Abstract :
In this paper we are dealing with the oscillatory and asymptotic behaviour of solutions of second order nonlinear difference equations of the form Δ(rnΔxn) + f(n, xn) = 0, n ∈ N(n0). (1) We obtain the following results. (a) If ∑+∞k = n0 (l/rk) < + ∞ any nonoscillatory solution of (1) must belong to one of the following four types: Kβα, K∞α, Kβ0, K∞0. (b) If ∑+∞k = n0 (l/rk) = + ∞ any nonoscillatory solution of (1) must belong to one of the following three types: K0α, Kβ∞, K0∞. (c) Necessary and sufficient conditions for (1) to have a nonoscillatory solution which belongs to Kβα, Kα, Kβ0, K0α, or Kβ∞ are given depending on whether f is a superlinear or sublinear function. All these results include and improve B. Szmanda′s results in Bull. Polish Acad. Sci. Math.34, Nos. 3-4, 1986, 133-141.
