The global attractivity of the rational difference equation $y_n=A+\left (\frac {y_{n-k}}{y_{n-m}}\right )^p$
HTML articles powered by AMS MathViewer
- by Kenneth S. Berenhaut, John D. Foley and Stevo Steviฤ PDF
- Proc. Amer. Math. Soc. 136 (2008), 103-110 Request permission
Abstract:
This paper studies the behavior of positive solutions of the recursive equation \begin{eqnarray} y_n=A+\left (\frac {y_{n-k}}{y_{n-m}}\right )^p,\quad n=0,1,2,\ldots , \nonumber \end{eqnarray} with $y_{-s},y_{-s+1}, \ldots , y_{-1} \in (0, \infty )$ and $k,m \in \{1,2,3,4,\ldots \}$, where $s=\max \{k,m\}$. We prove that if $\mathrm {gcd}(k,m) = 1$, and $p\leq \min \{1,(A+1)/2\}$, then $y_n$ tends to $A+1$. This complements several results in the recent literature, including the main result in K. S. Berenhaut, J. D. Foley and S. Steviฤ, The global attractivity of the rational difference equation $y_{n}=1+\frac {y_{n-k}}{y_{n-m}}$, Proc. Amer. Math. Soc., 135 (2007) 1133โ1140.References
- R. M. Abu-Saris and R. DeVault, Global stability of $y_{n+1}=A+\frac {y_n}{y_{n-k}}$, Appl. Math. Lett. 16 (2003), no.ย 2, 173โ178. MR 1962312, DOI 10.1016/S0893-9659(03)80028-9
- A. M. Amleh, E. A. Grove, G. Ladas, and D. A. Georgiou, On the recursive sequence $x_{n+1}=\alpha +x_{n-1}/x_n$, J. Math. Anal. Appl. 233 (1999), no.ย 2, 790โ798. MR 1689579, DOI 10.1006/jmaa.1999.6346
- Kenneth S. Berenhaut, John D. Foley, and Stevo Steviฤ, The global attractivity of the rational difference equation $y_n=1+{y_{n-k}\over y_{n-m}}$, Proc. Amer. Math. Soc. 135 (2007), no.ย 4, 1133โ1140. MR 2262916, DOI 10.1090/S0002-9939-06-08580-7
- Kenneth S. Berenhaut and Stevo Steviฤ, A note on the difference equation $x_{n+1}=\frac {1}{x_nx_{n-1}}+\frac {1}{x_{n-3}x_{n-4}}$, J. Difference Equ. Appl. 11 (2005), no.ย 14, 1225โ1228. MR 2182249, DOI 10.1080/10236190500331370
- K. S. Berenhaut and S. Steviฤ, On Positive Nonoscillatory Solutions of the Difference Equation $x_{n+1}=\alpha + \frac {{x_{n-k}}^p}{{x_n}^p}$, J. Differ. Equations Appl., in press (2005).
- Lothar Berg, Asymptotische Darstellungen und Entwicklungen, Hochschulbรผcher fรผr Mathematik, Band 66, VEB Deutscher Verlag der Wissenschaften, Berlin, 1968 (German). MR 0241873
- L. Berg, On the asymptotics of nonlinear difference equations, Z. Anal. Anwendungen 21 (2002), no.ย 4, 1061โ1074. MR 1957315, DOI 10.4171/ZAA/1127
- Lothar Berg, Inclusion theorems for non-linear difference equations with applications, J. Difference Equ. Appl. 10 (2004), no.ย 4, 399โ408. MR 2047219, DOI 10.1080/10236190310001625280
- Lothar Berg, Corrections to: โInclusion theorems for non-linear difference equations with applicationsโ [J. Difference Equ. Appl. 10 (2004), no. 4, 399โ408; MR2047219], J. Difference Equ. Appl. 11 (2005), no.ย 2, 181โ182. MR 2114324, DOI 10.1080/10236190512331328370
- L. Berg and L. v. Wolfersdorf, On a class of generalized autoconvolution equations of the third kind, Z. Anal. Anwendungen 24 (2005), no.ย 2, 217โ250. MR 2174021, DOI 10.4171/ZAA/1238
- R. DeVault, C. Kent, and W. Kosmala, On the recursive sequence $x_{n+1}=p+{x_{n-k}\over x_n}$, J. Difference Equ. Appl. 9 (2003), no.ย 8, 721โ730. Special Session of the American Mathematical Society Meeting, Part II (San Diego, CA, 2002). MR 1992905, DOI 10.1080/1023619021000042162
