Oscillation criteria for third-order functional half-linear dynamic equations

Abstract

In this paper, we study the third-order functional dynamic equation $$ \bigl\{ r_{2}(t)\phi_{\alpha_{2}} \bigl( \bigl[ r_{1}(t) \phi _{\alpha _{1}} \bigl( x^{\Delta}(t) \bigr) \bigr] ^{\Delta} \bigr) \bigr\} ^{\Delta}+q(t)\phi_{\alpha} \bigl( x\bigl(g(t)\bigr) \bigr) =0, $$ on an upper-unbounded time scale $\mathbb{T}$ . We will extend the so-called Hille and Nehari type criteria to third-order dynamic equations on time scales. This work extends and improves some known results in the literature on third-order nonlinear dynamic equations and the results are established for a time scale $\mathbb{T}$ without assuming certain restrictive conditions on $\mathbb{T}$ . Some examples are given to illustrate the main results.