Necessary conditions for the existence of positive solutions of second-order linear difference equations and sufficient conditions for the oscillation of solutions

We consider the difference equation where m ∈ N , the functions p l : N → R + and τ l : N → N , lim k→ + ∞ τ l ( k ) = + ∞ , l = 1, …, m, are defined on the set of natural numbers, and the difference operator is defined as follows: Δ u ( k ) = u ( k + 1) − u ( k ) , Δ 2 = Δ ◦ Δ . We establish necessary conditions for the above equation to have a positive solution. We also obtain oscillation criteria of a new type that generalize some earlier known results.