On Delay Differential Inequalities of Higher Order

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Abstract
Consider the nth order (n ≥ 1) delay differential inequalities and and the delay differential equation , where q(t) ≥ 0 is a continuous function and p, τ are positive constants. Under the condition pτe ≥ 1 we prove that when n is odd (1) has no eventually positive solutions, (2) has no eventually negative solutions, and (3) has only oscillatory solutions and when n is even (1) has no eventually negative bounded solutions, (2) has no eventually positive bounded solutions, and every bounded solution of (3) is oscillatory. The condition pτe > 1 is sharp. The above results, which generalize previous results by Ladas and by Ladas and Stavroulakis for first order delay differential inequalities, are caused by the retarded argument and do not hold when τ = 0.