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Boundedness and global stability of a higher-order difference equation. (English) Zbl 1161.39011
This paper deals with the boundedness and global stability of following difference equation
\[ x_n=\frac{p+\sum_{j=1}^{m}\alpha_jx_{n-p_j}}{qx_{n-k} +\sum_{j=1}^{m}\alpha_jx_{n-p_j}}, \quad n\in {\mathbb N}_0, \]
where \(k, m, p_j, j=1, 2, \dots, m\) are natural numbers, \(p_1<\dots<p_m\), \(k\neq p_j\), \(j=1, 2, \dots, m\), \(p, q\in (0, \infty)\) and \(\alpha_j>0\), \(j=1, 2, \dots, m\) such that \(\sum_{j=1}^{m}\alpha_j=1\). This equation has a unique positive equilibrium \(\bar {x}\). The author divides the problem into three different cases \(p<q\), \(p>q\) and \(p=q\). In each case, the global stability of \(\bar{x}\) is studied. Finally, the author concludes that if \(p, q\in (0, \infty)\), then all positive solutions are bounded. These results extend those by R. DeVault, W. Kosmala, G. Ladas and S. W. Schultz [Nonlinear Anal., Theory Methods Appl. 47, No. 7, 4743–4751 (2001; Zbl 1042.39523)].

MSC:
39A11 Stability of difference equations (MSC2000)
39A20 Multiplicative and other generalized difference equations
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[1] DOI: 10.1016/S0362-546X(01)00586-7 · Zbl 1042.39523
[2] Kocic V.L., Global Behaviour of Nonlinear Difference Equations of Higher Order with Applications (1993) · Zbl 0787.39001
[3] DOI: 10.1080/0003681021000021114 · Zbl 1022.39005
[4] DOI: 10.1006/jmaa.2000.7032 · Zbl 0967.39004
[5] Kulenović M., Dynamics of Second Order Rational Difference Equations (2002) · Zbl 0981.39011
[6] DOI: 10.1080/10236190601069325 · Zbl 1113.39011
[7] Sun T., Discrete Dyn. Nat. Soc. 2006 pp 12– (2006)
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