The Periodicity of Positive Solutions of the Nonlinear Difference Equation
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.
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 .
Recently, in , 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.
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 .
A solution of (3) is said to be periodic with period if for all
The linearized equation for (1) about the positive equilibrium is
2. Main Results
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.
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.
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.
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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