×

The global attractivity of the rational difference equation \(y_{n}=1+\frac{y_{n-k}}{y_{n-m}}\). (English) Zbl 1109.39004

The authors prove that the solutions of the recursive equation \[ y_n=1+\displaystyle\frac{y_{n-k}}{y_{n-m}},\,\,\,n=0,1,2,\ldots,\eqno(1) \] with \(y_{-s},\,y_{-s+1},\ldots,y_{-1}\in (0,\infty)\), \(k,\,m\in \{1,2,\ldots,\}\), \(s=\text{max}\{k,\,m\}\), \(\text{gcd}(k,m)=1\) and \(k\) odd, converge exponentially to the unique equilibrium solution 2. By using a result due to E. A. Grove and G. Ladas [Periodicities in Nonlinear Difference Equations. Advances in Discrete Mathematics and Applications 4. (Boca Raton,) FL: Chapman & Hall/CRC. (2005; Zbl 1078.39009)], the periodic character of equation (1) is also established.

MSC:

39A11 Stability of difference equations (MSC2000)
39A20 Multiplicative and other generalized difference equations

Citations:

Zbl 1078.39009
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] R. M. Abu-Saris and R. DeVault, Global stability of \?_{\?+1}=\?+\frac{\?_{\?}}\?_{\?-\?}, Appl. Math. Lett. 16 (2003), no. 2, 173 – 178. · Zbl 1049.39002 · doi:10.1016/S0893-9659(03)80028-9
[2] A. M. Amleh, E. A. Grove, G. Ladas, and D. A. Georgiou, On the recursive sequence \?_{\?+1}=\?+\?_{\?-1}/\?_{\?}, J. Math. Anal. Appl. 233 (1999), no. 2, 790 – 798. · Zbl 0962.39004 · doi:10.1006/jmaa.1999.6346
[3] K. S. BERENHAUT, J. D. FOLEY AND S. STEVIC, Quantitative bounds for the recursive sequence \( y_{n+1}=A+\frac{y_{n}}{y_{n-k}}\), Appl. Math. Lett., in press, (2005).
[4] E. Camouzis, G. Ladas, and H. D. Voulov, On the dynamics of \?_{\?+1}=\?+\?\?_{\?-1}+\?\?_{\?-2}\over\?+\?_{\?-2}, J. Difference Equ. Appl. 9 (2003), no. 8, 731 – 738. Special Session of the American Mathematical Society Meeting, Part II (San Diego, CA, 2002). · Zbl 1050.39003 · doi:10.1080/1023619021000042153
[5] H. El-Metwally, E. A. Grove, G. Ladas, and H. D. Voulov, On the global attractivity and the periodic character of some difference equations, J. Differ. Equations Appl. 7 (2001), no. 6, 837 – 850. On the occasion of the 60th birthday of Calvin Ahlbrandt. · Zbl 0993.39008 · doi:10.1080/10236190108808306
[6] C. H. Gibbons, M. R. S. Kulenović, G. Ladas, and H. D. Voulov, On the trichotomy character of \?_{\?+1}=(\?+\?\?_{\?}+\?\?_{\?-1})/(\?+\?_{\?}), J. Difference Equ. Appl. 8 (2002), no. 1, 75 – 92. · Zbl 1005.39017 · doi:10.1080/10236190211940
[7] E. A. Grove and G. Ladas, Periodicities in nonlinear difference equations, Advances in Discrete Mathematics and Applications, vol. 4, Chapman & Hall/CRC, Boca Raton, FL, 2005. · Zbl 1078.39009
[8] V. L. Kocić and G. Ladas, Global behavior of nonlinear difference equations of higher order with applications, Mathematics and its Applications, vol. 256, Kluwer Academic Publishers Group, Dordrecht, 1993. · Zbl 0787.39001
[9] W. T. Patula and H. D. Voulov, On the oscillation and periodic character of a third order rational difference equation, Proc. Amer. Math. Soc. 131 (2003), no. 3, 905 – 909. · Zbl 1014.39010
[10] Stevo Stević, Behavior of the positive solutions of the generalized Beddington-Holt equation, PanAmer. Math. J. 10 (2000), no. 4, 77 – 85. · Zbl 1039.39005
[11] Stevo Stević, A note on periodic character of a difference equation, J. Difference Equ. Appl. 10 (2004), no. 10, 929 – 932. · Zbl 1057.39005 · doi:10.1080/10236190412331272616
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.