DOI QR코드

DOI QR Code

THE EXPRESSIONS OF SOLUTIONS AND THE PERIODICITY OF SOME RATIONAL DIFFERENCE EQUATIONS SYSTEMS

  • AHMED, A.M. (Department of Mathematics, Faculty of Science, Al-Azhar University) ;
  • ELSAYED, E.M. (King Abdulaziz University, Faculty of Science, Mathematics Department, Department of Mathematics, Faculty of Science, Mansoura University)
  • Received : 2015.02.23
  • Accepted : 2015.06.08
  • Published : 2016.01.30

Abstract

In this paper we deal with the expressions of solutions and the periodicity nature of some systems of nonlinear difference equations with order three with nonzero real numbers initial conditions.

Keywords

1. Introduction

In this paper, we investigate the periodic character and the form of the solutions of some rational difference equations systems of order three

with initial conditions x−2, x−1, x0, y−2, y−1 and y0 are nonzero real numbers.

In recent years, rational difference equations have attracted the attention of many researchers for varied reasons. On the one hand, they provide examples of nonlinear equations which are, in some cases, treatable but whose dynamics present some new features with respect to the linear case. On the other hand, rational equations frequently appear in some biological models, and, hence, their study is of interest also due to their applications. The periodicity of the positive solutions of the rational difference equations systems

has been obtained by Cinar [4].

Elabbasy et al. [7] has studied the solutions of particular cases of the following general system of difference equations

The behavior of positive solutions of the following system

has been studied by Kurbanli et al. [24].

Kurbanli [25] investigated the behavior of the solution of the difference equation system

Özban [26] has investigated the positive solution of the system of rational difference equations

Also, Touafek et al. [28] studied the periodicity and gave the form of the solutions of the following systems

In [29] Yalçınkaya investigated the sufficient condition for the global asymptotic stability of the following system of difference equations

In [35] Zhang et al. studied the boundedness, the persistence and global asymptotic stability of the positive solutions of the system of difference equations

Similar to difference equations and nonlinear systems of rational difference equations were investigated, see [1]-[38].

Definition (Periodicity).

A sequence is said to be periodic with period p if xn+p = xn for all n ≥ −k.

 

2. The First System :

In this section, we get the form of the solutions of the system of the difference equations

where n = 0, 1, 2, ... and the initial conditions x−2, x−1, x0, y−2, y−1 and y0 are arbitrary nonzero real numbers with x0y−1 ≠ 1, x−1y−2 ≠ 1.

Theorem 1. If {xn, yn} are solutions of difference equation system (1). Then for n = 0, 1, 2, ...,

where x−2 = c, x−1 = b, x0 = a, y−2 = f, y−1 = e and y0 = d.

Proof. For n = 0 the result holds. Now suppose that n > 1 and that our assumption holds for n − 1. that is,

Now we obtain from Eq.(1) that

Also, we see from Eq.(1) that

and

Also, we can prove the other relations. This completes the proof. □

 

3. The Second System :

In this section, we get the solutions of the system of the difference equations

where n = 0, 1, 2, ... and the initial conditions x−2, x−1, x0, y−2, y−1 and y0 are arbitrary nonzero real numbers with x−2y−1, x0y−1, x−1y0, x−1y−2 ≠ 1.

Theorem 2. If {xn, yn} are solutions of difference equation system (2). Then for n = 0, 1, 2, ...,

Proof. For n = 0 the result holds. Now suppose that n > 0 and that our assumption holds for n − 1. that is,

Now it follows from Eq.(2) that

Also, we see from Eq.(2) that

By the same way we can prove the other relations. The proof is complete. □

Lemma 1. The solution of system (2) is unbounded except in the following case.

Theorem 3. System (2) has a periodic solution of period four iff d = f, a = c and it will be taken the following form

Proof. First suppose that there exists a prime period four solution

of system (2), we see from the form of the solution of system (2) that

Then we get d = f, a = c. Second assume that d = f, a = c. Then we see from the form of the solution of system (1) that

Thus we have a periodic solution of period four and the proof is complete. □

Lemma 2. System (2) has a periodic solution of period two iff d = f, a = c, bd = ce = 2 and it will be taken the following form {xn} = {c, b, c, b,...}, {yn} = {f, e, f, e,...} .

