LAUSR

In this paper, we study the third-order functional dynamic equation {r2(t)ϕα2([r1(t)ϕα1(xΔ(t))]Δ)}Δ+q(t)ϕα(x(g(t)))=0,$$ \bigl\{ r_{2}(t)\phi_{\alpha_{2}} \bigl( \bigl[ r_{1}(t) \phi _{\alpha _{1}} \bigl( x^{\Delta}(t) \bigr) \bigr] ^{\Delta} \bigr) \bigr\} ^{\Delta}+q(t)\phi_{\alpha} \bigl( x\bigl(g(t)\bigr) \bigr) =0, $$ on an upper-unbounded time scale T$\mathbb{T}$. 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 T$\mathbb{T}$ without assuming certain restrictive conditions on T$\mathbb{T}$. Some examples are given to illustrate the main results.