Oscillatory and Asymptotic Behavior of Second Order Nonlinear Difference Equations

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.