## Stability of the recursive sequence $$x_{n+1}=(\alpha-\beta x_n)/(\gamma+x_{n-1})$$.(English)Zbl 0990.39009

Consider the recursive sequence $x_{n+1}= {\alpha+ \beta x_n \over \gamma+x_{n-1}},\;n=0,1,\dots \tag{*}$ where $$\alpha,\beta$$ and $$\gamma$$ are nonnegative and the initial conditions $$x_1$$ and $$x_0$$ are arbitrary. Equation (*) has two equilibrium points positive and negative.
If there exists $$k\geq 2$$ such that $$\gamma\geq k\alpha/ \beta$$ and $$\alpha\geq k\beta^2$$, then the positive equilibrium point is a global attractor with some given basin. The asymptotic properties in the case $$\alpha=0$$, $$\beta<0$$, $$\gamma >0$$ are investigated in details.

### MSC:

 39A11 Stability of difference equations (MSC2000) 39B05 General theory of functional equations and inequalities
Full Text:

### References:

 [1] Feuer, J.; Janowski, E.J.; Ladas, G., Lyness-type equations in the third quadrant, Nonlinear anal., 30, 1183-1189, (1997) · Zbl 0893.39004 [2] Kocic, V.L.; Ladas, G., Global attractivity in a second-order nonlinear difference equation, J. math. anal. appl., 180, 144-150, (1993) · Zbl 0802.39001 [3] Kocic, V.L.; Ladas, G., Global behavior of nonlinear difference equations of higher order with applications, (1993), Kluwer Academic Dordrecht · Zbl 0787.39001 [4] Kocic, V.L.; Ladas, G., Permanence and global attractivity in nonlinear difference equations, World congress of nonlinear analysis, tampa, florida, I-IV, 1992, (1996), de Gruyter Berlin, p. 1161-1172 · Zbl 0843.39010 [5] Kocic, V.L.; Ladas, G.; Rodrigues, I.W., On rational recursive sequences, J. math. anal. appl., 173, 127-157, (1993) · Zbl 0777.39002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.