DYNAMICS AND GLOBAL BEHAVIOR FOR A FOURTH-ORDER RATIONAL DIFFERENCE EQUATION

Abstract

In this paper we study the behavior of the rational difference equationof the fourth orderx n+1 = ax n + bx n x n−2 cx n−2 + dx n−3 , n = 0, 1, ...,where the initial conditions x−3 , x −2 , x −1 , x are arbitrary positive realnumbers and a, b, c, d are positive constants. Also, we give the solutionof some special cases of this equation.

Keywords

• difference equations
• stability
• boundedness
• periodicity
• solution of differ

References

1. Agarwal, R. P. Difference Equations and Inequalities, 1 st edition, (Marcel Dekker, New York, 1992), 2 nd edition, 2000.
2. Agarwal, R. P. and Elsayed, E. M. Periodicity and stability of solutions of higher order rational difference equation, Advanced Studies in Contemporary Mathematics 17 No 2, 181–201, 2008.
3. Agarwal, R. P. and Elsayed, E. M. On the Solution of Fourth-Order Rational Recursive Sequence, Advanced Studies in Contemporary Mathematics 20 No 4, 525–545, 2010.
4. Aloqeili, M. Dynamics of a rational difference equation, Appl. Math. Comp. 176 No 2, 768–774, 2006.
5. Battaloglu, N., Cinar, C. and Yal¸ cınkaya, I. The dynamics of the difference equation, ARS Combinatoria 97, 281–288, 2010.
6. Camouzis, E. and Ladas, G. The rational recursive sequence x n+1 = bx 2 n 1 + x 2 n−1 , Computers & Mathematics with Applications 28, 37–43, 1994.
7. Cinar, C. On the positive solutions of the difference equation x n+1 = ax n−1 1+bx n x n−1 , Appl. Math. Comp. 156, 587–590, 2004.
8. Ebru Das, S. Dynamics of a nonlinear rational difference equations, Hacettepe Journal of Mathematics and Statistics 42 No 1, 9–14, 2013.

