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Global stability of a higher order rational recursive sequence. (English) Zbl 1105.39004
The authors investigate stability, periodicity, and oscillatory behavior of solutions of the rational difference equation $$x_{n+1}=\frac{\beta ~x_{n-k+1}+\gamma~ x_{n-2k+1}}{A+B~x_{n-k+1}},\quad n=0,1,2,....$$ or equivalently, via the transformation $x_n=Ay_n/B$, $$y_{n+1}=\frac{p~y_{n-k+1}+q~y_{n-2k+1}}{1+y_{n-k+1}},\quad n=0,1,2,...$$ where the initial conditions $x_{-2k+1},...,x_{-1},x_0$ are positive, $k\in\{1,2,...\}$, and the parameters $\beta,~\gamma,~A,~B$ are positive. It is worth mentioning that if we introduce the change of variables $$z_m=y_{mk},$$ then $z_m$ satisfies the difference equation $$z_{m+1}=y_{(m+1)k}=\frac{p~y_{mk}+q~y_{(m-1)k}}{1+y_{mk}}=\frac{p~z_m+q~z_{m-1}}{1+z_m},$$ which has been investigated by several authors, as mentioned by the authors of the paper under review.

##### MSC:
 39A11 Stability of difference equations (MSC2000) 39A20 Generalized difference equations
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##### References:
 [1] Camouzis, E.; Devault, R.; Ladas, G.: On the recursive sequence xn+1=-1+xn-1xn. Journal of difference equations and applications 7, 477-482 (2001) · Zbl 1081.39503 [2] Cunningham, K.; Kulenovic, M. R. S.; Ladas, G.; Valicenti, S.: On the recursive sequence $xn+1=\alpha +\beta$xnBxn+Cxn-1. Nonlinear analysis, theory, methods & applications 47, 4603-4614 (2001) · Zbl 1042.39522 [3] Dehghan, M.; Saadatmandi, A.: Bounds for solutions of a six-point partial-difference scheme. Computers and mathematics with applications 47, 83-89 (2004) · Zbl 1054.65094 [4] M. Dehghan, M. Jaberi Douraki, The oscillatory character of the recursive sequence xn+1=\alpha +\beta xn-2k+1A+Bxn-k+1, Applied Mathematics and Computation, in press, doi:10.1016/j.amc.2005.07.012. · Zbl 1094.39006 [5] Dehghan, M.; Douraki, M. Jaberi: On the recursive sequence $xn+1=\alpha +\beta$n-$k+1+\gamma$xn-2k+1Bxn-k+1+Cxn-2k+1. Applied mathematics and computation 170, 1045-1066 (2005) · Zbl 1090.39006 [6] Devault, R.; Kosmala, W.; Ladas, G.; Schaultz, S. W.: Global behavior of yn+1=p+yn-kqyn+yn-k. Nonlinear analysis, theory, methods & applications 47, 4743-4751 (2001) · Zbl 1042.39523 [7] El-Metwally, H.; Grove, E. A.; Ladas, G.; Levins, R.; Radin, M.: On the difference equation $xn+1=\alpha +\beta$xn-1e-xn. Nonlinear analysis, theory, methods & applications 47, 4623-4634 (2001) · Zbl 1042.39506 [8] Franke, J. E.; Hong, J. T.; Ladas, G.: Global attractivity and convergence to the two-cycle in a difference equation. Journal of difference equations and applications 5, No. 2, 203-209 (1999) · Zbl 0927.39005 [9] Gibbons, C. H.; Kulenovic, M. R. S.; Ladas, G.: On the recursive sequence $xn+1=\alpha +\beta$xn$\gamma +xn$. Mathematical sciences research hot-line 4, No. 2, 1-11 (2000) · Zbl 1039.39004 [10] Grove, E. A.; Kent, C. M.; Levins, R.; Ladas, G.; Valicenti, S.: Global stability in some population models. Proceedings of the fourth international conference on difference equations and applications, August 27 -- 31, 1998, Poznań, Poland, 149-176 (2000) · Zbl 0988.39018 [11] Grove, E. A.; Ladas, G.; Mcgrath, L. C.; Teixeira, C. T.: Existence and behavior of solutions of a rational system. Communications on applied nonlinear