Non-oscillatory behaviour of higher order functional differential equations of neutral type

Abstract

In this paper, we obtain sufficient conditions so that the neutral functional differential equation [r(t)[y(t) - p(t)y(τ(t))]′](n-1) + q(t)G(y(h(t))) = ƒ(t) has a bounded and positive solution. Here n ≥ 2; q, τ, h are continuous functions with q(t) ≥ 0; h(t) and τ(t) are increasing functions which are less than t, and approach infinity as t → ∞. In our work, r(t) ≡ 1 is admissible, and neither we assume that G is non-decreasing, that xG(x) > 0 for x ≠ 0, nor that G is Lipschitzian. Hence the results of this paper generalize many results in [1] and [4]-[8].