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Asymptotics of some classes of higher-order difference equations. (English) Zbl 1180.39009
Summary: We present some methods for finding asymptotics of some classes of nonlinear higher-order difference equations. Among others, we confirm a conjecture posed by S. Stević [Rostocker Math. Kolloq. 59, 3–10 (2005; Zbl 1083.39011)]. Monotonous solutions of the equation $$y_n= A+(y_{n-k}/\sum_{j=1}^m \beta_jy_{n-q_j})^p$$, $$n\in\mathbb N_0$$, where $$p,A\in(0,\infty)$$, $$k,m\in\mathbb N$$, $$q_j$$, $$j\in\{1,\dots,m\}$$, are natural numbers such that $$q_1<q_2<\dots<q_m$$, $$\beta_j\in(0,+\infty)$$, $$j\in \{1,\dots,m\}$$, $$\sum_{j=1}^m \beta_j=1$$, and $$y_{-s},y_{-s+1},\dots,y_{-1}\in(0,\infty)$$, where $$s= \max\{k,q_m\}$$, are found. A new inclusion theorem is proved. Also, some open problems and conjectures are posed.

##### MSC:
 39A10 Additive difference equations
Full Text:
##### References:
 [1] K. S. Berenhaut, J. D. Foley, and S. Stević, “The global attractivity of the rational difference equation yn=1+yn - k/yn - m,” Proceedings of the American Mathematical Society, vol. 135, no. 4, pp. 1133-1140, 2007. · Zbl 1109.39004 · doi:10.1090/S0002-9939-06-08580-7 [2] K. S. Berenhaut and S. Stević, “A note on the difference equation xn+1=1/(xnxn - 1)+1/(xn - 3xn - 4),” Journal of Difference Equations and Applications, vol. 11, no. 14, pp. 1225-1228, 2005. · Zbl 1088.39017 · doi:10.1080/10236190500331370 [3] K. S. Berenhaut and S. Stević, “The behaviour of the positive solutions of the difference equation xn=A+(xn - 2/xn - 1)p,” Journal of Difference Equations and Applications, vol. 12, no. 9, pp. 909-918, 2006. · Zbl 1111.39003 · doi:10.1080/10236190600836377 [4] K. S. Berenhaut and S. Stević, “The difference equation xn+1=\alpha +(xn - k/(\sum i=0k - 1cixn - i)) has solutions converging to zero,” Journal of Mathematical Analysis and Applications, vol. 326, no. 2, pp. 1466-1471, 2007. · Zbl 1113.39003 · doi:10.1016/j.jmaa.2006.02.088 [5] L. Berg, Asymptotische Darstellungen und Entwicklungen, Hochschulbücher für Mathematik, Band 66, VEB Deutscher Verlag der Wissenschaften, Berlin, Germany, 1968. · Zbl 0165.36901 [6] L. Berg, “On the asymptotics of nonlinear difference equations,” Zeitschrift für Analysis und ihre Anwendungen, vol. 21, no. 4, pp. 1061-1074, 2002. · Zbl 1030.39006 · doi:10.4171/ZAA/1127 [7] L. Berg, “Inclusion theorems for non-linear difference equations with applications,” Journal of Difference Equations and Applications, vol. 10, no. 4, pp. 399-408, 2004. · Zbl 1056.39003 · doi:10.1080/10236190310001625280 [8] L. Berg, “Oscillating solutions of rational difference equations,” Rostocker Mathematisches Kolloquium, no. 58, pp. 31-35, 2004. · Zbl 1096.39002 · ftp.math.uni-rostock.de [9] L. Berg, “Corrections to: “Inclusion theorems for non-linear difference equations with applications”,” Journal of Difference Equations and Applications, vol. 11, no. 2, pp. 181-182, 2005. · Zbl 1080.39002 · doi:10.1080/10236190512331328370 [10] L. Berg and L. von Wolfersdorf, “On a class of generalized autoconvolution equations of the third kind,” Zeitschrift für Analysis und ihre Anwendungen, vol. 24, no. 2, pp. 217-250, 2005. · Zbl 1104.45001 [11] R. DeVault, C. Kent, and W. Kosmala, “On the recursive sequence xn+1=p+(xn - k/xn),” Journal of Difference Equations and Applications, vol. 9, no. 8, pp. 721-730, 2003. · Zbl 1049.39026 · doi:10.1080/1023619021000042162 [12] H. M. El-Owaidy, A. M. Ahmed, and M. S. Mousa, “On asymptotic behaviour of the difference equation xn+1=\alpha +(xn - 1p/xnp),” Journal of Applied Mathematics & Computing, vol. 12, no. 1-2, pp. 31-37, 2003. · Zbl 1052.39005 · doi:10.1007/BF02936179 [13] G. Karakostas, “Convergence of a difference equation via the full limiting sequences method,” Differential Equations and Dynamical Systems, vol. 1, no. 4, pp. 289-294, 1993. · Zbl 0868.39002 [14] G. Karakostas, “Asymptotic 2-periodic difference equations with diagonally self-invertible responses,” Journal of Difference Equations and Applications, vol. 6, no. 3, pp. 329-335, 2000. · Zbl 0963.39020 · doi:10.1080/10236190008808232 [15] V. L. Kocić and G. Ladas, Global Behavior of Nonlinear Difference Equations of Higher Order with Applications, vol. 256 of Mathematics and Its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1993. · Zbl 0787.39001 [16] W. Kosmala and C. Teixeira, “More on the difference equation yn+1=(p+yn)/(qyn+yn - 1),” Applicable Analysis, vol. 81, no. 1, pp. 143-151, 2002. · Zbl 1022.39005 · doi:10.1080/0003681021000021114 [17] N. Kruse and T. Nesemann, “Global asymptotic stability in some discrete dynamical systems,” Journal of Mathematical Analysis and Applications, vol. 235, no. 1, pp. 151-158, 1999. · Zbl 0933.37016 · doi:10.1006/jmaa.1999.6384 [18] H. Levy and F. Lessman, Finite Difference Equations, Dover, New York, NY, USA, 1992. · Zbl 0092.07702 [19] G. Ladas, “Open problems and conjectures,” Journal of Difference Equations and Applications, vol. 4, pp. 497-499, 1998. [20] “Putnam exam,” American Mathematical Monthly, pp. 734-736, 1965. [21] S. Stević, “On stability results for a new approximating fixed points iteration process,” Demonstratio Mathematica, vol. 34, no. 4, pp. 873-880, 2001. · Zbl 1011.47050 [22] S. Stević, “Asymptotic behavior of a sequence defined by iteration with applications,” Colloquium Mathematicum, vol. 93, no. 2, pp. 267-276, 2002. · Zbl 1029.39006 · doi:10.4064/cm93-2-6 [23] S. Stević, “A global convergence results with applications to periodic solutions,” Indian Journal of Pure and Applied Mathematics, vol. 33, no. 1, pp. 45-53, 2002. · Zbl 1002.39004 [24] S. Stević, “Asymptotic behavior of a nonlinear difference equation,” Indian Journal of Pure and Applied Mathematics, vol. 34, no. 12, pp. 1681-1687, 2003. · Zbl 1049.39012 [25] S. Stević, “On the recursive sequence xn+1=A/(\prod i=0kxn - i)+1/(\prod j=k+22(k+1)xn - j),” Taiwanese Journal of Mathematics, vol. 7, no. 2, pp. 249-259, 2003. · Zbl 1054.39008 [26] S. Stević, “A note on periodic character of a difference equation,” Journal of Difference Equations and Applications, vol. 10, no. 10, pp. 929-932, 2004. · Zbl 1057.39005 · doi:10.1080/10236190412331272616 [27] S. Stević, “Periodic character of a difference equation,” Rostocker Mathematisches Kolloquium, no. 59, pp. 3-10, 2005. · Zbl 1083.39011 [28] S. Stević, “On the recursive sequence xn+1=\alpha +(xn - 1p/xnp),” Journal of Applied Mathematics & Computing, vol. 18, no. 1-2, pp. 229-234, 2005. · Zbl 1078.39013 · doi:10.1007/BF02936567 [29] S. Stević, “On the recursive sequence xn+1=(\alpha +\beta xn - k/f(xn,\cdots ,xn - k+1)),” Taiwanese Journal of Mathematics, vol. 9, no. 4, pp. 583-593, 2005. · Zbl 1100.39014 [30] S. Stević, “Global stability and asymptotics of some classes of rational difference equations,” Journal of Mathematical Analysis and Applications, vol. 316, no. 1, pp. 60-68, 2006. · Zbl 1090.39009 · doi:10.1016/j.jmaa.2005.04.077 [31] S. Stević, “On monotone solutions of some classes of difference equations,” Discrete Dynamics in Nature and Society, vol. 2006, Article ID 53890, p. 9, 2006. · Zbl 1109.39013 · doi:10.1155/DDNS/2006/53890 · eudml:127142 [32] S. Stević, “On positive solutions of a (k+1)th order difference equation,” Applied Mathematics Letters, vol. 19, no. 5, pp. 427-431, 2006. · Zbl 1095.39010 · doi:10.1016/j.aml.2005.05.014 [33] S. Stević, “Existence of nontrivial solutions of a rational difference equation,” Applied Mathematics Letters, vol. 20, no. 1, pp. 28-31, 2007. · Zbl 1131.39009 · doi:10.1016/j.aml.2006.03.002 [34] S. Stević, “On the recursive sequence xn+1=A+(xnp/xn - 1p),” Discrete Dynamics in Nature and Society, vol. 2007, Article ID 34517, p. 9, 2007. · Zbl 1180.39007 · doi:10.1155/2007/34517 [35] S. Stević, “Nontrivial solutions of a higher-order rational difference equation,” submitted. · Zbl 1219.39007 · doi:10.1134/S0001434608110138 [36] T. Sun and H. Xi, “Global asymptotic stability of a family of difference equations,” Journal of Mathematical Analysis and Applications, vol. 309, no. 2, pp. 724-728, 2005. · Zbl 1080.39019 · doi:10.1016/j.jmaa.2004.11.040 [37] T. Sun, H. Xi, and H. Wu, “On boundedness of the solutions of the difference equation xn+1=xn - 1/(p+xn),” Discrete Dynamics in Nature and Society, vol. 2006, Article ID 20652, p. 7, 2006. · Zbl 1149.39301 · doi:10.1155/DDNS/2006/20652 [38] S.-E. Takahasi, Y. Miura, and T. Miura, “On convergence of a recursive sequence xn+1=f(xn-1,xn),” Taiwanese Journal of Mathematics, vol. 10, no. 3, pp. 631-638, 2006. · Zbl 1100.39001 [39] X. Yang, “Global asymptotic stability in a class of generalized Putnam equations,” Journal of Mathematical Analysis and Applications, vol. 322, no. 2, pp. 693-698, 2006. · Zbl 1104.39012 · doi:10.1016/j.jmaa.2005.09.049 [40] X.-X. Yan, W.-T. Li, and Z. Zhao, “On the recursive sequence xn+1=\alpha - (xn/xn - 1),” Journal of Applied Mathematics & Computing, vol. 17, no. 1-2, pp. 269-282, 2005. · Zbl 1068.39030 · doi:10.1007/BF02936054
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