Necessary and sufficient conditions for the oscillatory and asymptotic behaviour of solutions to neutral delay dynamic equations

This article concerns the asymptotic behaviour of solutions to nonlinear first-order neutral delay dynamic equations involving coefficients with opposite signs. We present necessary and sufficient conditions for the solutions to oscillate or to converge to zero. The coefficient associated with the neutral part is considered in three distinct ranges, in one of which the coefficient is allowed to oscillate. Illustrative examples show that the existing results do not apply to these examples and hence they show the significance of our results. The results of this article are also new for the particular choices of the time scale $mathbb{T}=mathbb{R}$ and $mathbb{T}=mathbb{Z}$.