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Discrete Dynamics in Nature and Society
Volume 2013 (2013), Article ID 742912, 3 pages
http://dx.doi.org/10.1155/2013/742912
Research Article

The Periodicity of Positive Solutions of the Nonlinear Difference Equation

Department of Mathematics, Faculty of Science and Arts, Bülent Ecevit University, 67100 Zonguldak, Turkey

Received 30 November 2012; Revised 4 February 2013; Accepted 6 February 2013

Academic Editor: M. De la Sen

Copyright © 2013 Mehmet Gümüş. 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 give a remark about the periodic character of positive solutions of the difference equation , , where is an odd integer, and the initial conditions are arbitrary positive numbers.

1. Introduction

In this paper, we consider the difference equation where is an odd integer, is positive, , and the initial conditions are arbitrary positive numbers. Equation (1) was studied by many authors for different cases of .

In [1], the authors studied the special case of of (1), that is, the recursive sequence where are positive and the initial values are positive numbers.

Recently, in [2], the authors obtained the periodicity results of positive solutions of (1). They investigated the existence of a prime periodic solution of (1). But they did not investigate the positive solutions of (1) which converge to a prime two-periodic solution.

There exist many other papers related with (1) and on its extensions (see [38]).

Our aim in this paper is to give a remark about the periodic character of all positive solutions of (1). We show that all positive solutions of (1), for is odd, converge to a prime two-periodic solution.

We believe that difference equations, also referred to as recursive sequence, are a hot topic. There has been an increasing interest in the study of qualitative analysis of difference equations and systems of difference equations. Difference equations appear naturally as discrete analogues and as numerical solutions of differential and delay differential equations having applications in biology, ecology, economics, physics, computer sciences, and so on.

Here, we recall some notations and results which will be useful in our proofs.

Let be some interval of real numbers and let be continuous function defined on . Then, for initial conditions , it is easy to see that the difference equation has a unique solution .

We say that the equilibrium point of (3) is the point that satisfies the condition That is, for is a solution of (3), or equivalently, is a fixed point of .

A solution of (3) is said to be periodic with period if for all

A solution of (3) is said to be periodic with prime period , or a -cycle if it is periodic with period and is the least positive integer for which (5) holds.

The linearized equation for (1) about the positive equilibrium is

2. Main Results

In this section, we find solutions, for is odd of (1), which converge to a prime two-periodic solution. The following three results are essentially proved in [1, 2]. Hence, we omit their proofs.

Lemma 1. Suppose that then every positive solution of (1) is bounded.

Lemma 2. Assume that k is odd. Then, (1) has prime two-periodic solutions if and only if and there exists a sufficient small positive number , such that

Lemma 3. If either or hold, then (1) has a unique equilibrium point .

Theorem 4. Consider (1) where is odd, for every positive solution of (1) which satisfies any of the following initial conditions: (i) for  all and for  all .(ii) for  all and for  all.(iii)for  all and for  all.(iv)for  all and for  all.

Then, the sequences and are eventually monotone.

Proof. We have If for  all and for  all, from (12) and (13) we obtain and consequently . By induction we obtain Similarly if for  all and   for  all, from (12) and (13) using induction we obtain If for  all and for  all and we can obtain from (12) and (13) Therefore, the result follows immediately.

The following result is essentially proved in [1] for of (1). We obtain the same result, for is an odd integer.

Lemma 5. Consider (1) where (8) and (9) hold and is odd. Let be a solution of (1) such that either or Then if (17) holds, we have Also if (18) holds, we have

Proof. Firstly, assume that (17) holds. Then, from (9) we have Working inductively, we can get Therefore, we can easily prove relations (19). Similiarly if (18) holds, we can prove that (20) is satisfied.

Lemma 6. If , then every positive solution of (1) satisfies the following inequalities: and here the cases hold.

Proof. We have for  all. By induction, we obtain (23) and (24). If here , we also see that

Now, we are ready for the main result of this paper.

Theorem 7. Consider (1) where (8) and (9) hold and is odd. Suppose that then every positive solution with initial values , which satisfy conditions of Theorem 4 and either (17) or (18), converges to a prime two-periodic solution.

Proof. Let be a solution with initial values , which satisfy conditions of Theorem 4, and either (17) or (18). Using Lemma 1 and Theorem 4, we have that there exist Besides, from Lemma 5 we have that either or belongs to the interval . Furthermore, from Lemma 3 we have that (1) has a unique equilibrium such that . Therefore, from (27) we have that . So converges to a prime two-periodic solution. The proof is complete.

Acknowledgments

The authors would like to thank the referees for their helpful suggestions. This research is supported by TUBITAK and Bulent Ecevit University Research Project Coordinatorship.

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