Asymptotics of some classes of higher-order difference equations

Abstract

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 (2005). Monotonous solutions of the equation yn=A+(yn−k/∑j=1mβjyn−qj)p, n∈ℕ0, where p,A∈(0,∞), k,m∈ℕ, qj,j∈{1,…,m}, are natural numbers such that q1<q2<⋯<qm,βj∈(0,+∞), j∈{1,…,m}, ∑j=1mβj=1, and y−s,y−s+1,…,y−1∈(0,∞), where s=max{k,qm}, are found. A new inclusion theorem is proved. Also, some open problems and conjectures are posed.