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In this paper the nonlinear second-order neutral difference equation of the following form: \(\Delta ( a_{n}\Delta(x_{n}-p_{n}x_{n-1}) ) + q_{n}f(x_{n-\tau})=0\) is considered. By suitable substitution the above equation is transformed into a new one, which is a third-order non-neutral difference equation. Using results obtained for the new equation, the asymptotic properties of the neutral difference equation are studied. Some classification of nonoscillatory solutions is presented, as well as an estimation of the solutions. Finally, we present necessary and sufficient conditions for the existence of solutions to both considered equations being asymptotically equivalent to the given sequences.

Trichotomy of nonoscillatory solutions to second-order neutral difference equation with quasi-difference