, and global stability of nonnegative solutions of the difference equation

where , and is a continuous function nondecreasing in each variable such that .

It is worthwhile to note that the above mentioned references ([4, 10–12]), especially [4, 11], only discussed the dynamical behavior of positive solutions of difference equation. Furthermore, inspired by the above work and [6, 7], the main goal of this paper is to study the global behavior of negative solutions of the difference equation

where is a positive integer, , and the initial conditions .

In fact, it is easy to see that (5) is an extension of an open problem introduced by and Ladas in [9] and also is a special case of (4) by a simple change. However, here we establish some results regarding the global stability, invariant interval, and periodic character of negative solutions of (5).

2. Linearized Stability and Period 2 Solutions

The aim of this section is to discuss the local stability of the unique negative equilibrium of (5). The period 2 solutions of (5), in addition, will be verified.

In this section, we need the following lemma.

Lemma 1 (see [3]). Assume that and . Then

is a sufficient condition for the asymptotic stability of the difference equation

Suppose in addition that one of the following two cases holds.

(a) is odd and .(b) is even and .

Then (6) is also a necessary condition for the asymptotic stability of (7).

Assume that is an equilibrium of (5). Then it satisfies the equation , which implies that the unique negative equilibrium of (5) is

Let

Then the linearized equation of (5) at is

where

Straightforward calculations yield

From here and Lemma 1, we obtain the following result.

Theorem 2. The unique negative equilibrium of (5) is locally asymptotically stable if .

Proof. If , then

It follows from Lemma 1 that is locally asymptotically stable. Thus the proof is complete.

Now, we will examine the existence of period 2 solutions of (5).

Theorem 3. Let be a negative solution of (5). Then the following statements hold.

(a)Assume that is odd. Then,

(i)equation (5) has a negative prime period 2 solution if and only if ;(ii)if , then the values and of all negative prime period 2 solutions are given by ;(iii)if , then (5) has no negative solutions with prime period 2.

(b)Assume that is even. Then (5) has no negative solutions with prime period 2.

Proof. Let be a negative solution of (5).

(a)Assume that is odd. Then,

(i)suppose that is a negative prime period 2 solution. Then from (5), we have

From the above relations, we derive that

Since , then .The reverse part is clear by a simple computation. So it is omitted.(ii)Applying the relations (14), we get that and ; namely, and . As , then .(iii)If , then it follows from (15) that .

(b)Assume that is even. Let be a negative prime period 2 solution; then

It follows from the above equations that . As , then , which contradicts the hypothesis that .

3. Invariant Interval

In this section, we will consider the invariant interval of negative solutions of (5).

Let

Lemma 4. The following statements are true.

(a)Assume that . Then,

(i);(ii)if , then and .

(b)Assume that is defined by (17) and . Then is nonincreasing in and nondecreasing in .

Proof. The proofs of (a) and (b) are as follows:

(a)if , then

(ii), clearly, ;(iii)the condition implies that . Then

which leads to . Note that ; thus . Also

Clearly, .

(b)Note that

So the result holds.

Theorem 5. Assume that . Then is an invariant interval of (5).

Proof. Suppose that is a solution to (5) with initial conditions . Since , it follows by a direct computation that . By Lemma 4(b), we immediately get that the function is nonincreasing in and nondecreasing in with . Then

which implies that . It follows by induction that for all . Thus, the proof is complete.

4. Global Stability

Recall that (5) has a unique negative equilibrium , which is locally asymptotically stable if by Theorem 2. In this section, we will show that (a) is globally asymptotically stable under the conditions and ; (b) every negative solution converges to when .

Lemma 6 (see [3]). Consider the difference equation

where . Let be some interval of real numbers and assume that

is a continuous function satisfying the following properties:

(a) is nonincreasing in and nondecreasing in ;(b)if is a solution of the system

 then .

Then (22) has a unique equilibrium and every solution of (22) converges to .

