ON CONVERGENCE OF A RECURSIVE SEQUENCE $x_{n+1} = f(x_{n-1}, x_n)$

Abstract

C. H. Gibbons, M. R. S. Kulenovic and G. Ladas [1] have posed the following problem: Is there a solution of the difference equation: $$ x_{n+1} = \frac{\beta x_{n-1}}{\beta + x_n}, \quad x_{-1}, x_0 \gt 0, \beta \gt 0 \quad (n = 0, 1, 2, \dots) $$ such that $\displaystyle \lim_{n \to \infty} x_n = 0$? S. Stevic [2] gives an affirmative answer to this open problem and generalize this result to the equation of the form: $$ x_{n+1} = \frac{x_{n-1}}{g(x_n)}, \quad x_{-1}, x_0 \gt 0 \quad (n = 0, 1, 2, \dots) $$ by using his ingenious device. In this note, we generalize the result of Stevic to the equation of the form: $$ x_{n+1} = f(x_{n-1}, x_n), \quad x_{-1}, x_0 \gt 0 \quad (n = 0, 1, 2, \dots). $$ However our proof is simple and short.