Title:
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On the rational recursive sequence $ \ x_{n+1}=\Big ( A+\sum _{i=0}^k\alpha _ix_{n-i}\Big ) \Big / \sum _{i=0}^k\beta _ix_{n-i} $ (English) |
Author:
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Zayed, E. M. E. |
Author:
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El-Moneam, M. A. |
Language:
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English |
Journal:
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Mathematica Bohemica |
ISSN:
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0862-7959 (print) |
ISSN:
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2464-7136 (online) |
Volume:
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133 |
Issue:
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3 |
Year:
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2008 |
Pages:
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225-239 |
Summary lang:
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English |
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Category:
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math |
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Summary:
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The main objective of this paper is to study the boundedness character, the periodic character, the convergence and the global stability of positive solutions of the difference equation \[ x_{n+1}=\bigg ( A+\sum _{i=0}^k\alpha _ix_{n-i}\bigg ) \Big / \sum _{i=0}^k\beta _ix_{n-i},\ \ n=0,1,2,\dots \] where the coefficients $A$, $\alpha _i$, $\beta _i$ and the initial conditions $x_{-k},x_{-k+1},\dots ,x_{-1},x_0$ are positive real numbers, while $k$ is a positive integer number. (English) |
Keyword:
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difference equations |
Keyword:
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boundedness character |
Keyword:
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period two solution |
Keyword:
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convergence |
Keyword:
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global stability |
MSC:
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34C99 |
MSC:
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39A10 |
MSC:
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39A11 |
MSC:
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39A20 |
MSC:
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39A22 |
MSC:
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39A23 |
MSC:
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39A30 |
MSC:
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39A99 |
idZBL:
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Zbl 1199.39025 |
idMR:
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MR2494777 |
DOI:
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10.21136/MB.2008.140612 |
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Date available:
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2010-07-20T17:26:29Z |
Last updated:
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2020-07-29 |
Stable URL:
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http://hdl.handle.net/10338.dmlcz/140612 |
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Reference:
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[1] Aboutaleb, M. T., El-Sayed, M. A., Hamza, A. E.: Stability of the recursive sequence $x_{n+1}=(\alpha -\beta x_n)/(\gamma +x_{n-1})$.J. Math. Anal. Appl. 261 (2001), 126-133. Zbl 0990.39009, MR 1850961 |
Reference:
|
[2] Agarwal, R.: Difference Equations and Inequalities.Theory, Methods and Applications, Marcel Dekker, New York (1992). Zbl 0925.39001, MR 1155840 |
Reference:
|
[3] Amleh, A. M., Grove, E. A., Ladas, G., Georgiou, D. A.: On the recursive sequence $x_{n+1}=\allowbreak\alpha +(x_{n-1}/x_n)$.J. Math. Anal. Appl. 233 (1999), 790-798. MR 1689579 |
Reference:
|
[4] Vault, R. De, Kosmala, W., Ladas, G., Schultz, S. W.: Global behavior of $\thickmuskip1mu plus 2mu y_{n+1}=(p+y_{n-k})/\allowbreak(qy_n+y_{n-k})$.Nonlinear Analysis 47 (2001), 4743-4751. MR 1975867 |
Reference:
|
[5] Vault, R. De, Ladas, G., Schultz, S. W.: On the recursive sequence $x_{n+1}=A/x_n+1/x_{n-2}$.Proc. Amer. Math. Soc. 126 (1998), 3257-3261. MR 1473661, 10.1090/S0002-9939-98-04626-7 |
Reference:
|
