Oscillation criteria for second-order functional difference equations with “summation small” coefficient

We consider the second-order functional difference equation Δ(a(n)Δ(n)) + p(n)x(g(n))=0, where n), p(n) are sequences of integers with a(n) > 0, 0 ≤ g(n) ≤ n, limn→∞g(n) = ∞, p(n) ≥ 0, and p(n) 0. We obtain some necessary conditions for equation (1) exists nonoscillatory solutions and some sufficient conditions for equation (1) is oscillatory.