Oscillation criteria for neutral half-linear differential equations without commutativity in deviating arguments

Abstract

We study the half-linear neutral differential equation \begin{equation*} \Bigl[r(t)\Phi(z'(t))\Bigr]'+c(t)\Phi(x(\sigma(t)))=0, \qquad z(t)=x(t)+b(t)x(\tau(t)), \end{equation*} where $\Phi(t)=|t|^{p-2}t$. We present new oscillation criteria for this equation in case when $\sigma(\tau(t))\neq \tau(\sigma(t))$ and $\int^\infty {r^{1-q}}(t)\mathrm{d}t<\infty$, $q=p/(p-1)$, $p\geq 2$ is a real number. The results of this paper complement our previous results in case when the above integral is divergent and/or the deviations $\tau$, $\sigma$ commute with respect to their composition.

Keywords

• half-linear differential equation
• delay equation
• oscillation criteria
• neutral equation