Global asymptotic properties of third-order difference equations

The third-order nonlinear difference equations Δ(pnΔ(rnΔxn))+qnf(xn+p)=0,p {0,1,2},(Ep) where (Pn), (rn), and (qn) are sequences of positive real numbers for n ε , ƒ : → is a continuous function such that ƒ(u)u > 0 for u ≠ 0, are investigated. All nonoscillatory solutions of these equations are classified according to the sign of their quasidifferences to classes Ni, i = 0, 1, 2, 3, and sufficient conditions ensuring Ni = , i ε {1,2,3} are given. Special attention is paid to equation (E1) for which the generalized zeros of solutions are studied and an energy function F is introduced. The relation between the class N0 and solutions for which Fn < 0 for n ε is established.