## On monotone solutions of some classes of difference equations.(English)Zbl 1109.39013

Assume that $$\alpha_{i} \geq 0$$ $$i = 0,\ldots,k-1,$$ $$\sum_{i=0}^{k-1}\alpha_{i} = 1.$$ The main result of this paper is the proof that
1) if $$p > -1,$$ the equation
$x_{n+1} = p + \frac{x_{n-k}}{\sum_{i=0}^{k-1}\alpha_{i}x_{n-i}} \quad (n = 0,1,\ldots)$
has a positive solution which remains above the equilibrium $$\overline x_{1} = p+1$$;
2) the equation $x_{n+1} = \frac{1+x_{n-k}}{\sum_{i=0}^{k-1}\alpha_{i}x_{n-i}} \quad (n = 0,1,\ldots)$ has a nontrivial positive solution which decreases to the equilibrium $$\overline x_{2} = (1 + \sqrt 5)/2;$$
3) if $$\alpha > 0,$$ the equation $x_{n+1} =\frac{\alpha + x_{n-k}}{\sum_{i=0}^{k-1}\alpha_{i}x_{n-i}} \quad (n = 0,1,\ldots)$ has a nontrivial positive solution which decreases to the equilibrium $$\overline x_{3} = \sqrt \alpha$$.
This solves positively some earlier conjectures.

### MSC:

 39A11 Stability of difference equations (MSC2000) 39A20 Multiplicative and other generalized difference equations

### Keywords:

rational difference equations; positive solution
Full Text:

### References:

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