Proof. The proof is consequently from the previous theorem and so, will be omitted. □

 

4. Other Systems

Here we study some systems of difference equations and the proof of all the Theorems are similar to above systems and so, will be omitted and in all cases we suppose that x−2 = c, x−1 = b, x0 = a, y−2 = f, y−1 = e and y0 = d.

Theorem 4. The solutions of the following system (3) with x0y-1 ≠ 1, x-1y-2 ≠ 1 are given by the following formula for n = 0, 1, 2, ...,

Theorem 5. If {xn, yn} are solutions of the difference equation system (4) where the initial conditions x−2, x−1, x0, y−2, y−1 and y0 are arbitrary nonzero real numbers with x0y−1 ≠ −1, x−1y−2 ≠ −1. Then for n = 0, 1, 2, ...,

Theorem 6. If {xn, yn} are solutions of the difference equations system (5) where the initial conditions x−2, x−1, x0, y−2, y−1 and y0 are arbitrary nonzero real numbers with x0y−1 ≠ −1, x−1y−2 ≠ −1: Then for n = 0, 1, 2, ...,

Theorem 7. Assume that {xn, yn} are solutions of the system (6) with the initial conditions are arbitrary nonzero real numbers with x−2y−1, x−1y0 ≠ −1, x0y−1, x−1y−2 ≠ 1: Then for n = 0, 1, 2, ...,

Lemma 3. The solution of equation system (6) is unbounded except in the following case.

Theorem 8. System (6) has a periodic solution of period four iff d = −f, a = −c and it will be taken the following form

Theorem 9. For n = 0, 1, 2, ..., the solutions of system (7) with x−2y−1, x−1y0 ≠ 1, x0y−1, x−1y−2 ≠ −1 are given by the following relations

Lemma 4. The solution of equation system (7) is unbounded except in the following case.

Theorem 10. System (7) has a periodic solution of period four iff d = −f, a = −c and it will be taken the following form

Theorem 11. Suppose that the sequences are solutions of system equations (8) with x−2y−1, x−1y0, x0y−1, x−1y−2 ≠ −1, then we obtain the following expressions of the solutions for n = 0, 1, 2, ...,

Lemma 5. The solution of equation system (8) is unbounded except in the following case.

Theorem 12. System (8) has a periodic solution of period four iff d = f, a = c and it will be taken the following form

Lemma 6. System (8) has a periodic solution of period two iff d = f, a = c, bd = ce = −2 and it will be taken the following form {xn} = {c, b, c, b,...} , {yn} = {f, e, f, e,...}.

 

5. Numerical Examples

In order to illustrate the results of the previous sections and to support our theoretical discussions, we consider several interesting numerical examples in this section. These examples represent different types of qualitative behavior of solutions to nonlinear difference equations.

Example 1. We consider numerical example for the difference system (1) with the initial conditions x−2 = 3:07, x−1 = 0.13, x0 = 0.4, y−2 = 0.02, y−1 = 0.7 and y0 = 0.03. (See Figure 1).

Figure 1

Example 2. We consider interesting example for the difference system (1) with the initial conditions x−2 = 0.07, x−1 = 0:4, x0 = −0.04, y−2 = 0.02, y−1 = −0.07 and y0 = 0.03. (See Figure 2).

Figure 2

Example 3. We consider numerical example for the difference system (2) with the initial conditions x−2 = 0.8, x−1 = 0.4, x0 = 0.9, y−2 = 0.2, y−1 = 0.7 and y0 = 0.3. (See Figure 3).

Figure 3

Example 4. See Figure (4) when we take system (2) with the initial conditions x−2 = 9, x−1 = 0.5, x0 = 7, y−2 = 8, y−1 = 2 and y0 = 4.

Figure 4

Example 5. We assume the difference equations system (2)when we put the initial conditions x−2 = 9, x−1 = 7, x0 = 9, y−2 = 5, y−1 = 2 and y0 = 5. See figure 5.