9. Elabbasy, E. M., El-Metwally, H. and Elsayed, E. M. On the difference equation x n+1 = ax n − bx n cx n −dx n−1 , Adv. Differ. Equ. Article ID 82579, 1–10, 2006.
10. Elabbasy, E. M., El-Metwally, H. and Elsayed, E. M. On the difference equations x n+1 = αx n−k β+γ k i=0 x n−i , J. Conc. Appl. Math. 5 No 2, 101–113, 2007.
11. Elabbasy, E M., El-Metwally, H. and Elsayed, E. M. Qualitative behavior of higher order difference equation, Soochow Journal of Mathematics 33 No 4, 861–873, 2007.
12. Elabbasy, E. M., El-Metwally, H. and Elsayed, E. M. Global behavior of the solutions of difference equation, Advances in Difference Equations 2011, 2011:28.
13. Elabbasy, E. M., El-Metwally, H. and Elsayed, E. M. Some properties and expressions of solutions for a class of nonlinear difference equation, Utilitas Mathematica 87, 93–110, 20 El-Metwally, H. and Elsayed, E. M. Solution and behavior of a third rational difference equation, Utilitas Mathematica 88, 27–42, 2012.
14. El-Metwally, H. and Elsayed, E. M. Form of solutions and periodicity for systems of difference equations, J. Comp. Anal. Appl. 15 No 5, 852–857, 2013.
15. Elsayed, E. M. Qualitative behavior of difference equation of order two, Mathematical and Computer Modelling 50, 1130–1141, 2009.
16. Elsayed, E. M. Solutions of rational difference system of order two, Mathematical and Computer Modelling 55, 378–384, 2012.
17. Elsayed, E. M. Dynamics of recursive sequence of order two, Kyungpook Mathematical Journal 50, 483–497, 2010.
18. Elsayed, E. M. Qualitative behavior of difference equation of order three, Acta Scientiarum Mathematicarum (Szeged) 75 No 1-2, 113–129, 2009.
19. Elsayed, E. M. Solution and attractivity for a rational recursive sequence, Discrete Dynamics in Nature and Society 2011, Article ID 982309, 2011.
20. Elsayed, E. M. On the solution of some difference equations, European J. Pure Appl. Math. 4 No 3, 287–303, 2011.
21. Elsayed, E. M. Solution of a recursive sequence of order ten, General Mathematics 19 No 1, 145–162, 2011.
22. Elsayed, E. M. Behavior and expression of the solutions of some rational difference equations, J. Comp. Anal. Appl. 15 No 1, 73–81, 2013.
23. Elsayed, E. M. and El-Dessoky, M. M. Dynamics and behavior of a higher order rational recursive sequence, Advances in Difference Equations 2012, 2012:69, 2012.
24. Gelisken, A., Cinar, C. and Yalcinkaya, I. On a max–type difference equation, Advances in Difference Equations 2010, Article ID 584890, 2010.
25. Gelisken, A., Cinar, C. and Yalcinkaya, I. On the periodicity of a difference equation with maximum, Discrete Dynamics in Nature and Society 2008, Article ID 820629, 2008.
26. Kocic V. L. and Ladas, G. Global Behavior of Nonlinear Difference Equations of Higher Order with Applications, (Kluwer Academic Publishers, Dordrecht, 1993).
27. Kulenovic M. R. S. and Ladas, G. Dynamics of Second Order Rational Difference Equations with Open Problems and Conjectures, (Chapman & Hall / CRC Press, 2001).
28. Li, D., Li, P. and Li, X. Dynamical properties for a class of fourth-order nonlinear difference equations, Advances in Difference Equations 2008, Article ID 648702, 2008.
29. Li, X. The rule of trajectory structure and global asymptotic stability for a nonlinear difference equation, Indian J. Pure Appl. Math. 38 No 6, 1–9, 2007.
30. Li, X. Existence of solutions with a single semicycle for a general second order rational difference equation, J. Math. Anal. Appl. 334, 528–533, 2007.
31. Li, X. Qualitative properties for a fourth-order rational difference equation, J. Math. Anal. Appl. 311, 103–111, 2005.
32. Li, X. and Zhu, D. Global asymptotic stability of a nonlinear recursive sequence, Appl. Math. Letters 17, 833–838, 2004.
33. Li, X. and Zhu, D. Two rational recursive sequence, Comput. Math. Appl. 47 No 10-11, 1487–1494, 2004.
34. Touafek, N. On a second order rational difference equation, Hacettepe Journal of Mathematics and Statistics, 41 No 6, 867–874, 2012.
35. Touafek, N. and Elsayed, E. M. On the solutions of systems of rational difference equations, Mathematical and Computer Modelling 55, 1987–1997, 2012.
36. Touafek, N. and Elsayed, E. M. On the periodicity of some systems of nonlinear difference equations, Bull. Math. Soc. Sci. Math. Roumanie, Tome 55 (103), No. 2, 217–224, 2012. Wang, C., Gong, F., Wang, S., Li, L. and Shi, Q. Asymptotic behavior of equilibrium point for a class of nonlinear difference equation, Advances in Difference Equations 2009, Article ID 214309, 2009.
37. Yal¸ cınkaya, I., Cinar, C. and Atalay, M. On the solutions of systems of difference equations, Advances in Difference Equations 2008, Article ID 143943, 2008.
38. Yal¸ cınkaya, I., Iricanin, B. D. and Cinar, C. On a max-type difference equation, Discrete Dynamics in Nature and Society 2007, Article ID 47264, 2007.
39. Yal¸ cınkaya, I. On the global asymptotic stability of a second-order system of difference equations, Discrete Dynamics in Nature and Society 2008, Article ID 860152, 2008.
40. Yal¸ cınkaya, I. On the difference equation x n+1 = α + x n−m x k n , Discrete Dynamics in Nature and Society 2008, Article ID 805460, 2008.
41. Yal¸ cınkaya, I. On the global asymptotic behavior of a system of two nonlinear difference equations, ARS Combinatoria 95, 151–159, 2010.
42. Zayed, E. M. E. Dynamics of the nonlinear rational difference equation x n+1 = Ax n + Bx n−k + px n +x n−k q+x n−k , European Journal of Pure and Applied Mathematics 3 No 2, 254–268, 20