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Discrete Dynamics in Nature and Society
Volume 2012 (2012), Article ID 475038, 8 pages
http://dx.doi.org/10.1155/2012/475038
Research Article

The Dynamics of the Solutions of Some Difference Equations

1Department of Mathematics, College of Science and Arts, King Abdulaziz University, Rabigh Campus, P.O. Box 344, Rabigh 21911, Saudi Arabia
2Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt
3Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia

Received 22 December 2011; Revised 17 May 2012; Accepted 17 May 2012

Academic Editor: Ibrahim Yalcinkaya

Copyright © 2012 H. El-Metwally 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 is devoted to investigate the global behavior of the following rational difference equation: , where and with the initial conditions ,. We will find and classify the equilibrium points of the equations under studying and then investigate their local and global stability. Also, we will study the oscillation and the permanence of the considered equations.

1. Introduction

The aim of this paper is to study the dynamics of the solutions of the following recursive sequence: where and , where , with the initial conditions ,  ,  ,. We deal with the classification of the equilibrium points of (1.1) in terms of being stable or unstable, where we investigate the global attractor of the solutions of (1.1) as well as the permanence of the equation. Also, we establish some appropriate conditions, which grantee the oscillation of the solutions of (1.1). For more results in the direction of this study, see, for example, [123] and the papers therein.

In the sequel, we present some well-known results and definition that will be useful in our investigation of (1.1). 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 1.1 (permanence). The difference equation (1.2) is said to be permanent if there exist numbers and with such that, for any initial conditions , there exists a positive integer which depends on the initial conditions such that for all .

Definition 1.2 (periodicity). A sequence is said to be periodic with period if for all .

Definition 1.3 (semicycles). A positive semicycle of a sequence consists of a “string” of terms all greater than or equal to the equilibrium point , with and such that either or and ; either or and . A negative semicycle of a sequence consists of a “string” of terms all less than the equilibrium point , with and such that either or and ; either or and .

Definition 1.4 (oscillation). A sequence is called nonoscillatory about the point if there exists such that either for all or for all . Otherwise, is called oscillatory about .

2. Dynamics of (1.1)

The change of variables reduces (1.1) to the following difference equation where .

In this section, we study the local stability character and the global stability of the equilibrium points of the solutions of (2.1). Also, we give some results about the oscillation and the permanence of (2.1).

Recall that the equilibrium points of (2.1) are given by Then, whenever , (2.1) has the only equilibrium point , and, while at , (2.1) possesses the unique positive equilibrium point .

The following theorem deals with the local stability of the equilibrium point of (2.1).

Theorem 2.1. The following statements are true. (i)If , then the equilibrium point of (2.1) is locally asymptotically stable.(ii)If , then the equilibrium point of (2.1) is a saddle point.(iii)If , then the equilibrium point of (2.1) is nonhyperbolic with and .

Proof. The linearized equation of (2.1) about is . Then, the associated eigenvalues are and . Then, the proof is complete.

Theorem 2.2. Assume that , then the equilibrium point of (2.1) is globally asymptotically stable.

Proof. Let be a solution of (2.1). It was shown in Theorem 2.1 that the equilibrium point   of (2.1) is locally asymptotically stable. So it suffices to show that Now, it follows from (2.1) that Now, assume that . Then, it follows from (1.1) and after some simple computations are achieved that , . Therefore as , and this completes the proof.

Theorem 2.3. Assume that .  Then, every solution of (2.1) is either oscillatory or tends to the equilibrium point  .

Proof. Let be a solution of (2.1). Without loss of generality, assume that is a nonoscillatory solution of (2.1), then it is suffices to show that . Assume that for (the case where for is similar and will be omitted). It follows from (2.1) that Hence, each subsequence ,  , of is decreasing sequence and therefore it has a limit. Let for some , , and, for the sake of contradiction, assume that . Then, by taking the limit of both sides of we obtain , which contradicts the hypothesis that is the only positive solution of (2.2). Therefore, , for all . This means that all the subsequences ,  , of have the same limit, , and therefore , which completes the proof.

Theorem 2.4. Assume that , , and let be a solution of (2.1) which is strictly oscillatory about the positive equilibrium point of (2.1). Then, the extreme point in any semicycle occurs in one of the first terms of the semicycle.

Proof. Assume that is a strictly oscillatory solution of (2.1). Let , and let be a positive semicycle followed by the negative semicycle . Now, it follows from (2.1) that Then, for all .
Similarly, we see from (2.1) that
Therefore, for all . The proof is complete.

Theorem 2.5. Assume that , and let be a solution of (2.1) which is strictly oscillatory about the positive equilibrium point of (2.1). Then, the extreme point in any semicycle occurs in one of the first terms of the semicycle.

Proof. Assume that is a strictly oscillatory solution of (2.1). Let , and  let be a positive semicycle followed by the negative semicycle . Now, it follows from (2.1) that Then, for all .
Similarly, we see from (2.1) that Therefore, for all . The proof is complete.

Theorem 2.6. Assume that , , and let be a solution of (2.1) which is strictly oscillatory about the positive equilibrium point of (2.1). Then, the extreme point in any semicycle occurs in one of the first terms of the semicycle.

Proof. Assume that be a strictly oscillatory solution of (2.1). Let and let be a positive semicycle followed by the negative semicycle . Now it follows from (2.1) that Then, for all .
Similarly, we see from (2.1) that Therefore for all . The proof is complete.

Theorem 2.7. Equation (2.1) is permanent.

Proof. Let be a solution of (2.1). There are two cases to consider.(i) is a nonoscillatory solution of (2.1). Then, it follows from Theorem 2.3 that that is, there is a sufficiently large positive integer such that for all and for some . So , this means that there are two positive real numbers, say and , such that .(ii) is strictly oscillatory about .
Now, let be a positive semicycle followed by the negative semicycle . If and are the extreme values in these positive and negative semicycles, respectively, with the smallest possible indices and , then by Theorem 2.4 we see that and . Now, for any positive indices and with , it follows from (2.1) for that Therefore, for and , we obtain Again whenever and , we see that That is . It follows from (i) and (ii) that . Then, the proof is complete.

Acknowledgments

This paper was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. The authors, therefore, acknowledge with thanks the DSR technical and financial support.

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