Figure 5

Example 6. Figure (6) shows the periodicity of the solution of the difference system (2) with the initial conditions x−2 = −3, x−1 = 5, x0 = −3, y−2 = 0.4, y−1 = −2/3 and y0 = 0.4.

Figure 6

References

  1. R.P. Agarwal and E.M. Elsayed, On the solution of fourth-order rational recursive sequence, Advanced Studies in Contemporary Mathematics, 20 (2010), 525-545.
  2. M. Aloqeili, Dynamics of a rational difference equation, Appl. Math. Comp., 176 (2006), 768-774. https://doi.org/10.1016/j.amc.2005.10.024
  3. N. Battaloglu, C. Cinar and I. Yalçınkaya, The dynamics of the difference equation, ARS Combinatoria, 97 (2010), 281-288.
  4. C. Cinar,On the positive solutions of the difference equation system xn+1 = 1/yn, yn+1 = yn/xn-1yn-1, Appl. Math. Comput., 158 (2004), 303-305. https://doi.org/10.1016/j.amc.2003.08.073
  5. S.E. Das and M. Bayram, On a system of rational difference equations, World Applied Sciences Journal, 10 (2010), 1306-1312.
  6. E.M. Elabbasy, H. El-Metwally and E.M. Elsayed, On the solutions of a class of difference equations systems, Demonstratio Mathematica, 41 (2008), 109-122.
  7. E.M. Elabbasy, H. El-Metwally and E.M. Elsayed, Global behavior of the solutions of difference equation, Adv. Differ. Equ., (2011), 2011-2028.
  8. E.M. Elabbasy, H. El-Metwally and E.M. Elsayed, Some properties and expressions of solutions for a class of nonlinear difference equation, Utilitas Mathematica, 87 (2012), 93-110.
  9. H. El-Metwally and E.M. Elsayed, Solution and behavior of a third rational difference equation, Utilitas Mathematica, 88 (2012), 27-42.
  10. H. El-Metwally and E.M. Elsayed, Form of solutions and periodicity for systems of differenc equations, J. Comp. Anal. Appl., 15 (2013), 852-857.
  11. E.M. Elsayed, On the solutions of higher order rational system of recursive sequences, Mathematica Balkanica, 21 (2008), 287-296.
  12. E.M. Elsayed, On the solutions of a rational system of difference equations, Fasciculi Mathematici, 45 (2010), 25-36.
  13. E.M. Elsayed, Behavior and expression of the solutions of some rational difference equations, J. Comp. Anal. Appl., 15 (2013), 73-81.
  14. E.M. Elsayed, Behavior of a rational recursive sequences, Studia Univ. ” Babes-Bolyai ”, Mathematica, LVI (1) (2011), 27-42.
  15. E.M. Elsayed, M. Mansour and M.M. El-Dessoky, Solutions of fractional systems of difference equations, Ars Combinatoria, 110 (2013), 469-479.
  16. E.M. Elsayed, Solution and attractivity for a rational recursive sequence, Dis. Dyn. Nat. Soc., Volume 2011, Article ID 982309, 17 pages.
  17. E.M. Elsayed, On the solution of some difference equations, European J. Pure Appl. Math., 4 (2011), 287-303.
  18. E.M. Elsayed and H.A. El-Metwally, On the solutions of some nonlinear systems of difference equations, Adv. Differ. Equ., (2013), 2013-2016.
  19. E.M. Elsayed, Solutions of rational difference system of order two, Math. Comput. Mod., 55 (2012), 378-384. https://doi.org/10.1016/j.mcm.2011.08.012
  20. E.M. Elsayed and M.M. El-Dessoky, Dynamics and behavior of a higher order rational recursive sequence, Advances in Difference Equations, (2012), 2012-2069.
  21. E.M. Elsayed, M.M. El-Dessoky and A. Alotaibi, On the solutions of a general system of difference equations, Dis. Dyn. Nat. Soc., Volume 2012, Article ID 892571, 12 pages.
