On oscillation and asymptotic behaviour of a neutral differential equation of first order with positive and negative coefficients

Abstract

In this paper sufficient conditions are obtained so that every solution of (y(t) - p(t)y(t - τ))′ + Q(t)G(y(t - σ)) - U(t)G(y(t - α)) = ƒ(t) tends to zero or to ±∞ as t tends to ∞, where τ, σ, α are positive real numbers, p, ƒ ∈ C([0, ∞), R), Q, U ∈ C([0, ∞), [0, ∞)), and G ∈ C(R, R), G is non decreasing with xG(x) > 0 for x ≠ 0. The two primary assumptions in this paper ∫^∞_t0 Q(t) = ∞ and ∫∞t0 U(t) < ∞. The results hold when G is linear, super linear, or sublinear and also hold when ƒ(t) ≡ 0. This paper generalizes and improves some of the recent results in [5, 7, 8, 10].