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