Classification and Existence of Nonoscillatory Solutions of the Second-Order Neutral Delay Dynamic Equations on Time Scales

Abstract

In the present paper, we give a classification of nonoscillatory solutions of the second-order neutral delay dynamic equation on time scales $$ \begin{array}{*{20}{c}} {{{{\left[ {x(t)-c(t)x\left( {t-\tau } \right)} \right]}}^{{\varDelta \varDelta }}}+f\left( {t,x\left( {{g_1}(t)} \right),\ldots,x\left( {{g_m}(t)} \right)} \right)=0,} \hfill & {t\in T.} \hfill \end{array} $$ Some existence results for each kind of nonoscillatory solutions are also established.