Abstract and Applied AnalysisVolume 2012 (2012), Article ID 248291, 16 pagesdoi:10.1155/2012/248291

Research Article

Qualitative Study of Solutions of Some Difference Equations

Hamdy El-Metwally1,2 and E. M. Elsayed2,3

¹Department of Mathematics, Rabigh College of Science and Art, King Abdulaziz University, P.O. Box 344, Rabigh 21911, Saudi Arabia²Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt³Mathematics Department, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia

Received 14 February 2012; Accepted 15 March 2012

Academic Editor: Chaitan Gupta

Copyright © 2012 Hamdy El-Metwally and 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.

We obtain in this paper the solutions of the following recursive sequences , where the initial conditions are arbitrary real numbers and we study the behaviors of the solutions and we obtained the equilibrium points of the considered equations. Some qualitative behavior of the solutions such as the boundedness, the global stability, and the periodicity character of the solutions in each case have been studied. We presented some numerical examples by giving some numerical values for the initial values and the coefficients of each case. Some figures have been given to explain the behavior of the obtained solutions in the case of numerical examples by using the mathematical program Mathematica to confirm the obtained results.

1. Introduction

In this paper, we obtain the solutions of the following recursive sequences: where the initial conditions are arbitrary real numbers. Also, we study the behavior of the solutions.

Recently, there has been a great interest in studying the qualitative properties of rational difference equations. For the systematical studies of rational and nonrational difference equations, one can refer to the papers [1–31] and references therein.

The study of rational difference equations of order greater than one is quite challenging and rewarding because 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. Therefore, the study of rational difference equations of order greater than one is worth further consideration.

Recently, Agarwal and Elsayed [4] investigated the global stability and periodicity character and gave the solution of some special cases of the difference equation: Aloqeili [5] has obtained the solutions of the difference equation:Çinar [8, 9] investigated the solutions of the following difference equations: Elabbasy et al. [10, 12] investigated the global stability and periodicity character and gave the solution of special case of the following recursive sequences: Ibrahim [19] got the solutions of the rational difference equation: Karatas et al. [20] got the form of the solution of the difference equation: Simsek et al. [26] obtained the solutions of the following difference equations: Here, we recall some notations and results which will be useful in our investigation.

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 [21].

Definition 1.1 (Equilibrium Point). A point is called an equilibrium point of (1.10) if That is, , for is a solution of (1.10), or equivalently, is a fixed point of .

Definition 1.2 (Stability). (i) The equilibrium point of (1.10) is locally stable if, for every there exists such that for all with we have(ii) The equilibrium point of (1.10) is locally asymptotically stable if is locally stable solution of (1.10) and there exists , such that for all withwe have(iii) The equilibrium point of (1.10) is global attractor if, for all we have(iv) The equilibrium point of (1.10) is globally asymptotically stable if is locally stable, and is also a global attractor of (1.10).(v) The equilibrium point of (1.10) is unstable if is not locally stable.The linearized equation of (1.10) about the equilibrium is the linear difference equation

Theorem A (see [22]). Assume that and. Then, is a sufficient condition for the asymptotic stability of the difference equation:

Definition 1.3 (Periodicity). A sequence is said to be periodic with period if for all .

2. On the Equation

In this section, we give a specific form of the solution of the first equation in the form: where the initial values are arbitrary positive real numbers.

Theorem 2.1. Let be a solution of (2.1). Then, for , where.

Proof. For , the result holds. Now, suppose that and that our assumption holds for . That is, Now, it follows from (2.1) that Hence, we have Similarly, Hence, we have Similarly, one can easily obtain the other relations. Thus, the proof is completed.

Theorem 2.2. Equation (2.1) has a unique equilibrium point which is the number zero and this equilibrium point is not locally asymptotically stable.

Proof. For the equilibrium points of (2.1), we can write Then, we have or Thus the equilibrium point of (2.1) is . Let be a function defined by Therefore, it follows that we see that The proof follows by using Theorem A.

Numerical ExamplesFor confirming the results of this section, we consider numerical examples which represent different types of solutions to (2.1).Example 2.3. We assume , (see Figure 1).Example 2.4. See Figure 2, since .

Figure 1

Figure 2

3. On the Equation

In this section, we obtain the solution of the second equation in the form: where the initial values are arbitrary nonzero real numbers with .

Theorem 3.1. Let be a solution of (3.1). Then, (3.1) has unboundedness solution and, for , where.

Proof. For the result holds. Now, suppose that and that our assumption holds for . That is, Now, it follows from (3.1) that Similarly, one can easily obtain the other relations. Thus, the proof is completed.

Theorem 3.2. Equation (3.1) has a periodic solution of period six iff and will take the form:   .

Proof. First suppose that there exists a prime period six solution: of (3.1), we see from the form of the solution of (3.1) that or Then, Second, assume that . Then, we see from the form of the solution of (3.1) that Thus, we have a periodic solution of period six and the proof is complete.

Theorem 3.3. Equation (3.1) has two equilibrium points which are and these equilibrium points are not locally asymptotically stable.

Proof. For the equilibrium points of (3.1), we can write Then, we have or Thus, the equilibrium points of (3.1) are .Let be a function defined by Therefore, it follows that we see that The proof follows by using Theorem A.

Numerical ExamplesHere, we will represent different types of solutions of (3.1).Example 3.4. We consider , (see Figure 3).Example 3.5. See Figure 4 since .

Figure 3

Figure 4

The following cases can be proved similarly.

4. On the Equation

In this section, we get the solution of the third following equation: where the initial values are arbitrary positive real numbers.

Theorem 4.1. Let be a solution of (4.1). Then, for ,

Theorem 4.2. Equation (4.1) has a unique equilibrium point which is the number zero and this equilibrium point is not locally asymptotically stable.

Example 4.3. Assume that see Figure 5).

Figure 5

Example 4.4. See Figure 6 since .

Figure 6

5. On the Equation

Here, we obtain a form of the solutions of the equation where the initial values are arbitrary nonzero real numbers with .

Theorem 5.1. Let be a solution of (5.1). Then, (5.1) has unboundedness solution and, for ,

Theorem 5.2. Equation (5.1) has a periodic solution of period six if and only if and will take the form:.

Theorem 5.3. Equation (5.1) has a unique equilibrium point which is and this equilibrium point is not locally asymptotically stable.

Example 5.4. Consider see Figure 7).

Figure 7

Example 5.5. Figure 8 shows the solutions when .

Figure 8

Acknowledgments

This paper was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, Saudi Arabia under Grant no. (1431/662/415). The authors, therefore, acknowledge with thanks DSR technical and financial support. Last, but not least, sincere appreciations are dedicated to all our colleagues at the Faculty of Science, Rabigh branch, for their nice wishes.

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