Theorem 7. Assume that and . Then the following statements are true:

(i)the unique negative equilibrium of (5) is a global attractor with a basin ;(ii)the unique negative equilibrium of (5) is globally asymptotically stable.

Proof. Suppose that is a solution to (5) with initial values . By Lemma 4(b), we immediately get that the function is continuous and nonincreasing in and nondecreasing in on the invariant interval . Furthermore, let be a solution of the system , ; then by Lemma 4(a)(i) and (ii), we have and , and by Theorem 3, we obtain . Finally, it follows by Lemma 6 that . Hence, the proof is complete.

Lemma 8. Let . Then (5) has no nontrivial negative periodic solutions of (not necessarily prime) period .

Proof. If , then substituting (8), it becomes that . Suppose that is a negative solution to (5) satisfying for all ; then

Simplifying the above equation, it follows that . Clearly, for all . The proof is complete.

Theorem 9. Assume that . Then every negative solution to (5) converges to the unique negative equilibrium .

Proof. Assume that . It follows from Lemma 8 that . Then

Case  1. If , then , so .

Case  2. If , then , so .

Next, we consider Case 1 (Case 2 is similar and thus it is omitted). If for all , then it is clear that for there exists such that

But then is a periodic solution of (not necessarily prime) period . By Lemma 8 the result holds.

5. Examples

To illustrate the main results in Section 4, here we present two examples.

Example 1. Consider (1) with , and . Then . As and , it follows from Theorem 7(ii) that the unique negative equilibrium is globally asymptotically stable. For the initial conditions , Figure 1 exhibits how evolves with .

Figure 1: Evolution of for (1) with parameters and initial condition given in Example 1.

Example 2. Consider (1) with , and . Then . As , it follows from Theorem 9 that the unique negative equilibrium is globally attractive. For the initial conditions , Figure 2 exhibits how evolves with .

Figure 2: Evolution of for (1) with parameters and initial condition given in Example 2.

Acknowledgments

The authors are greatly indebted to the referees for their valuable suggestions. This work is financially supported by the National Natural Science Foundation of China (no. 10771227) and the Fundamental Research Funds for the Central Universities (no. CDJXS10181130).

References

M. T. Aboutaleb, M. A. El-Sayed, and A. E. Hamza, “Stability of the recursive sequence ,” Journal of Mathematical Analysis and Applications, vol. 261, no. 1, pp. 126–133, 2001.

X. Yang, Y. Yang, and J. Luo, “On the difference equation ,” Applied Mathematics and Computation, vol. 189, no. 1, pp. 918–926, 2007.

R. DeVault, W. Kosmala, G. Ladas, and S. W. Schultz, “Global behavior of ,” Nonlinear Analysis: Theory, Methods & Applications, vol. 47, no. 7, pp. 4743–4751, 2001.

C. H. Gibbons, M. R. S. Kulenović, and G. Ladas, “On the recursive sequence ,” Mathematical Sciences Research Hot-Line, vol. 4, no. 2, pp. 1–11, 2000.

F. Sun, X. Yang, and C. Zhang, “On the recursive sequence ,” Discrete Dynamics in Nature and Society, vol. 2009, Article ID 608976, 8 pages, 2009.

X.-M. Jia and L.-X. Hu, “Global attractivity of a higher-order nonlinear difference equation,” International Journal of Difference Equations, vol. 5, no. 1, pp. 95–101, 2010.

M. Saleh and M. Aloqeili, “On the difference equation with ,” Applied Mathematics and Computation, vol. 176, no. 1, pp. 359–363, 2006.

S. Stević, “On the recursive sequence ,” Taiwanese Journal of Mathematics, vol. 6, no. 3, pp. 405–414, 2002.

M. R. S. Kulenović and G. Ladas, Dynamics of Second Order Rational Difference Equations with Open Problems and Conjecture, Chapman and Hall/CRC, Boca Raton, Fla, USA, 1st edition, 2001.

S. Stević, “On the recursive sequence ,” Indian Journal of Pure and Applied Mathematics, vol. 33, no. 12, pp. 1767–1774, 2002.

