Classifying non-oscillatory solutions and oscillation of a neutral difference equation

The authors examine the oscillatory and non-oscillatory behavior of solutions of a class of second order difference equations of neutral type that includes half-linear equations as a special case. They classify the possible non-oscillatory solutions according to their asymptotic behavior and then prove the existence of solutions of each type using fixed point techniques. Criteria for almost oscillation, i.e., solutions either oscillate or converge to zero, are also proved. The results are illustrated with examples.