Nonexistence of periodic solutions and S-asymptotically periodic solutions in fractional difference equations

- Nonexistence of periodic solution is proved for periodic right-hand side.
- Existence of S-asymptotically periodic solutions is discussed.
- Existence and uniqueness results are obtained for S-asymptotically periodic solution.
- Nontrivial estimations of a ratio of gamma functions are used.

In this paper, it is shown that a fractional difference equation with a periodic right-hand side can not possess a periodic solution but it can have an S-asymptotically periodic solution. Sufficient conditions are proved for the existence of a unique S-asymptotically periodic solution. An example is also given illustrating the obtained results.