Oscillation of third-order quasilinear neutral dynamic equations on time scales with distributed deviating arguments


Authors

M. Tamer Senel - Department of Mathematics, Faculty of Sciences, Erciyes University, 38039, Kayseri, Turkey. Nadide Utku - Institute of Sciences, Erciyes University, 38039, Kayseri, Turkey.


Abstract

The aim of this paper is to give oscillation criteria for the third-order quasilinear neutral delay dynamic equation \begin{equation*} \bigg[r(t)\big([x(t)+p(t)x(\tau_{0}(t))]^{\Delta\Delta}\big)^{\gamma}\bigg]^{\Delta}+\int_{c}^{d}q_{1}(t)x^{\alpha}(\tau_{1}(t,\xi))\Delta\xi+\int_{c}^{d}q_{2}(t)x^{\beta}(\tau_{2}(t,\xi))\Delta\xi=0, \end{equation*} on a time scale \(\mathbb{T}\), where \(0<\alpha<\gamma<\beta\). By using a generalized Riccati transformation and integral averaging technique, we establish some new sufficient conditions which ensure that every solution of this equation oscillates or converges to zero.


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