`International Journal of Mathematics and Mathematical SciencesVolume 2008, Article ID 391265, 15 pageshttp://dx.doi.org/10.1155/2008/391265`
Research Article

On the Rational Recursive Sequence

1Mathematics Department, Faculty of Science, Taif University, El-Taif 5700, El-Hawiyah, Kingdom of Saudi Arabia
2Mathematics Department, Faculty of Science, Zagazig University, Zagazig 4419, Egypt
3Mathematics Department, Faculty of Science, El-Minia University, El Minia 61519, Egypt

Received 1 November 2007; Accepted 11 May 2008

Copyright © 2008 E. M. E. Zayed 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 study the global stability, the periodic character, and the boundedness character of the positive solutions of the difference equation ,, in the two cases: (i) ; (ii) , where the coefficients and, and the initial conditions are real numbers. We show that the positive equilibrium of this equation is a global attractor with a basin that depends on certain conditions posed on the coefficients of this equation.

1. Introduction

The asymptotic stability of the rational recursive sequence, was investigated when the coefficients are nonnegative real numbers (see ). Studying the asymptotic behavior of the rational sequence (1.1) when some of the coefficients are negative was suggested in . Recently, Aboutaleb et al.  studied the rational recursive sequence, where are nonnegative real numbers and obtained sufficient conditions for the global attractivity of the positive equilibria. Yan et al.  studied recently the rational recursive sequence, where are real numbers while is an integer number, and the initial conditions are arbitrary real numbers. They proved that the positive equilibrium of (1.3) is a global attractor with a basin that depends on certain conditions of the coefficients. He et al.  studied recently the rational recursive sequence, where are real numbers while is an integer number and the initial conditions are arbitrary real numbers. They proved the global attractivity and periodic character of the positive solution of (1.4). Stević  studied recently the rational recursive sequence, where the parameters are nonnegative real numbers and is an integer number while the initial conditions are arbitrary real numbers. Other related results can be found in .

Our aim in this paper is to study the global attractivity, the periodicity, and the boundedness of the positive solution of the following rational recursive sequence: in the two cases (i) (ii) where the coefficients are real numbers and is an integer number, while the initial conditions are arbitrary real numbers. We will prove that the positive equilibrium of (1.6) is a global attractor with a basin that depends on certain conditions of these coefficients.

2. Local Stability and Permanence

We first recall some results which will be useful in the sequel. Let be some real interval and let be a continuous function defined on . Then, for initial conditions , it is easy to see that the difference equation, has a unique solution .

Definition 2.1. A point is called an equilibrium of (2.1), if That is, for is a solution of (2.1), or equivalently, is a fixed point of .

Definition 2.2. An interval is called an invariant interval of (2.1) if the initial conditions imply that the solution for . That is, every solution of (2.1) with initial conditions in remains in .

Definition 2.3. The difference equation (2.1) 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 .

The linearized equation associated with (2.1) about the equilibrium is Its characteristic equation is

Theorem 2.4 (see ). Assume that is a -function and let be an equilibrium of (2.1). Then, the following statements are true:
(a) if all the roots of (2.3) lie in the open unit disk then the equilibrium of (2.1) is locally asymptotically stable; (b) if at least one root of (2.3) has absolute value greater than one, then the equilibrium of (2.1) is unstable.

Theorem 2.5 (see [3, 8]). 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: (i) k is odd and , or (ii) is even and . Then, (2.4) is also a necessary condition for the asymptotic stability of (2.5) (see ).

First, we study the rational recursive sequence together with the conditions

The unique positive equilibrium point of (2.6) is the solution of the equation which is given by where If (2.7) holds and then (2.6) has a unique positive equilibrium . If (2.7) holds and then (2.6) has two positive equilibria given by (2.9).

The linearized equation of (2.6) about the equilibrium is given by The characteristic equation associated with (2.6) about is Now, we have the following results:

(a)if , then and hence the equilibrium of (2.6) is unstable (see Figure 1);(b)if , then Thus, the linearized stability analysis fails. On the other hand, the characteristic equation associated with (2.6) about is Now, we have the following results:

(a)if , then it is obvious that hence the equilibrium of (2.6) is unstable;(b)if , then it is easy to see that and consequently, we have and hence the equilibrium of (2.6) is unstable.

For the positive equilibrium , in view of conditions (2.7) and , we have Hence, if then Consequently, we have which by Theorem 2.5 implies that is locally asymptotically stable (see Figure 2).

Lemma 2.6. Let and assume that conditions (2.7) and (2.18) hold. Then, the following statements are true:
(a) (b) is a strictly decreasing function in (c) let then the function is a strictly decreasing function in and strictly increasing function in .

Proof. We prove (a) only. The proofs of (b) and (c) are omitted here. In view of (2.7) and (2.18), we have From (2.8) and (2.21), we have and so . Also, in view of (2.7) and (2.18), we have and so . Consequently, which implies that . The proof is completed.

Theorem 2.7. Assume that the conditions (2.7) and (2.18) hold. Let be any solution of (2.6). If for and if , then That is the solution is bounded.

Proof. By part (c) of Lemma 2.6, we have From (2.18), we deduce that , and then we have Also, we have Thus, The result (2.23) now follows by induction. The proof is completed.

3. Global Attractivity

In this section, we will study the global attractivity of positive solutions of (2.6). We show that the positive equilibrium of (2.6) is a global attractor with a basin that depends on certain conditions imposed on the coefficients.

Theorem 3.1. Assume that the conditions (2.7) and (2.18) hold. Then, the equilibrium point of (2.6) is globally asymptotically stable.