- R. DeVault, G. Ladas, and S. W. Schultz, On the recursive sequence $x_{n+1}=A/x_nx_{n-1}+(1/x_{n-3}x_{n-4})$, J. Differ. Equations Appl. 6 (2000), no.ย 4, 481โ483. MR 1785161, DOI 10.1080/10236190008808242
- 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.ย 6, 837โ850. On the occasion of the 60th birthday of Calvin Ahlbrandt. MR 1870725, DOI 10.1080/10236190108808306
- H. M. El-Owaidy, A. M. Ahmed, and M. S. Mousa, On asymptotic behaviour of the difference equation $x_{n+1}=\alpha +\frac {x_{n-1}{}^p}{x_n{}^p}$, J. Appl. Math. Comput. 12 (2003), no.ย 1-2, 31โ37. MR 1976801, DOI 10.1007/BF02936179
- 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. MR 2193366
- V. L. Kociฤ 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. MR 1247956, DOI 10.1007/978-94-017-1703-8
- 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.ย 3, 905โ909. MR 1937429, DOI 10.1090/S0002-9939-02-06611-X
- Stevo Steviฤ, Asymptotic behavior of a sequence defined by iteration with applications, Colloq. Math. 93 (2002), no.ย 2, 267โ276. MR 1930804, DOI 10.4064/cm93-2-6
- Stevo Steviฤ, Asymptotic behavior of a nonlinear difference equation, Indian J. Pure Appl. Math. 34 (2003), no.ย 12, 1681โ1687. MR 2030114
- Stevo Steviฤ, On the recursive sequence $x_{n+1}=\frac {A}{\prod ^k_{i=0}x_{n-i}}+\frac {1}{\prod ^{2(k+1)}_{j=k+2}x_{n-j}}$, Taiwanese J. Math. 7 (2003), no.ย 2, 249โ259. MR 1978014, DOI 10.11650/twjm/1500575062
- Stevo Steviฤ, A note on periodic character of a difference equation, J. Difference Equ. Appl. 10 (2004), no.ย 10, 929โ932. MR 2079642, DOI 10.1080/10236190412331272616
- S. Steviฤ, Some open problems and conjectures on difference equations, http://www.mi.sanu.ac.yu, April 29, 2004.
- Stevo Steviฤ, On the recursive sequence $x_{n+1}=\alpha +{x^p_{n-1}\over x^p_n}$, J. Appl. Math. Comput. 18 (2005), no.ย 1-2, 229โ234. MR 2137703, DOI 10.1007/BF02936567
- Stevo Steviฤ, Global stability and asymptotics of some classes of rational difference equations, J. Math. Anal. Appl. 316 (2006), no.ย 1, 60โ68. MR 2201749, DOI 10.1016/j.jmaa.2005.04.077
- Stevo Steviฤ, On positive solutions of a $(k+1)$th order difference equation, Appl. Math. Lett. 19 (2006), no.ย 5, 427โ431. MR 2213143, DOI 10.1016/j.aml.2005.05.014
Additional Information
- Kenneth S. Berenhaut
- Affiliation: Department of Mathematics, Wake Forest University, Winston-Salem, North Carolina 27109
- Email: berenhks@wfu.edu
- John D. Foley
- Affiliation: Department of Mathematics, Wake Forest University, Winston-Salem, North Carolina 27109
- Email: folejd4@wfu.edu
- Stevo Steviฤ
- Affiliation: Mathematical Institute of The Serbian Academy of Science, Knez Mihailova 35/I 11000 Beograd, Serbia
- Email: sstevic@ptt.yu, sstevo@matf.bg.ac.yu
- Received by editor(s): April 18, 2006
- Received by editor(s) in revised form: July 31, 2006
- Published electronically: September 24, 2007
- Additional Notes: The first author acknowledges financial support from a Sterge Faculty Fellowship.
- Communicated by: Carmen C. Chicone
- © Copyright 2007
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 136 (2008), 103-110
- MSC (2000): Primary 39A10, 39A11
- DOI: https://doi.org/10.1090/S0002-9939-07-08860-0
- MathSciNet review: 2350394