Asymptotic properties of fourth-order nonlinear difference equations

We consider a class of fourth-order nonlinear difference equations of the form Δ ( a n Δ ( b n Δ ( c n Δ y n ) ) ) + f ( n , y n ) = 0 , where a, b, c : N → ℛ + and (an) is bounded away from zero, an ≥ cn, the series Σi=1∞ (1/ai) is convergent and Σj=1∞ (1/cj) Σi=j∞ (1/bi) is divergent. The classification of nonoscillatory solutions is given. Sufficient conditions for the above equation are obtained to admit the existence of nonoscillatory solutions with special asymptotic properties.