Proof. In Section 2, we have shown under the assumptions (2.7) and (2.18) that the equilibrium is locally asymptotically stable. It remains to prove that the equilibrium is a global attractor. To this end, set and which by Theorem 2.7 exist and are positive numbers. Then, from (2.6) we deduce that Consequently, we have from which it follows that Thus, the the proof of Theorem 3.1 is completed.

Lemma 3.2 (see ). Consider the difference equation Let be some interval of real numbers, and assume that is a continuous function satisfying the following properties:
(a) is a nonincreasing function in u, and a nondecreasing function in ; (b) if is a solution of the system then, Then, (3.3) has a unique equilibrium point and every solution of (3.3) converges to .

Theorem 3.3. Assume that the conditions (2.7) and (2.18) hold. Then, the positive equilibrium of (2.6) is a global attractor with a basin

Proof. For , set We claim that In fact, if we set , then and in view of the condition (2.18), we have Since is decreasing in and increasing in , it follows that for all , which implies that our assertion is true. On the other hand, conditions (a) and (b) of Lemma 3.2 are clearly true. Let be a solution of (2.6) with the initial conditions . By Lemma 3.2, we have . The proof is completed.

Theorem 3.4. Assume that conditions (2.7) and (2.18) hold. Then, the positive equilibrium of (2.6) is a global attractor with a basin

Proof. Let be a solution of (2.6) with the initial conditions . Then, by Theorem 2.7, we have By Theorem 3.3, we have and so . The proof is completed.

Theorem 3.5. Assume that conditions (2.7) hold with . Also, assume that k is an odd positive integer. Then, the necessary and sufficient condition for (2.6) to have positive solutions of prime period two is that

Proof. First, suppose that there exist distinctive positive solutions of prime period two, of the difference equation (2.6).
If is odd, then . It follows from the difference equation (2.6) that Consequently, we obtain Thus, we deduce that Now it is clear that are two positive distinct real roots of the quadratic equation Therefore, we have From (3.13) and (3.15) we obtain condition (3.9). Conversely, suppose that the condition (3.9) is valid. Then, we deduce that (3.13) and (3.15) hold. Consequently, there exists two positive distinct real numbers and such that where is given by Thus, and given by (3.16) and (3.17) represent two positive distinct real roots of the quadratic equation (3.14). Now, we are going to prove that and given by (3.16) and (3.17) are positive solutions of prime period two of the difference equation (2.6). To this end, we assume that We wish to prove that and
It follows from the difference equation (2.6) and the formulas (3.16) and (3.17) that After some reduction, we deduce that Similarly, we can show that, By using the induction, we have Thus, the difference equation (2.6) has positive solutions of prime period two. Hence, the proof of Theorem 3.5 is completed.

Theorem 3.6. Assume that conditions (2.7) hold. If k is even, then (2.6) has no positive solutions of prime period two.

Proof. Suppose that there exists distinctive positive solutions of prime period two, of the difference equation (2.6).
If is even, then . It follows from the difference equation (2.6) that From which we have and by using (2.7), we deduce that . This is a contradiction. Thus, the proof of Theorem 3.6 is completed.

4. The Case

Secondly, we study the rational recursive sequence where are real numbers and By putting , (4.1) yields where . Equation (4.2) has two equilibrium points The linearized equation associated with (4.2) about the equilibria is The characteristic equation of (4.4) about the equilibrium is Now, we deduce from (4.5) the following results:

(a)if and since then the equilibrium is unstable (see );(b)if and since then the equilibrium is unstable;(c)if then Now, we have the following results from case (c): (i) if , then the equilibrium is unstable; (ii) if , then the equilibrium is unstable; (iii) if , then the linearized stability analysis fails;

(d)if , and hence the equilibrium is unstable;(e)if and hence the equilibrium is unstable. The characteristic equation of (4.4) about the equilibrium is This equation has two roots Now, we deduce from (4.10) the following results:

(i)if then the equilibrium is locally asymptotically stable (see Figure 3);(ii)if then the equilibrium is unstable (see Figure 4);(iii)if then the linearized stability analysis fails.

In the following results, we assume that where

Lemma 4.1. Assume that the initial conditions for and Then, is nonnegative and monotonically decreasing to zero, while is nonpositive and monotonically increasing to zero.

Proof. Suppose that for and Clearly, and . By induction, we can see that and for
If we have and hence Similarly, we can show that The proof of Lemma 4.1 is completed.

On using arguments similar to that used in Lemma 4.1, we can easily prove the following lemma.

Lemma 4.2. Assume that the initial conditions for and Then, is nonpositive and monotonically increasing to zero, while is nonnegative and monotonically decreasing to zero.

Corollary 4.3. The equilibrium point of (4.1) is a global attractor with a basin

Theorem 4.4. The equilibrium point of (4.1) is a global attractor with a basin where

Proof. Assuming that the initial conditions . If with then we deduce that By induction, it follows that for . Thus, the proof of Theorem 4.4 follows from Corollary 4.3.

Theorem 4.5. If then the equilibrium point of (4.2) is globally asymptotically stable.

Finally, on using arguments similar to that used in Theorems 3.5 and 3.6, we can prove easily the following results.

Theorem 4.6. Assume that If k is an odd positive integer, then the necessary and sufficient condition for (4.2) to have positive solutions of prime period two is that (see Figure 5)

Theorem 4.7. If k is an even positive integer, then (4.2) has no positive solutions of prime period two (see Figure 6).

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