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Global asymptotic stability for a higher order nonlinear rational difference equations. (English) Zbl 1110.39011
Consider the rational difference equation $$x_{n+1}=\frac{ax_{n}+bx_{n-k}}{A+Bx_{n}}~\ \ ,~\ n=0,1,\dots \tag {*}$$ where $a~,b~,A,B$ are positive real numbers, $k\geq 1$ is a positive integer, and the initial conditions $x_{-k},~\dots,~x_{-1},~x_{0}$ are nonnegative real numbers. The authors solve an open problem posed by {\it M. R. S. Kulenović} and {\it G. Ladas} [Dynamics of second order rational difference equations, Chapman & Hall / CRC, Boca Raton, FL (2002; Zbl 0981.39011), p. 129]. They prove the following {Theorem}: (a) If $b\leq A-a,$ then the zero equilibrium of Eq. ($*$) is globally asymptotically stable. (b) If $A-a<b<A+a,$ then the positive equilibrium of Eq. ($*$) is globally asymptotically stable. The boundedness, periodic character, invariant intervals of all nonnegative solutions of ($*$) are investigated.

##### MSC:
 39A11 Stability of difference equations (MSC2000) 39A20 Generalized difference equations
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##### References:
 [1] Amleh, A. M.; Grove, E. A.; Georgiou, D. A.; Ladas, G.: On the recursive sequence $xn+1=\alpha$+(xn - 1/xn). J. math. Anal. appl. 233, 790-798 (1999) · Zbl 0962.39004 [2] Devault, R.; Kosmala, W.; Ladas, G.; Schultz, S. W.: Global behavior of yn+1=(p+yn - k)/(qyn+yn - k). Nonlin. analysis TMA 47, 4743-4751 (2001) · Zbl 1042.39523 [3] Grove, E. A.; Kent, C. M.; Ladas, G.; Valicenti, S.; Levins, R.: Global attractivity 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 [4] Kocic, V. L.; Ladas, G.: Global behavior of nonlinear difference equations of higher order with application. (1993) · Zbl 0787.39001 [5] Kocic, V. L.; Ladas, G.; Rodrigues, I. W.: On rational recursive sequences. J. math. Anal. appl. 173, 127-157 (1993) · Zbl 0777.39002 [6] Kosmala, W. A.; Kulenovic, M. R. S.; Ladas, G.; Teixeira, C. T.: On the recursive sequence yn+1=(p+yn - 1)/(qyn+yn - 1). J. math. Anal. appl. 251, 571-586 (2000) · Zbl 0967.39004 [7] Kulenovi, M. R. S.; Ladas, G.: Dynamics of second order rational difference equations. (2002) · Zbl 0981.39011 [8] Kulenovic, M. R. S.; Ladas, G.; Prokup, N. R.: On the recursive sequence $xn+1=\alpha xn+\beta$xn-11+xn. J. differ. Eq. appl. 6, No. 5, 563-576 (2000) · Zbl 0966.39003 [9] Li, W. T.; Sun, H. R.: Global attractivity of a rational recursive sequence. Dyn. syst. Appl. 11, No. 3, 339-345 (2002) · Zbl 1019.39007 [10] Su, Y. H.; Li, W. T.: Global asymptotic stability of a second order nonlinear difference equation. Appl. math. Comput. 168, 981-989 (2005) · Zbl 1098.39005 [11] Su, Y. H.; Li, W. T.: Global behavior of a higher order nonlinear rational difference equation. J. differ. Eq. appl. 11, No. 10, 947-958 (2005) · Zbl 1081.39005 [12] Su, Y. H.; Li, W. T.; Stevic, S.: Dynamics of a higher order nonlinear rational difference equation. J. differ. Eq. appl. 11, No. 2, 133-150 (2005) · Zbl 1071.39017 [13] Yan, X. X.; Li, W. T.; Sun, H. R.: Global attractivity in a higher order nonlinear difference equation. Appl. math. E-notes 2, 51-58 (2002) · Zbl 1004.39010 [14] Yan, X. X.; Li, W. T.: Global attractivity in the recursive sequence xn+1=($\alpha - \beta$xn)/($\gamma$ - xn - 1). Appl. math. Comput. 138, 415-423 (2003) · Zbl 1030.39024 [15] Yan, X. X.; Li, W. T.: Global attractivity in a rational recursive sequence. Appl. math. Comput. 145, 1-12 (2003) · Zbl 1044.39013 [16] Yan, X. X.; Li, W. T.: Global attractivity for a class of higher order nonlinear difference equations. Appl. math. Comput. 149, 533-546 (2004) · Zbl 1040.39009 [17] Yan, X. X.; Li, W. T.: Global attractivity for a class of nonlinear difference equations. Soochow J. Math. 29, No. 3, 327-338 (2003) · Zbl 1045.39010 [18] Yan, X. X.; Li, W. T.: Dynamic behavior of a recursive sequence. Appl. math. Comput. 157, 713-727 (2004) · Zbl 1069.39025 [19] Yan, X. X.; Li, W. T.; Zhao, Z.: On the recursive sequence $xn+1=\alpha$ - (xn/xn - 1). J. appl. Math. comput. 17, No. 1 -- 2, 269-282 (2005) · Zbl 1068.39030