#### Vol. 2, No. 2, 2009

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On the existence of unbounded solutions for some rational equations

### Gabriel Lugo

Vol. 2 (2009), No. 2, 237–247
##### Abstract

We resolve several conjectures regarding the boundedness character of the rational difference equation

${x}_{n}=\frac{\alpha +\delta {x}_{n-3}}{A+B{x}_{n-1}+C{x}_{n-2}+E{x}_{n-4}},\phantom{\rule{1em}{0ex}}n\in ℕ.$

We show that whenever parameters are nonnegative, $A<\delta$, and $C,E>0$, unbounded solutions exist for some choice of nonnegative initial conditions. We also partly resolve a conjecture regarding the boundedness character of the rational difference equation

${x}_{n}=\frac{{x}_{n-3}}{B{x}_{n-1}+{x}_{n-4}},\phantom{\rule{1em}{0ex}}n\in ℕ.$

We show that whenever $B>{2}^{5}$, unbounded solutions exist for some choice of nonnegative initial conditions.

##### Keywords
difference equation, periodic convergence, boundedness character, unbounded solutions, periodic behavior of solutions of rational difference equations, nonlinear difference equations of order greater than one, global asymptotic stability
##### Mathematical Subject Classification 2000
Primary: 39A10, 39A11