Boundedness Character of the System of Recursive Difference Equations

Abstract

In this paper, we take into consideration the boundedness character of positive solutions of the difference system $x_{n}=\alpha +\prod \limits_{i=1}^{k}y_{n-i}^{a_{i}} $ $y_{n}=\beta +\prod \limits_{i=1}^{k}x_{n-i}^{b_{i}},$ where $a_{i},b_{i}\in \mathbb{R}$, $i=\overline{1,k},$ $a_{k}\neq 0,$ $b_{k}\neq 0$ and $\alpha $ and $\beta $ are nonnegative real numbers.

Keywords

• System of nonlinear difference equations
• bounded solutions
• positive solutions

References

1. Alzahrani, E.O., El-Dessoky, M.M., Elsayed, E.M., Kuang,Y., Solutions and properties of some degenerate systems of difference equations, J. of Comp. Anal. App., 18(2)(2015), 321–333.
2. El-Dessoky, M.M., Elsayed, E.M. On the solutions and periodic nature of some systems of rational difference equations, J. of Comp. Anal. App., 18(2)(2015), 206–218.
3. El-Metwally, H., Yalcinkaya, I., Cinar,.C., Global stability of an economic model, Utilitas Mathematica, 95(2014), 235–244.
4. Elsayed, E.M., The expressions of solutions and periodicity for some nonlinear systems of rational difference equations, Advanced Studies in Cont. Math., 25(3)(2015), 345–371.
5. Elsayed, E.M., On the solutions and periodic nature of some systems of difference equations, International J. of Biomath., 7(6)(2014).
6. Elsayed, E.M., Cinar, C., On the solutions of some systems of difference equations, Utilitas Mathematica, 93(2014), 279–289.
7. Elsayed, E.M., Ibrahim, T.F. Periodicity and solutions for some systems of nonlinear rational difference equations, Hacettepe J. of Math. and Statis., 44(6)(2015,) 1361–1390.
8. Ergin, S., Karatas, R., On the dynamics of a recursive sequence, ARS Combinatoria, 109(2013), 353–360.

9. Gelisken, A., Kara, M.,Some general systems of rational Difference Equations, J. of Dif. Eq., 2015(2015).
10. Gelisken, A., Cinar, C., Yalcinkaya, I., On a max-type difference equation, Advances in Dif. Eq., 2010(2010).
11. Gelisken, A., Cinar,C., Kurbanli, A.S., On the asymptotic behavior and periodic nature of a difference equation with maximum, Comp. & Math. with App., 59(2)(2010), 898–902.
12. Gelisken, A., Cinar, C., On the global attractivity of a max-type difference equation, Disc. Dyn. Nat. Soc., 2009(2009).
13. Halim, Y., Touafek,N., Yazlik, Y., Dynamic behavior of a second-order nonlinear rational difference equation, Turkish Journal of Mathematics, 39(2015), 1004–1018.
14. Hatir, E., Mansour, T., Yalcinkaya, I., On a fuzzy difference equation, Utilitas Mathematica, 93(2014), 135–151.
15. Iricanin, B.,On a higher-order nonlinear diffrence equation, Abstr. Appl. Anal., 2010(2010).
16. Karatas, R., Global behavior of a higher order difference equation, Comp. & Math. with App., 60(3)(2010), 830–839.
17. Karatas, R., Global behavior of a rational recursive sequence, Ars Combinatoria, 97(A)(2010), 421–428.
18. Kurbanli, A.S., Cinar, C., Yalcinkaya, I., On the behavior of positive solutions of the system of rational difference equations $x_{n+1}=x_{n-1}/y_{n}x_{n-1}+1$, $y_{n+1}=y_{n-1}/x_{n}y_{n-1}+1$, Math. and Comp. Model., 53(5-6)(2011), 1261–1267.
19. Papaschinopoulos, G., Radin, M., Schinas, C.J., Study of the asymptotic behavior of the solutions of three systems of difference equations of exponential form, Appl. Math. Comput., 218(2012), 5310–5318.
20. Papaschinopoulos, G., Schinas, C.J., Stefanidou, G., On the nonautonomous difference equation $x_{n+1}=A_{n}+(x_{n-1}^{p}/x_{n}^{q}),$ Appl. Math. Comput., 217(2011), 5573–5580.
21. Simsek, D., Demir, B., Cinar, C., On the solutions of the system of difference equations $x_{n+1}=\max\{A/x_{n},y_{n}/x_{n}\}$, $y_{n+1}=\max\{A/y_{n},x_{n}/y_{n}\}$, 2009(2009).
22. Simsek, D., Cinar, C., Yalcinkaya, I., On the recursive sequence $x(n+1)=x(n-(5k+9))/(1+x(n-4)x(n-9)...x(n-(5k+4)))$, Taiwanese J. of Math., 12(5)(2008), 1087–1099.
23. Stefanidou, G., Papaschinopoulos, G., Schinas, C.J., On a system of two exponential type difference equations, Commun. Appl. Nonlinear Anal., 17(2)(2010), 1-13.
24. Stevic,S., Alghamdi, M.A., Alotaibi, A., Boundedness character of the recursive sequence $x_{n}=\alpha +\prod\limits_{j=1}^{k}x_{n-j}^{a_{j}}$, Applied Mathematics Letters, 50(2015), 83–90.
25. Stevic, S., On the recursive sequence $x_{n+1}=\alpha+(x_{n-1}^{p}/x_{n}^{p})$, J. Appl. Math. Comput., 18 (1-2)(2005), 229–234.
26. Stevic, S., On the recursive sequence $x_{n+1}=A+(x_{n}^{p}/x_{n-1}^{r})$, Discrete Dyn. Nat. Soc., 2007(2007).
27. Stevic, S., On a class of higher-order difference equations,Chaos Solutions Fractals, 42(2009), 138–145.
28. Tollu, D.T., Yazlik, Y., Taskara, N., On fourteen solvable systems of difference equations, Appl. Math. Comput., 233(2014), 310–319.
29. Touafek, N., Elsayed, E.M., On a second order rational systems of difference equations, Hokkaido Mathematical Journal, 44(1)(2015), 29–45.
30. Yalcinkaya, I., On the global asymptotic behavior of a system of two nonlinear difference equations, ARS Combinatoria, 95(2010), 151–159.
31. Yazlik, Y., Tollu, D.T., Taskara, N., On the solutions of a max-type difference equation system, Mathematical Methods in the Applied Sciences, 38(17)(2015), 4388–4410.
32. Yazlik, Y., Tollu, D.T. Taskara, N., On the behaviour of solutions for some systems of difference equations, J. of Comp. Anal. App., 18(1)(2015), 166–178.
33. Yazlik, Y., Elsayed, E.M., Taskara, N., On the behaviour of the solutions of difference equation systems, J. of Comp. Anal. App., 16(5)(2014), 932–941.