On the recursive sequence \(x_{n+1}=\frac A{x_n}+\frac 1{x_{n-2}}\). (English) Zbl 0904.39012

DeVault, R.; Ladas, G.; Schultz, S. W.

Proc. Am. Math. Soc. 126, No. 11, 3257-3261 (1998).

The authors establish that every positive solution of the equation \[ x_{n+1}= \frac{A}{x_n}+ {1}{x_{n-2}},\quad n= 0,1,\ldots, \] where \(x_{-1}\),\(x_{-2}\), \(A\in(0,\infty)\), converges to a period two solution. This proves Conjecture 2.4.2 of G. Ladas [J. Differ. Equ. Appl. 2, 449–452 (1996; 10.1080/10236199608808079)].

Reviewer: E.Thandapani (Salem)

MSC:

39A12	Discrete version of topics in analysis
39A10	Additive difference equations
39A30	Stability theory for difference equations

Keywords:

recursive sequence; global asymptotic stability; period two solution; positive solution