Improved oscillation criteria for second order quasilinear dynamic equations of noncanonical type

Abstract

AbstractThe present discussion is to study the following second order nonlinear delay dynamic equation of the form: $$\begin{aligned}{}[r(\theta )(\mathcal {W}^{\Delta }(\theta ))^{\alpha }]^{\Delta } +\mathcal {P}(\theta )\mathcal {W}^{\beta }(\eta (\theta ))=0,\;\theta \in \mathbb {T}_{0}=[\theta _{0},\infty )\cap \mathbb {T} \end{aligned}$$ [ r ( θ ) ( W Δ ( θ ) ) α ] Δ + P ( θ ) W β ( η ( θ ) ) = 0 , θ ∈ T 0 = [ θ 0 , ∞ ) ∩ T under the assumption $$\begin{aligned} \int _{\theta _{0}}^{\theta }r^{-1/\alpha }(s)\Delta s<\infty . \end{aligned}$$ ∫ θ 0 θ r - 1 / α ( s ) Δ s < ∞ . We divide the research into two halves, $$\alpha >\beta $$ α > β and $$\alpha <\beta $$ α < β , and look for some $$\limsup $$ lim sup type conditions that cause all solutions to oscillate. In addition, we extend the Philos-type oscillation criteria. To illustrate the analytical findings, two examples are provided.