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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.

MSC:
39A10 Additive difference equations
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