## Asymptotics of some classes of higher-order difference equations.(English)Zbl 1180.39009

Summary: We present some methods for finding asymptotics of some classes of nonlinear higher-order difference equations. Among others, we confirm a conjecture posed by S. Stević [Rostocker Math. Kolloq. 59, 3–10 (2005; Zbl 1083.39011)]. Monotonous solutions of the equation $$y_n= A+(y_{n-k}/\sum_{j=1}^m \beta_jy_{n-q_j})^p$$, $$n\in\mathbb N_0$$, where $$p,A\in(0,\infty)$$, $$k,m\in\mathbb N$$, $$q_j$$, $$j\in\{1,\dots,m\}$$, are natural numbers such that $$q_1<q_2<\dots<q_m$$, $$\beta_j\in(0,+\infty)$$, $$j\in \{1,\dots,m\}$$, $$\sum_{j=1}^m \beta_j=1$$, and $$y_{-s},y_{-s+1},\dots,y_{-1}\in(0,\infty)$$, where $$s= \max\{k,q_m\}$$, are found. A new inclusion theorem is proved. Also, some open problems and conjectures are posed.