On Delay Differential Inequalities of Higher Order

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.

Oscillations of higher order neutral differential equations

“…This implies that l/n /I \ ' [10] and, since a = a n , the last inequality, in view of (18) It is easy to see that w(t) is a solution of Equation (1) which belongs to Class (19) Class II 2n and that …”

Oscillations of certain functional differential equations

“…From the results in [5], we see that all bounded solutions of (E * ) are oscillatory if the following condition holds:…”

Necessary and sufficient conditions for oscillations of delay differential inequalities

“…We mention in particular the papers by Koplatadze and Chanturia [1], Ladas [2], and Ladas and Stavroulakis [3]. for a survey of results concerned with oscillations we refer to Zhang [5] and the references therein.…”