Existence of solutions with a single semicycle for a general second-order rational difference equation

By making use of inclusion theorem, we show in this paper the existence of solutions with a single semicycle for a general second-order rational difference equation. As a corollary, our results positively confirm Conjectures 4.8.3 and 5.4.6 in [M.R.S. Kulenovic, G. Ladas, Dynamics of Second-Order Rational Difference Equations, with Open Problems and Conjectures, Chapman and Hall/CRC, 2002].