On the Behavior of Solutions of a Third-Order Nonlinear Dynamic Equation on Time Scales

Abstract

We study oscillatory and asymptotic properties of the third-order nonlinear dynamic equation $${{\left[ {{{{\left( {\frac{1}{{{r_2}(t)}}{{{\left( {{{{\left( {\frac{1}{{{r_1}(t)}}{x^{\varDelta }}(t)} \right)}}^{{{\gamma_1}}}}} \right)}}^{\varDelta }}} \right)}}^{{{\gamma_2}}}}} \right]}^{\varDelta }}+f\left( {t,{x^{\sigma }}(t)} \right)=0,\quad t\in \mathbb{T}.$$ By using the Riccati transformation, we present new criteria for the oscillation or certain asymptotic behavior of solutions of this equation. It is supposed that the time scale T is unbounded above.