The main results of the paper are the following two theorems.
Theorem 1. Every positive solution to the difference equation
where , , , , and , converges to one.
Theorem 2. Assume that , , , , and , , , . Then every positive solution to
converges to .
Moreover, Theorem 2 solves the conjecture proposed by the author of this paper [Appl. Math. Comput. 210, No. 2, 525–529 (2009; Zbl 1167.39007)] and by F. Sun [Discrete Dyn. Nat. Soc. 2008, Article ID 243291, 6 p. (2008; Zbl 1155.39008)].