×

On the difference equation \(x_{n+1}=\frac{a+bx_{n-k}^{cx_n-m}}{1+g(x_{n-1})}\). (English) Zbl 1137.39009

The second author investigated global attractivity, boundedness and periodic solutions of the rational difference equation \(x_{n+1}=(\alpha+ \beta x_{n-1})/(1+g(x_{n}))\). In this paper the authors generally consider the equation \(x_{n+1}=(a+ b x_{n-k}-cx_{n-m})/(1+g(x_{n-l}))\), where \(a,b,c\) are nonnegative real numbers, \(k,l,m\) are nonnegative integers and \(g\) is a nonnegative real function. They discuss its oscillation and periodic solutions and study the boundedness and stability of its positive solutions. They also generalize their results to
\[ x_{n+1}=\frac{a+\sum_{j=0}^k b_j x_{n-j} -\sum_{i=0}^m c_i x_{n-i}}{f(x_n,\dots,x_{n-l})}. \]

MSC:

39A11 Stability of difference equations (MSC2000)
39A20 Multiplicative and other generalized difference equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] M. T. Aboutaleb, M. A. El-Sayed and A. E. Hamza,Stability of the recursive sequence x n+1=({\(\alpha\)}x n )/({\(\gamma\)}+x n), J. Math. Anal. Appl.261 (2001), 126–133. · Zbl 0990.39009 · doi:10.1006/jmaa.2001.7481
[2] D. C. Chang and S. Stević,On the recursive sequence \(x_{n + 1} = \alpha + \frac{{\beta x_{n - 1} }}{{1 + g(x_n )}}\) , Appl. Anal.82(2) (2003), 145–156. · Zbl 1033.39007 · doi:10.1080/0003681031000063810
[3] C. H. Gibbons, M. R. S. Kulenović and G. Ladas,On the recursive sequence \(x_{n + 1} = \frac{{\alpha + \beta x_{n - 1} }}{{\gamma + x_n }}\) , ath. Sci. Res. Hot-Line4(2), (2000), 1–11. · Zbl 1039.39004
[4] V. L. Kocić and G. Ladas,Global Behavior of Nonlinear Difference Equations of Higher Order with Application, Kluwer Academic Publishers, Dordrecht, 1993. · Zbl 0779.34057
[5] M. R. S. Kulenović and G. Ladas,Dynamics of Second Order Rational Difference Equa- tions, Chapman & Hall/CRC, (2001).
[6] C. H. Ou, H. S. Tang and W. Lou,Global stability for a class of difference equations, Appl. Math. J. Chinese Univ. Ser. B15 (2000), 33–36. · Zbl 0959.39013 · doi:10.1007/s11766-000-0006-7
[7] S. Stević,Behaviour of the positive solutions of the generalized Beddington-Holt equation, Panamer. Math. J.10(4) (2000), 77–85. · Zbl 1039.39005
[8] S. Stević,A generalization of the Copson’s theorem concerning sequences which satisfy a linear inequality, Indian J. Math.43(3) (2001), 277–282. · Zbl 1034.40002
[9] S. Stević,A note on bounded sequences satisfying linear inequality, Indian J. Math.43(2) (2001), 223–230. · Zbl 1035.40002
[10] S. Stević,On the recursive sequence \(x_{n + 1} = - \frac{1}{{x_n }} + \frac{A}{{x_{n - 1} }}\) , Int. J. Math. Math. Sci.27(1) (2001), 1–6. · Zbl 1005.39016 · doi:10.1155/S0161171201010614
[11] S. Stević,A note on the difference equation \(x_{n + 1} = \sum\nolimits_{i = 0}^k {\frac{{\alpha _i }}{{x_{n - i}^{p_i } }}} \) , J. Differ. Equations Appl.8(7) (2002), 641–647. · Zbl 1008.39005 · doi:10.1080/10236190290032507
[12] S. Stević,A global convergence result, Indian J. Math.44(3) (2002), 361–368. · Zbl 1034.39002
[13] S. Stević,A global convergence results with applications to periodic solutions, Indian J. pure appl. Math.33(1) (2002), 45–53. · Zbl 1002.39004
[14] S. Stević,Asymptotic behaviour of a sequence defined by iteration with applications, Col- loq. Math.93(2) (2002), 267–276. · Zbl 1029.39006 · doi:10.4064/cm93-2-6
[15] S. Stević,On the recursive sequence x n+1 =g(x n ,x n-1 )/(A + x n ), Appl. Math. Lett.15 (2002), 305–308. · Zbl 1029.39007 · doi:10.1016/S0893-9659(01)00135-5
[16] S. Stević,On the recursive sequence x n+1 =x n-1 /(x n ), Taiwanese J. Math.6(3) (2002), 405–414. · Zbl 1019.39010
[17] S. Stević,On the recursive sequence \(x_{n + 1} = \frac{{\alpha + \beta x_{n - 1} }}{{1 + g(x_n )}}\) Indian J. pure appl. Math.33(12) (2002), 1767–1774. · Zbl 1019.39011
[18] S. Stević,Boundedness and persistence of solutions of a nonlinear difference equation, Demonstratio Math. 36(1) (2003), 99–104. · Zbl 1028.39002
[19] S. Stević,On the recursive sequence \(x_{n + 1} = \frac{A}{{\Pi _{i = 0}^k x_{n - i} }} + \frac{1}{{\Pi _{j = k + 2}^{2(k + 1)} x_{n - j} }}\) , Taiwanese J. Math.7(2) (2003), 249–259. · Zbl 1054.39008
[20] G. Zhang and L. J. Zhang,Periodicity and Attractivity of A Nonlinear Higher Order Difference Equation, Appl. Math. Comput.126(2)(2005), 395–401. · Zbl 1068.39034 · doi:10.1016/j.amc.2003.12.034
[21] L. J. Zhang, G. Zhang and H. Liu,Periodicity and attractivity of a nonlinear higher order difference equation, J. Applied Mathematics & Computing19(1-2) (2005), 191–201. · Zbl 1083.39015 · doi:10.1007/BF02935798
[22] X. X. Yan and W. T. Li,Global attractivity in the recursive sequence xn+1 =({\(\alpha\)} + {\(\beta\)}xn)/({\(\gamma\)}-xn-1), Appl. Math. Comput.128 (2003), 415–423. · Zbl 1030.39024 · doi:10.1016/S0096-3003(02)00145-5
[23] X. X. Yan, W. T. Li and H. R. Sun,Global attractivity in a higher order nonlinear differ- ence equation, Appl. Math. E-Notes.2 (2002), 51–58. · Zbl 1004.39010
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.