**5. Conclusions**

Based on Gronwall–Bellman inequality, we have proved the Ulam stability of *n*th order delay integro-differential equations. By applying Lipschitz conditions and the Banach contraction principle, the existence and uniqueness theorem of a solution was given. In addition, the expression of the solution played a great role in the proof of the main theorems. The Ulam stability of the *n*-th order delay integro-differential equation is related to many applications, such as the effect of tire dynamics on vehicle shimmy and optimal control of a size-structured population, and the research in this field is still open. In future work, we recommend that interested scholars extend their work to the Ulam stability of a fractional delay integro-differential equation with a Caputo derivative.

**Author Contributions:** Conceptualization, S.W.; writing—original draft preparation, S.W.; writing—review and editing, S.W. and F.M. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Acknowledgments:** The authors thank the reviewers for their helpful suggestions and comments.

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