On Asymptotic Behavior of Solutions of Generalized Emden-Fowler Differential Equations with Delay Argument

Abstract

The following differential equation $u^{(n)}(t)+p(t)|u(\sigma (t)){|}^{\mu (t)}\text{\hspace{0.17em}sign}u(\sigma (t))=\mathrm{0}$ is considered. Here $p\in L_{\mathrm{\text{loc}}}(R_{+};R_{+})$, $\mu \in C(R_{+};(\mathrm{0},+\mathrm{\infty }))$, $\sigma \in C(R_{+};R_{+})$, $\sigma (t)\le t$, and ${\lim }_{t\to +\mathrm{\infty }}\sigma (t)=+\mathrm{\infty }$. We say that the equation is almost linear if the condition ${\lim }_{t\to +\mathrm{\infty }}\mu (t)=\mathrm{1}$ is fulfilled, while if ${\lim \sup }_{t\to +\mathrm{\infty }}\mu (t)\ne \mathrm{1}$ or ${\lim \inf }_{t\to +\mathrm{\infty }}\mu (t)\ne \mathrm{1}$, then the equation is an essentially nonlinear differential equation. In the case of almost linear and essentially nonlinear differential equations with advanced argument, oscillatory properties have been extensively studied, but there are no results on delay equations of this sort. In this paper, new sufficient conditions implying Property A for delay Emden-Fowler equations are obtained.