analysis 8, 1-25 (2001) · Zbl 1035.39013 [12] Jaroma, J. H.: On the global asymptotic stability of $xn+1=\alpha +\beta$xnA+Cxn-1. Proceedings of the first international conference on difference equations and applications, May 25 -- 28, 1994, san antonio, TX, 281-294 (1995) [13] Karakostas, G.: Asymptotic 2-periodic difference equations with diagonally self-invertible responses. Journal of difference equations and applications 6, No. 3, 329-335 (2000) · Zbl 0963.39020 [14] Kocic, V. L.; Ladas, G.: Global behavior of nonlinear difference equations of higher order with applications. (1993) · Zbl 0787.39001 [15] Kocic, V. L.; Ladas, G.: Global attractivity in nonlinear delay difference equations. Proceedings of American mathematical society 115, 1083-1088 (1992) · Zbl 0756.39005 [16] Kocic, V. L.; Ladas, G.; Rodrigues, I. W.: On the rational recursive sequences. Journal of mathematical analysis and applications 173, 127-157 (1993) · Zbl 0777.39002 [17] Kosmala, W.; Kulenovic, M. R. S.; Ladas, G.; Teixeira, C. T.: On the recursive sequence yn+1=p+yn-1qyn+yn-1. Journal of mathematical analysis and applications 251, 571-586 (2000) · Zbl 0967.39004 [18] Kuang, Y. K.; Cushing, J. M.: Global stability in a nonlinear difference-delay equation model of flour beetle population growth. Journal of difference equations and applications 2, No. 1, 31-37 (1996) · Zbl 0862.39005 [19] Kulenovic, M. R. S.; Ladas, G.: Dynamics of second order rational difference equations with open problems and conjectures. (2002) [20] Kulenovic, M. R. S.; Ladas, G.; Prokup, N. R.: On the recursive sequence $xn+1=\alpha xn+\beta$xn-11+xn. Journal of difference equations and applications 6, No. 5, 563-576 (2000) · Zbl 0966.39003 [21] Kulenovic, M. R. S.; Ladas, G.; Prokup, N. R.: On a rational difference equation. Computers and mathematics with applications 41, 671-678 (2001) · Zbl 0985.39017 [22] Kulenovic, M. R. S.; Ladas, G.; Sizer, W. S.: On the recursive sequence $xn+1=\alpha xn+\beta$xn-$1\gamma$xn+Cxn-1. Mathematical science and researches hot-line 2, No. 5, 1-16 (1998) [23] Kuruklis, S. A.; Ladas, G.: Oscillation and global attractivity in a discrete delay logistic model. Quarterly of applied mathematics 50, 227-233 (1992) · Zbl 0799.39004 [24] Murray, J. D.: Mathematical biology. (1993) · Zbl 0779.92001 [25] Saaty, T. L.: Modern nonlinear equations. (1967) · Zbl 0148.28202 [26] Sedaghat, H.: Geometric stability conditions for higher order difference equations. Journal of mathematical analysis and applications 224, 225-272 (1998) · Zbl 0911.39003 [27] Sedaghat, H.: Nonlinear difference equations, theory with applications to social science models. (2003) · Zbl 1020.39007 [28] Majid Jaberi Douraki, The study of some classes of nonlinear difference equations, M.Sc. Thesis, Department of Applied Mathematics, Amirkabir University of Technology (Tehran Polytechnic), July 2004. · Zbl 1185.37025 [29] Dehghan, M.; Douraki, M. Jaberi: Dynamics of a rational difference equation using both theoretical and computational approaches. Applied mathematics and computation 168, 756-775 (2005) · Zbl 1085.39006 [30] Nasri, M.; Dehghan, M.; Douraki, M. Jaberi: Study of a system of non-linear difference equations arising in a deterministic model for HIV infection. Applied mathematics and computation 171, No. 2, 634-658 (2005) · Zbl 1087.92054