M. J. Douraki, M. Dehghan, and M. Razzaghi, “The qualitative behavior of solutions of a nonlinear difference equation,” Applied Mathematics and Computation, vol. 170, no. 1, pp. 485–502, 2005.

S. Stević, “On the recursive sequence ,” Taiwanese Journal of Mathematics, vol. 9, no. 4, pp. 583–593, 2005.

K. C. Cunningham, M. R. S. Kulenović, G. Ladas, and S. V. Valicenti, “On the recursive sequence ,” Nonlinear Analysis: Theory, Methods & Applications, vol. 47, no. 7, pp. 4603–4614, 2001.

L.-X. Hu, W.-T. Li, and S. Stević, “Global asymptotic stability of a second order rational difference equation,” Journal of Difference Equations and Applications, vol. 14, no. 8, pp. 779–797, 2008.

V. L. Kocić and G. Ladas, Global Behavior of Nonlinear Difference Equations of Higher Order with Applications, vol. 256 of Mathematics and its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1993.

M. R. S. Kulenović and O. Merino, “Global attractivity of the equilibrium of for ,” Journal of Difference Equations and Applications, vol. 12, no. 1, pp. 101–108, 2006.

W.-T. Li and H.-R. Sun, “Global attractivity in a rational recursive sequence,” Dynamic Systems and Applications, vol. 11, no. 3, pp. 339–345, 2002.

W. Li, Y. Zhang, and Y. Su, “Global attractivity in a class of higher-order nonlinear difference equation,” Acta Mathematica Scientia: Series B, vol. 25, no. 1, pp. 59–66, 2005.

W. Liu and X. Yang, “Quantitative bounds for positive solutions of a Stević difference equation,” Discrete Dynamics in Nature and Society, vol. 2010, Article ID 235808, 14 pages, 2010.

W. Liu, X. Yang, and J. Cao, “On global attractivity of a class of nonautonomous difference equations,” Discrete Dynamics in Nature and Society, vol. 2010, Article ID 364083, 13 pages, 2010.

S. Stević, “On the difference equation ,” Computers & Mathematics with Applications, vol. 56, no. 5, pp. 1159–1171, 2008.

S. Stević, “Global stability of a max-type difference equation,” Applied Mathematics and Computation, vol. 216, no. 1, pp. 354–356, 2010.

S. Stević, “On a generalized max-type difference equation from automatic control theory,” Nonlinear Analysis: Theory, Methods & Applications, vol. 72, no. 3-4, pp. 1841–1849, 2010.

Y.-H. Su, W.-T. Li, and S. Stević, “Dynamics of a higher order nonlinear rational difference equation,” Journal of Difference Equations and Applications, vol. 11, no. 2, pp. 133–150, 2005.

Y.-H. Su and W.-T. Li, “Global attractivity of a higher order nonlinear difference equation,” Journal of Difference Equations and Applications, vol. 11, no. 10, pp. 947–958, 2005.

X.-X. Yan, W.-T. Li, and H.-R. Sun, “Global attractivity in a higher order nonlinear difference equation,” Applied Mathematics E-Notes, vol. 2, pp. 51–58, 2002.

X.-X. Yan and W.-T. Li, “Global attractivity in the recursive sequence ,” Applied Mathematics and Computation, vol. 138, no. 2-3, pp. 415–423, 2003.

X. Yang, W. Liu, and J. Liu, “Global attractivity of a family of max-type difference equations,” Discrete Dynamics in Nature and Society, vol. 2011, Article ID 506373, 12 pages, 2011.

X. Yang, Y. Y. Tang, and J. Cao, “Global asymptotic stability of a family of difference equations,” Computers & Mathematics with Applications, vol. 56, no. 10, pp. 2643–2649, 2008.

Y. Yang and X. Yang, “On the difference equation ,” Applied Mathematics and Computation, vol. 203, no. 2, pp. 903–907, 2008.