Necessary and sufficient conditions for oscillation of a neutral differential equations with positive and negative coefficients

Abstract

In this paper, necessary and sufficient conditions have been obtained for every solutin of [y(t) − py(t − τ)] 0 + Q(t)G(y(t − σ)) − P(t)H(y(t − α)) = f(t) (NH) under the condition 0 ≤ p(t) < 1 or p < 0 and p 6= −1 with the assumption Z0∞ f(t)dt < ∞ is oscillatory or tends to zero as t → ∞. Moreover, necessary and sufficient conditions have been obtained for every bounded solution of (NH) is oscillatory or tends to zero as t → ∞.