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The characteristics of a higher-order rational difference equation. (English) Zbl 1108.39006
The authors study the semicycles, periodicity, global stability and boundedness of the solutions of the rational difference equation $x_{n+1}=\frac{\alpha+\beta x_{n-k+1} + \gamma x_{n-2k+1}}{A+Bx_{n-k+1}},\quad n=0,1,2,\dots$ where $$A,~B$$ and $$\alpha,~\beta,~\gamma$$ are positive, $$k\in \{1,2,3,\dots\}$$, and the initial conditions $$x_{-2k+1},\dots,x_{-1},x_0$$ are positive real numbers.

##### MSC:
 39A11 Stability of difference equations (MSC2000) 39A20 Multiplicative and other generalized difference equations, e.g., of Lyness type
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##### References:
 [1] Gibbons, C.H.; Kulenovic, M.R.S.; Ladas, G.; Voulov, H.D., On the trichotomy character of $$x_{n + 1} = \frac{\alpha + \beta x_n + \gamma x_{n - 1}}{A + x_n}$$, Journal of difference equations and applications, 8, 2, 1-11, (2000) · Zbl 1005.39017 [2] Dehghan, M.; Jaberi Douraki, M., On the recursive sequence $$x_{n + 1} = \frac{\alpha + \beta x_{n - k + 1} + \gamma x_{n - 2 k + 1}}{\mathit{Bx}_{n - k + 1} + \mathit{Cx}_{n - 2 k + 1}}$$, Applied mathematics and computation, 170, 1045-1066, (2005) · Zbl 1090.39006 [3] Su, You-Hui; Li, Wan-Tong, Global asymptotic stability of a second order nolinear difference equation, Applied mathematics and computation, 168, 981-989, (2005) · Zbl 1098.39005 [4] Kulenovic, M.R.S.; Ladas, G., Dynamics of second order rational difference equations with open problems and conjectures, (2002), Chapman and Hall/CRC Boca Raton · Zbl 0981.39011 [5] Sedaghat, H., Nonlinear difference equations, theory with applications to social science models, (2003), Kluwer Academic Publishers Dordrecht · Zbl 1020.39007 [6] Kuang, Y.K.; Cushing, J.M., Global stability in a nonlinear difference-delay equation model of flour beetle population growth, Journal of difference equation and application, 2, 1, 31-37, (1996) · Zbl 0862.39005 [7] Kocic, V.L.; Ladas, G., Global attractivity in nonlinear delay difference equations, Proceedings of American mathematical society, 115, 1083-1088, (1992) · Zbl 0756.39005 [8] Kocic, V.L.; Ladas, G., Global behavior of nonlinear difference equations of higher order with applications, (1993), Kluwer Academic Publishers Dordrecht · Zbl 0787.39001 [9] Dehghan, M.; Jaberi Douraki, M., Dynamics of a rational difference equation using both theoretical and computational approaches, Applied mathematics and computation, 168, 756-775, (2005) · Zbl 1085.39006 [10] Devault, R.; Kosmala, W.; Ladas, G.; Schaultz, S.W., Global behavior of $$y_{n + 1} = \frac{p + y_{n - k}}{\mathit{qy}_n + y_{n - k}}$$, Nonlinear analysis, theory, methods & applications, 47, 83-89, (2004) · Zbl 1042.39523 [11] Kuruklis, S.A.; Ladas, G., Oscilation and global attractivity in a discrete delay logistic model, Quarterly of applied mathematics, 50, 227-233, (1992) · Zbl 0799.39004 [12] Sandefur, J.T., Discrete dynamical systems, theory and applications, (1990), Clarendon Press Oxford · Zbl 0715.58001 [13] Kelly, W.G.; Peterson, A.C., Difference equations, an introduction with applications, (2001), Academic press [14] Mickens, R.E., Difference equations, theory and applications, (1990), Chapman & Hall New York, London · Zbl 1235.34118 [15] Kulenovic, M.R.S.; Merino, O., Discrete dynamical and difference equations with Mathematica, (2002), Chapman and Hall/CRC Boca Raton · Zbl 1107.39007 [16] El-Metwally, H.; Grove, E.A.; Ladas, G.; Levins, R.; Radin, M., On the difference equation $$x_{n + 1} = \alpha + \beta x_{n - 1} \operatorname{e}^{- x_n}$$, Nonlinear analysis, theory, methods & application, 47, 4623-4634, (2001) · Zbl 1042.39506 [17] Grove, E.A.; Kent, C.M.; Levins, R.; Ladas, G.; Valisenti, S., Global stability in some population models, (), 149-176 · Zbl 0988.39018 [18] Saaty, T.L., Modern nonlinear equation, (1967), McGraw-Hill New York · Zbl 0148.28202 [19] Sedaght, H., Geometric stability conditions for higher order difference equations, Journal of mathematical analysis and applications, 224, 225-272, (1998) [20] Murray, J.D., Mathematical biology, (1993), Springer-Verlag Berlin, Heidelberg · Zbl 0779.92001 [21] 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, 2, 203-209, (1999) · Zbl 0927.39005 [22] Gibbons, C.H.; Kulenovic, M.R.S.; Ladas, G., On the recursive sequence $$x_{n + 1} = \frac{\alpha + \beta x_{n - 1}}{\gamma + x_n}$$, Mathematical sciences research hot-line, 4, 2, 1-11, (2000) · Zbl 1039.39004 [23] Jaroma, J.H., On the global asymptotic stability of $$x_{n + 1} = \frac{\alpha + \beta x_n}{A + \mathit{Cx}_{n - 1}}$$, (), 281-294 [24] Sharkovski, A.N., Difference equations and boundary value problems, (), 3-22 · Zbl 1065.39040 [25] Iavernaro, F.; Mazzia, F.; Trigiante, D., On the discrete nature of physical laws, (), 35-50 · Zbl 1065.39036 [26] 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 [27] Chan, D.M.; Chang, E.R.; Dehghan, M.; Kent, C.M.; Mazrooei-Sebdani, R.; Sedaghat, H., Asymptotic stability for difference equations with decreasing arguments, Journal of difference equations and applications, 12, 2, 109-123, (2006) · Zbl 1093.39001 [28] R. Mazrooei-Sebdani, The convergence and stability of solutions of rational difference equations of second-order, M.Sc. Thesis, Department of Applied Mathematics, Amirkabir University of Technology, December 2005. · Zbl 1286.39005 [29] M. Dehghan, R. Mazrooei-Sebdani, Some results about the global attractivity of bounded solutions of difference equations with applications to periodic solutions, Chaos, Solitons & Fractals, in press. · Zbl 1138.39005 [30] Nasri, M.; Dehghan, M.; Jaberi Douraki, M., Study of a system of nonlinear difference equations arising in a deterministic model for HIV infection, Applied mathematics and computation, 171, 2, 1306-1330, (2005) · Zbl 1087.92054 [31] M. Dehghan, M. Nasri, M.R. Razvan, Global stability of a deterministic model for HIV infection in vivo, Chaos, Solitons & Fractals, in press. · Zbl 1142.92336
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