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(r_nΔx_n) + f(n, x_n) = 0, n ∈ N(n₀). (1) We obtain the following results. (a) If ∑^+∞_{k = n0} (l/r_k) < + ∞ any nonoscillatory solution of (1) must belong to one of the following four types: K^β_α, K^∞_α, K^β₀, K^∞₀. (b) If ∑^+∞_{k = n0} (l/r_k) = + ∞ any nonoscillatory solution of (1) must belong to one of the following three types: K⁰_α, K^β_∞, K⁰_∞. (c) Necessary and sufficient conditions for (1) to have a nonoscillatory solution which belongs to K^β_α, K_α, K^β₀, K⁰_α, or K^β_∞ 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.