Necessary and sufficient conditions on asymptotic behaviour of solutions of forced neutral delay dynamic equations (Q923965)

The authors establish some necessary and sufficient conditions on oscillation and asymptotic behavior of solutions of the delay dynamic equation \[ \left[ x\left( t\right) +A\left( t\right) x\left( \alpha\left( t\right) \right) \right] ^{\Delta}+B\left( t\right) F\left( x\left( \beta\left( t\right) \right) \right) -C\left( t\right) G\left( x\left( \gamma\left( t\right) \right) \right) =\varphi\left( t\right) ,\;t\in[ t_{0},\infty)_{\mathbb{T}}, \] where \(\mathbb{T}\) is a time scale unbounded from above. The study depends on the different ranges of the coefficient \(A.\) The theory is next extended to other types of equations They generalize, extend and improve to arbitrary time scales some recent results, which are obtained for the particular time scales \(\mathbb{T}=\mathbb{R}\) (continuous case) and \(\mathbb{T}=\mathbb{Z}\) (discrete case).