A class of first-order nonlinear difference equations of neutral type

The first-order nonlinear difference equations of neutral type Δ ( x ( n ) − p ( n ) x ( τ ( n ) ) ) + δ ∑ j = 1 m q j ( n ) f j ( x ( σ j ( n ) ) ) = 0 , ( E δ ) where δ = ±1, N(n0) = {n0, n0 + 1, }, p, qj : N(n0) → R+, τ,σj : N(n0) → N, and limn→∞ τ(n) = limn→∞ σj(n) = ∞, ƒj : R → R, j = 1,2,, m, are investigated. All nonoscillatory solutions of these equations are classified according to their asymptotic behavior. Criteria for oscillation of all solutions of these equations are established.