Necessary and sufficient conditions for oscillation of first-order nonlinear neutral difference equations

Abstract

In this paper, we prove that every solution of the first-order nonlinear neutral difference equation △ ( x n − p x n − τ ) + q n ∏ j = 1 m | x n − σ j | β j sign ( x n − σ 1 ) = 0 , n ≥ n 0 oscillates if and only if ∑ s = n 0 ∞ q s exp [ τ − 1 ln p ( ∑ j = 1 m β j − 1 ) s ] = ∞ , when ( ∑ j = 1 m β j − 1 ) ln p < 0 , and ∑ s = n 0 ∞ q s = ∞ , when ( ∑ j = 1 m β j − 1 ) ln p > 0 , where p , β j > 0 , τ > 0 and σ j ≥ 0 are integers, j = 1 , 2 , … , m , q n ≥ 0 , n ≥ 0 .