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Global asymptotic behavior of \(y_{n+1}=(py_{n}+y_{n - 1})/(r+qy_{n}+y_{n - 1})\). (English) Zbl 1149.39002

The authors study the global stability character of the equilibrium points and the period-two solutions of a certain rational difference equation with positive parameters and nonnegative initial conditions. It is shown that every solution of the equation converges to either the zero equilibrium, the positive equilibrium, or the period-two solution, for all values of parameters outside of a specific set. When the equilibrium points and period-two solution coexist, a precise description of the basins of attraction of all points are given.

MSC:

39A11 Stability of difference equations (MSC2000)
39A20 Multiplicative and other generalized difference equations
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[1] Kulenović MRS, Ladas G: Dynamics of Second Order Rational Difference Equations, with Open Problems and Conjectures. Chapman & Hall/CRC, Boca Raton, Fla, USA; 2002:xii+218. · Zbl 0981.39011
[2] Kocić VL, Ladas G, Rodrigues IW: On rational recursive sequences.Journal of Mathematical Analysis and Applications 1993,173(1):127-157. 10.1006/jmaa.1993.1057 · Zbl 0777.39002 · doi:10.1006/jmaa.1993.1057
[3] Gibbons CH, Kulenović MRS, Ladas G: On the recursive sequence .Mathematical Sciences Research Hot-Line 2000,4(2):1-11. · Zbl 1039.39004
[4] Kulenović MRS, Ladas G, Prokup NR: On the recursive sequence .Journal of Difference Equations and Applications 2000,6(5):563-576. 10.1080/10236190008808246 · Zbl 0966.39003 · doi:10.1080/10236190008808246
[5] Kulenović MRS, Ladas G, Prokup NR: A rational difference equation.Computers & Mathematics with Applications 2001,41(5-6):671-678. 10.1016/S0898-1221(00)00311-4 · Zbl 0985.39017 · doi:10.1016/S0898-1221(00)00311-4
[6] Kulenović MRS, Ladas G, Sizer WS: On the recursive sequence .Mathematical Sciences Research Hot-Line 1998,2(5):1-16. · Zbl 0960.39502
[7] Kulenović MRS, Merino O: Convergence to a period-two solution for a class of second order rational difference equations. In Proceedings of the 10th International Conference on Difference Equations, July 2007, Munich, Germany. World Scientific; 344-353. · Zbl 1128.39006
[8] Kulenović MRS, Merino O: Global attractivity of the equilibrium of for .Journal of Difference Equations and Applications 2006,12(1):101-108. 10.1080/10236190500410109 · Zbl 1099.39007 · doi:10.1080/10236190500410109
[9] Nussbaum RD: Global stability, two conjectures and Maple.Nonlinear Analysis: Theory, Methods & Applications 2007,66(5):1064-1090. 10.1016/j.na.2006.01.005 · Zbl 1121.39004 · doi:10.1016/j.na.2006.01.005
[10] Camouzis E, Ladas G: When does local asymptotic stability imply global attractivity in rational equations?Journal of Difference Equations and Applications 2006,12(8):863-885. 10.1080/10236190600772663 · Zbl 1105.39001 · doi:10.1080/10236190600772663
[11] Enciso GA, Sontag ED: Global attractivity, I/O monotone small-gain theorems, and biological delay systems.Discrete and Continuous Dynamical Systems. Series A 2006,14(3):549-578. · Zbl 1111.93071
[12] Kulenović MRS, Yakubu A-A: Compensatory versus overcompensatory dynamics in density-dependent Leslie models.Journal of Difference Equations and Applications 2004,10(13-15):1251-1265. · Zbl 1061.92047 · doi:10.1080/10236190410001652711
[13] Smith HL: The discrete dynamics of monotonically decomposable maps.Journal of Mathematical Biology 2006,53(4):747-758. 10.1007/s00285-006-0004-3 · Zbl 1118.65057 · doi:10.1007/s00285-006-0004-3
[14] Kulenović MRS, Merino O: A global attractivity result for maps with invariant boxes.Discrete and Continuous Dynamical Systems. Series B 2006,6(1):97-110. · Zbl 1092.37014
[15] Janowski EJ, Kulenović MRS: Attractivity and global stability for linearizable difference equations. · Zbl 1186.39025
[16] Kulenović MRS, Merino O: Competitive-exclusion versus competitive-coexistence for systems in the plane.Discrete and Continuous Dynamical Systems. Series B 2006,6(5):1141-1156. · Zbl 1116.37030 · doi:10.3934/dcdsb.2006.6.1141
[17] Smith HL: Planar competitive and cooperative difference equations.Journal of Difference Equations and Applications 1998,3(5-6):335-357. 10.1080/10236199708808108 · Zbl 0907.39004 · doi:10.1080/10236199708808108
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