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Русская версия

**A. M. Mahmoud**

Department of Mathematics, Faculty of Science,

New Valley Branch,

Assiut University. New Valley,

El-Khargah 72111, Egypt

In this paper, by using the Mawhin coincidence degree theory and analysis techniques, we establish new results on the existence and uniqueness of T-periodic solutions for a kind of third-order functional differential equation (FDE) with a time-delay. The obtained results are new and complement the related results of third-order (FDE) with a time-delay that have appeared in the literature. In the last section, we give an example to illustrate our main results of periodic solutions.

- Coincidence degree
- Existence and uniqueness
- Mawhin's continuation theorem
- Periodic solutions
- Third-order (FDE)

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