On the existence of unbounded solutions for some rational equations (Q2390648)

The author first broadens the proof for unbounded solutions of a \(k\)-th rational difference equation and generalizes the previous work due to \textit{F. J. Palladino} [J. Difference Equ. Appl. 15, No.~3, 253--260 (2009; Zbl 1169.39005)]. Then, based on this proof, he resolves several conjectures on the boundedness character of solutions of a 4th-order rational difference equation proposed by \textit{E. Camouzis} and \textit{G. Ladas} [Dynamics of third-order rational difference equations with open problems and conjectures. Boca Raton, FL: Chapman \& Hall (2008; Zbl 1129.39002)]. Finally, the author partially resolves the conjecture proposed by Palladino [loc. cit.] and shows that unbounded solutions of the special 4th-order rational difference equation \(x_n=\frac{x_{n-3}}{Bx_{n-1}+x_n-4}\) exist for some choice of the nonnegative initial conditions provided that \(B>2^5\).