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Discrete Dynamics in Nature and Society
Volume 2011 (2011), Article ID 982309, 18 pages
http://dx.doi.org/10.1155/2011/982309
Review Article

Solution and Attractivity for a Rational Recursive Sequence

1Mathematics Department, Faculty of Science, King Abdulaziz University, P. O. Box 80203, Jeddah 21589, Saudi Arabia
2Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt

Received 15 February 2011; Accepted 26 March 2011

Academic Editor: Ibrahim Yalcinkaya

Copyright © 2011 E. M. Elsayed. 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 concerned with the behavior of solution of the nonlinear difference equation , where the initial conditions , , are arbitrary positive real numbers and are positive constants. Also, we give specific form of the solution of four special cases of this equation.

1. Introduction

In this paper we deal with the behavior of the solution of the following difference equation: where the initial conditions are arbitrary positive real numbers and are positive constants. Also, we obtain the solution of some special cases of (1.1).

Let us introduce some basic definitions and some theorems that we need in the sequel.

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 [1].

Definition 1.1 (equilibrium point). A point is called an equilibrium point of (1.3) if That is, for is a solution of (1.3), or equivalently, is a fixed point of .

Definition 1.2 (stability). (i) The equilibrium point of (1.3) is locally stable if for every there exists such that for all with we have
(ii) The equilibrium point of (1.3) is locally asymptotically stable if is locally stable solution of (1.3) and there exists , such that for all with we have
(iii) The equilibrium point of (1.3) is global attractor if for all , we have
(iv) The equilibrium point of (1.3) is globally asymptotically stable if is locally stable, and is also a global attractor of (1.3).
(v) The equilibrium point of (1.3) is unstable if is not locally stable.The linearized equation of (1.3) about the equilibrium is the linear difference equation

Theorem A (see [2]). Assume that and . Then is a sufficient condition for the asymptotic stability of the difference equation

Remark 1.3. Theorem A can be easily extended to a general linear equations of the form where and . Then (1.13) is asymptotically stable provided that

Consider the following equation The following theorem will be useful for the proof of our results in this paper.

Theorem B (see [1]). Let be an interval of real numbers and assume that is a continuous function satisfying the following properties:(a) is nondecreasing in and in for each , and is nonincreasing in for each and in ;(b)if is a solution of the system then Then (1.15) has a unique equilibrium and every solution of (1.15) converges to .

Definition 1.4 (periodicity). A sequence is said to be periodic with period if for all .

Definition 1.5 (Fibonacci sequence). The sequence , that is, is called Fibonacci sequence.

Recently there has been a great interest in studying the qualitative properties of rational difference equations. Some prototypes for the development of the basic theory of the global behavior of nonlinear difference equations of order greater than one come from the results for rational difference equations.

However, there have not been any effective general methods to deal with the global behavior of rational difference equations of order greater than one so far. From the known work, one can see that it is extremely difficult to understand thoroughly the global behaviors of solutions of rational difference equations although they have simple forms (or expressions). One can refer to [323] for examples to illustrate this. Therefore, the study of rational difference equations of order greater than one is worth further consideration.

Many researchers have investigated the behavior of the solution of difference equations, for example, Aloqeili [24] has obtained the solutions of the difference equation Amleh et al. [25] studied the dynamics of the difference equationÇinar [26, 27] got the solutions of the following difference equation In [28], Elabbasy et al. investigated the global stability and periodicity character and gave the solution of special case of the following recursive sequence Elabbasy et al. [29] investigated the global stability, boundedness, and periodicity character and gave the solution of some special cases of the difference equation In [30], Ibrahim got the form of the solution of the rational difference equation Karatas et al. [31] got the solution of the difference equation Yalçinkaya and Çinar [32] considered the dynamics of the difference equation Yang [33] investigated the global asymptotic stability of the difference equation See also [1, 2, 30, 31, 3440]. Other related results on rational difference equations can be found in [32, 33, 4148].

2. Local Stability of (1.1)

In this section we investigate the local stability character of the solutions of (1.1). Equation (1.1) has a unique equilibrium point and is given by or if , then the unique equilibrium point is .

Let be a function defined by Therefore it follows that we see that The linearized equation of (1.1) about is

Theorem 2.1. Assume that Then the equilibrium point of (1.1) is locally asymptotically stable.

Proof. It follows from Theorem A that (2.6) is asymptotically stable if or and so, The proof is complete.

3. Global Attractor of the Equilibrium Point of (1.1)

In this section we investigate the global attractivity character of solutions of (1.1).

Theorem 3.1. The equilibrium point of (1.1) is global attractor if .

Proof. Let be real numbers and assume that is a function defined by , then we can easily see that the function is increasing in and decreasing in .Suppose that is a solution of the system Then from (1.1), we see that or then subtracting, we obtain Thus It follows from Theorem B that is a global attractor of (1.1), and then the proof is complete.

4. Boundedness of Solutions of (1.1)

In this section we study the boundedness of solutions of (1.1).

Theorem 4.1. Every solution of (1.1) is bounded if .

Proof. Let be a solution of (1.1). It follows from (1.1) that Then Then the subsequences , are decreasing and so are bounded from above by .

5. Special Cases of (1.1)

Our goal in this section is to find a specific form of the solutions of some special cases of (1.1) when , and are integers and give numerical examples of each case and draw it by using MATLAB 6.5.

5.1. On the Difference Equation

In this subsection we study the following special case of (1.1): where the initial conditions are arbitrary positive real numbers.

Theorem 5.1. Let be a solution of (5.1). Then for where.

Proof. For the result holds. Now suppose that and that our assumption holds for . That is, Now, it follows from (5.1) that Therefore Also, we see from (5.1) that Thus Hence, the proof is completed.

For confirming the results of this section, we consider numerical example for ,(See Figure 1).

982309.fig.001
Figure 1
5.2. On the Difference Equation

In this subsection we give a specific form of the solutions of the difference equation where the initial conditions are arbitrary positive real numbers with .

Theorem 5.2. Let be a solution of (5.8). Then for where .

Proof. For the result holds. Now suppose that and that our assumption holds for . That is, Now, it follows from (5.8) that Therefore Also, we see from (5.8) that Thus Hence, the proof is completed.

Assume that,. (See Figure 2), and for ,. (See Figure 3).

982309.fig.002
Figure 2
982309.fig.003
Figure 3

The following cases can be treated similarly.

5.3. On the Difference Equation

In this subsection we obtain the solution of the following difference equation where the initial conditions are arbitrary positive real numbers.

Theorem 5.3. Let be a solution of (5.15). Then for where .
Figure 4 shows the solution when .

982309.fig.004
Figure 4
5.4. On the Difference Equation

In this subsection we give the solution of the following special case of (1.1) where the initial conditions are arbitrary real numbers. with .

Theorem 5.4. Let be a solution of (5.17). Then every solution of (5.17) is periodic with period 12. Moreover takes the form where .
Figure 5 shows the solution when .

982309.fig.005
Figure 5

6. Conclusion

This paper discussed global stability, boundedness, and the solutions of some special cases of (1.1). In Section 2 we proved when, (1.1) local stability.In Section 3 we showed that the unique equilibrium of (1.1) is globally asymptotically stable if . In Section 4 we proved that the solution of (1.1) is bounded if . In Section 5 we gave the form of the solution of four special cases of (1.1) and gave numerical examples of each case and drew them by using Matlab 6.5.

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