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

Abstract

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(n 0) = {n 0, n 0 + 1, }, p, q j : N(n 0) → R +, τ,σ j : N(n 0) → N , and lim n→∞ τ( n) = lim n→∞ σ j(n) = ∞, ƒ j : 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.