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:
\[
x_{n+1}=f(x_{n-1}, x_n),\quad x_{-1}, x_0>0\quad (n=0,1,2,\cdots).
\]
Reviewer: Dingyong Bai (Guangzhou)
MSC:
39A10 | Additive difference equations |
39A20 | Multiplicative and other generalized difference equations |