Oscillatory and Asymptotic Behavior of Nonlinear Functional Dynamic Equations of Third Order

Abstract

The purpose of this work is to derive sufficient conditions for the oscillation of all solutions of the third-order functional dynamic equation ${\left\{p_2\left(\xi \right){\varphi }_{{\gamma }_2}\left({\left[p_1\left(\xi \right){\varphi }_{{\gamma }_1}\left(y^{\Delta }\left(\xi \right)\right)\right]}^{\Delta }\right)\right\}}^{\Delta }+p\left(\xi \right){\varphi }_{\beta }\left(y\left(g\left(\xi \right)\right)\right)=0,$ on a time scale $\mathbb{T}$ . In addition, we present some Hille-type conditions for generalized third-order dynamic equations that improve and extend significant contributions reported in the literature without imposing time-scale restrictions. An example is given to demonstrate the essential results.