  22. M.E. Erdoğan, C. Cinar and I. Yalçınkaya, On the dynamics of the recursive sequence, Comput. Math. Appl., 61 (2011), 533-537. https://doi.org/10.1016/j.camwa.2010.11.030
  23. E.A. Grove, G. Ladas, L.C. McGrath and C.T. Teixeira, Existence and behavior of solutions of a rational system, Commun. Appl. Nonlinear Anal., 8 (2001), 1-25.
  24. A.S. Kurbanli, C. Cinar and I. Yalçınkaya, On the behavior of positive solutions of the system of rational difference equations, Math. Comput. Mod., 53 (2011), 1261-1267. https://doi.org/10.1016/j.mcm.2010.12.009
  25. A.S. Kurbanli, On the behavior of solutions of the system of rational difference equations, Adv. Differ. Equ., (2011), 2011-2040.
  26. A.Y. Ozban, On the system of rational difference equations xn+1 = a/yn-3, yn+1 = byn-3/xn-qyn-q, Appl. Math. Comp., 188 (2007), 833-837. https://doi.org/10.1016/j.amc.2006.10.034
  27. M. Mansour, M.M. El-Dessoky and E.M. Elsayed, On the solution of rational systems of difference equations, J. Comp. Anal. Appl., 15 (2013), 967-976.
  28. N. Touafek and E.M. Elsayed, On the solutions of systems of rational difference equations, Math. Comput. Mod., 55 (2012), 1987-1997. https://doi.org/10.1016/j.mcm.2011.11.058
  29. N. Touafek and E.M. Elsayed, On the periodicity of some systems of nonlinear difference equations, Bull. Math. Soc. Sci. Math. Roumanie, Tome 55 (103), 2 (2012), 217-224.
  30. I. Yalcinkaya, On the global asymptotic stability of a second-order system of difference equations, Disc. Dyn. Nat. Soc., Vol. 2008, Article ID 860152 (2008), 12 pages.
  31. I. Yalçınkaya, On the global asymptotic behavior of a system of two nonlinear difference equations, ARS Combinatoria, 95 (2010), 151-159.
  32. I. Yalcinkaya, C. Cinar and M. Atalay, On the solutions of systems of difference equations, Adv. Differ. Equ., 2008 (2008) Article ID 143943, 9 pages.
  33. I. Yalcinkaya, C. Cinar, On the solutions of a systems of difference equations, International Journal of Mathematics and Statistics, Autumn 9 (S11) (2011), 62-67.
  34. X. Yang, Y. Liu and S. Bai, On the system of high order rational difference equations xn = a/yn-p, yn = byn-p/xn-qyn-q, Appl. Math. Comp., 171 (2005), 853-856. https://doi.org/10.1016/j.amc.2005.01.092
  35. Y. Zhang, X. Yang, G.M. Megson and D.J. Evans, On the system of rational difference equations, Appl. Math. Comp., 176 (2006), 403-408. https://doi.org/10.1016/j.amc.2005.09.039
  36. Y. Zhang, X. Yang, D.J. Evans and C. Zhu, On the nonlinear difference equation system, Comp. Math. Appl., 53 (2007), 1561-1566. https://doi.org/10.1016/j.camwa.2006.04.030
  37. C. Wang, Shu Wang, and W. Wang, Global asymptotic stability of equilibrium point for a family of rational difference equations, Applied Mathematics Letters, 24 (2011), 714-718. https://doi.org/10.1016/j.aml.2010.12.013
  38. C. Wang, F. Gong, S. Wang, L. LI and Q. Shi, Asymptotic behavior of equilibrium point for a class of nonlinear difference equation, Adv. Differ. Equ., Volume 2009, Article ID 214309, 8 pages.
  39. E.M.E. Zayed and M.A. El-Moneam, On the rational recursive sequence xn+1 = Ax + (β{{sub}}x + Υx{{sub}}n-k)/(Cxn + Dxn-k) , Comm. Appl. Nonlinear Analysis, 16 (2009), 91-106.

Cited by

  1. A generalized two-dimensional system of higher order recursive sequences vol.26, pp.2, 2016, https://doi.org/10.1080/10236198.2020.1718667