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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+φ n (y n-1 ,y n-2 ,,y n-m )+y n-s ), n 0 , with p0, r,q>0, s,m and positive initial values, and present some sufficient conditions for the parameters and maps φ n :( + ) m + , 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:
39A30Stability theory (difference equations)
39A20Generalized difference equations
39A22Growth, boundedness, comparison of solutions (difference equations)