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On convergence of a recursive sequence \(x_{n+1}= f(x_{n-1},x_n)\). (English) Zbl 1100.39001

C. H. Gibbons, M. R. S. Kulenovic and G. Ladas [Math. Sci. Res. Hot-Line, 4, 1–11 (2000; Zbl 1039.39004)] have posed the following problem: Is there a solution of the difference equation: \[ x_{n+1}=\frac{\beta x_{n-1}}{\beta+x_n},\quad x_{-1}, x_0>0, \beta>0\quad (n=0,1,2,\dots) \] such that \(\lim_{n\to \infty}x_n=0\)? S. Stevic [Taiwanese J. Math. 6, No. 3, 405–414 (2006; Zbl 1019.39010)] gives an affirmative answer to this open problem and generalizes this result to the equation of the form: \[ x_{n+1}=\frac{x_{n-1}}{g(x_n)},\quad x_{-1}, x_0>0\quad (n=0,1,2,\cdots) \] by using his ingenious device. In this note, the authors generalize the result of Stevic to the equation of the form: \[ x_{n+1}=f(x_{n-1}, x_n),\quad x_{-1}, x_0>0\quad (n=0,1,2,\cdots). \]

MSC:

39A10 Additive difference equations
39A20 Multiplicative and other generalized difference equations
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