**4. Conclusions**

In the current paper, the structure of non-oscillatory solutions for a class of second order dynamic equations on time scales is considered. Under the differentiable assumptions, we first establish two classifications of non-oscillatory solutions. Forthermore, by using the assumptions of superlinear and sublinear function, we obtain four sufficient and necessary conditions for existence of some kinds of non-oscillatory solutions.

**Author Contributions:** All authors have read and approved the final manuscript. Y.Z. finished the manuscript and B.A. and A.A. made the content correction and English language checking.

**Funding:** This research was funded by the National Natural Science Foundation of P.R. China (11671339).

**Conflicts of Interest:** The authors declare no conflict of interest.
