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On global attractivity of a class of nonautonomous difference equations. (English) Zbl 1203.39010

Summary: We mainly investigate the global behavior to the family of higher-order nonautonomous recursive equations given by \(y_n=(p+ry_{n-s})/(q+\phi_n(y_{n-1},y_{n-2},\dots,y_{n-m})+y_{n-s})\), \(n\in\mathbb N_0\), with \(p\geq 0\), \(r,q>0\), \(s,m\in\mathbb N\) and positive initial values, and present some sufficient conditions for the parameters and maps \(\phi_n:(\mathbb R^+)^m\to\mathbb R^+\), \(n\in\mathbb N_0\), under which every positive solution to the equation converges to zero or a unique positive equilibrium. Our main result in the paper extends some related results from the work of C. H. Gibbons, M. R. S. Kulenovic, and G. Ladas [Math. Sci. Res. Hot-Line 4, No. 2, 1–11(2000; Zbl 1039.39004)], B. D. Iričanin [Discrete Dyn. Nat. Soc. 2007, Article ID 73849 (2007; Zbl 1152.39005)], and S. Stević [Indian J. Pure Appl. Math. 33, No. 12, 1767–1774 (2002; Zbl 1019.39011); Taiwanese J. Math. 6, No. 3, 405–414 (2002; Zbl 1019.39010); ibid. 9, No. 4, 583–593 (2005; Zbl 1100.39014)]. Besides, several examples and open problems are presented in the end.

MSC:

39A30 Stability theory for difference equations
39A20 Multiplicative and other generalized difference equations
39A22 Growth, boundedness, comparison of solutions to difference equations
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