[6] Vault, R. De, Schultz, S. W.: On the dynamics of $ x_{n+1}=(\beta x_n+\gamma x_{n-1})/(Bx_n+Dx_{n-2})$.Comm. Appl. Nonlinear Analysis 12 (2005), 35-39. MR 2129054 |
Reference:
|
[7] El-Metwally, H., Grove, E. A., Ladas, G.: A global convergence result with applications to periodic solutions.J. Math. Anal. Appl. 245 (2000), 161-170. Zbl 0971.39004, MR 1756582, 10.1006/jmaa.2000.6747 |
Reference:
|
[8] El-Metwally, H., Ladas, G., Grove, E. A., Voulov, H. D.: On the global attractivity and the periodic character of some difference equations.J. Difference Equ. Appl. 7 (2001), 837-850. Zbl 0993.39008, MR 1870725, 10.1080/10236190108808306 |
Reference:
|
[9] EL-Owaidy, H. M., Ahmed, A. M., Mousa, M. S.: On asymptotic behavior of the difference equation $x_{n+1}=\alpha +(x_{n-1}^p/x_n^p)$.J. Appl. Math. & Comput. 12 (2003), 31-37. MR 1976801, 10.1007/BF02936179 |
Reference:
|
[10] EL-Owaidy, H. M., Ahmed, A. M., Elsady, Z.: Global attractivity of the recursive sequence $x_{n+1}=(\alpha -\beta x_{n-k})/(\gamma +x_n)$.J. Appl. Math. & Comput. 16 (2004), 243-249. MR 2080567, 10.1007/BF02936165 |
Reference:
|
[11] Karakostas, G.: Convergence of a difference equation via the full limiting sequences method.Diff. Equations and Dynamical. System 1 (1993), 289-294. Zbl 0868.39002, MR 1259169 |
Reference:
|
[12] Karakostas, G., Stević, S.: On the recursive sequences $x_{n+1}=A+f(x_n,\dots,x_{n-k+1})/x_{n-1}$.Commun. Appl. Nonlin. Anal. 11 (2004), 87-99. MR 2069821 |
Reference:
|
[13] Kocic, V. L., Ladas, G.: Global Behavior of Nonlinear Difference Equations of Higher Order with Applications.Kluwer Academic Publishers, Dordrecht (1993). Zbl 0787.39001, MR 1247956 |
Reference:
|
[14] Kulenovic, M. R. S., Ladas, G.: Dynamics of Second Order Rational Difference Equations with Open Problems and Conjectures.Chapman & Hall/CRC Press (2002). Zbl 0981.39011, MR 1935074 |
Reference:
|
[15] Kulenovic, M. R. S., Ladas, G., Sizer, W. S.: On the recursive sequence $x_{n+1}=(\alpha x_n+\beta x_{n-1})/(\gamma x_n+\delta x_{n-1})$.Math. Sci. Res. Hot-Line 2 (1998), 1-16. Zbl 0960.39502, MR 1623643 |
Reference:
|
[16] Kuruklis, S. A.: The asymptotic stability of $ x_{n+1}-ax_n+bx_{n-k}=0$.J. Math. Anal. Appl. 188 (1994), 719-731. MR 1305480 |
Reference:
|
[17] Ladas, G., Gibbons, C. H., Kulenovic, M. R. S., Voulov, H. D.: On the trichotomy character of $x_{n+1}=(\alpha +\beta x_n+\gamma x_{n-1})/(A+x_n)$.J. Difference Equations and Appl. 8 (2002), 75-92. Zbl 1005.39017, MR 1884593 |
Reference:
|
[18] Ladas, G., Gibbons, C. H., Kulenovic, M. R. S.: On the dynamics of $x_{n+1}=(\alpha +\beta x_n+\gamma x_{n-1})/(A+Bx_n)$.Proceeding of the Fifth International Conference on Difference Equations and Applications, Temuco, Chile, Jan. 3-7, 2000, Taylor and Francis, London (2002), 141-158. MR 2016061 |
Reference:
|
[19] Ladas, G., Camouzis, E., Voulov, H. D.: On the dynamic of $ x_{n+1}=(\alpha +\gamma x_{n-1}+\delta x_{n-2})/(A+x_{n-2})$.J. Difference Equ. Appl. 9 (2003), 731-738. MR 1992906 |
Reference:
|
[20] Ladas, G.: On the recursive sequence $x_{n+1}=(\alpha +\beta x_n+\gamma x_{n-1})/(A+Bx_n+Cx_{n-1})$.J. Difference Equ. Appl. 1 (1995), 317-321. MR 1350447 |
Reference:
|
[21] Li, W. T., Sun, H. R.: Global attractivity in a rational recursive sequence.Dyn. Syst. Appl. 11 (2002), 339-346. Zbl 1019.39007, MR 1941754 |
Reference:
|
[22] Lin, Yi-Zhong: Common domain of asymptotic stability of a family of difference equations.Appl. Math. E-Notes 1 (2001), 31-33. MR 1833834 |
Reference:
|
[23] Stevi'c, S.: On the recursive sequences $ x_{n+1}=x_{n-1}/g(x_n)$.Taiwanese J. Math. 6 (2002), 405-414. MR 1921603, 10.11650/twjm/1500558306 |
Reference:
|
[24] Stevi'c, S.: On the recursive sequences $ x_{n+1}=g(x_n,x_{n-1})/(A+x_n)$.Appl. Math. Letter 15 (2002), 305-308. MR 1891551 |
Reference:
|
[25] Stevi'c, S.: On the recursive sequences $ x_{n+1}=\alpha +(x_{n-1}^p/x_n^p)$.J. Appl. Math. Comput. 18 (2005), 229-234. MR 2137703, 10.1007/BF02936567 |
Reference:
|
[26] Zayed, E. M. E., El-Moneam, M. A.: On the rational recursive sequence $x_{n+1}=(D+\alpha x_n+\beta x_{n-1}+\gamma x_{n-2})/(Ax_n+Bx_{n-1}+Cx_{n-2})$.Commun. Appl. Nonlin. Anal. 12 (2005), 15-28. MR 2163175 |
Reference:
|
[27] Zayed, E. M. E., El-Moneam, M. A.: On the rational recursive sequence $x_{n+1}=(\alpha x_n+\beta x_{n-1}+\gamma x_{n-2}+\delta x_{n-3})/(Ax_n+Bx_{n-1}+Cx_{n-2}+Dx_{n-3})$.J. Appl. Math. Comput. 22 (2006), 247-262. MR 2248455, 10.1007/BF02896475 |
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