Asymptotics of some classes of higher-order difference equations

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. Stevic (2005). Monotonous solutions of the equation y(n) = A + (y(n-k)/Sigma(m)(j=1)beta(j)y(n-qj))p, n epsilon N-0, where p, A epsilon (0,infinity), k, m epsilon N, qj, j epsilon {1,..., m}, are natural numbers such that q(1) < q(2) < center dot center dot center dot < q(m), beta(j) epsilon (0,+infinity), j epsilon {1,..., m}, Sigma(m)(j=1)beta(j) = 1, and y(-s), y(-s+1),..., y(-1) epsilon (0,infinity), where s = max{k, q(m)}, are found. A new inclusion theorem is proved. Also, some open problems and conjectures are posed.