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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 $y_n=(p+ry_{n-s})/(q+\phi_n(y_{n-1},y_{n-2},\dots,y_{n-m})+y_{n-s})$, $n\in\Bbb N_0$, with $p\ge 0$, $r,q>0$, $s,m\in\Bbb N$ and positive initial values, and present some sufficient conditions for the parameters and maps $\phi_n:(\Bbb R^+)^m\to\Bbb R^+$, $n\in\Bbb N_0$, 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 {\it C. H. Gibbons, M. R. S. Kulenovic}, and {\it G. Ladas} [Math. Sci. Res. Hot-Line 4, No. 2, 1--11(2000; Zbl 1039.39004)], {\it B. D. Iričanin} [Discrete Dyn. Nat. Soc. 2007, Article ID 73849 (2007; Zbl 1152.39005)], and {\it 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:
39A30Stability theory (difference equations)
39A20Generalized difference equations
39A22Growth, boundedness, comparison of solutions (difference equations)
WorldCat.org
Full Text: DOI EuDML
References:
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