On Higher-Order Generalized Emden-Fowler Differential Equations with Delay Argument

Abstract

We consider a differential equation

\( {u}^{(n)}(t)+p(t){\left|u\left(\tau (t)\right)\right|}^{\mu (t)}\mathrm{sign}\left(\tau (t)\right)=0. \)

It is assumed that n ≥ 3, p ∈ L_loc(R₊;R₋), μ ∈ C(R₊;(0,+∞)), τ ∈ C(R₊;R₊), τ(t) ≤ t for t ∈ R₊ and lim_t→+∞τ(t) = +∞. In the case μ(t) ≡ const > 0, the oscillatory properties of equation (*) are extensively studied, whereas for μ(t) ≢ const, to the best of authors’ knowledge, problems of this kind were not investigated at all. We also establish new sufficient conditions for the equation (*) to have Property B.