Abstract: In this paper, we are concerned with the existence of periodic solution to a nonautonomous second order boundary value problem on time scales $\mathbb{T}$. By simultaneously utiliting the critical-point theorem and variational methods, we first transform the existence of solutions to the boundary value problem into the critical points of the associated functional by using variational methods. Some new results on the existence of at least one periodic solutions are established by means of the generalized mountain pass lemma. Our results are new even for the corresponding differential, difference equations as well as in general time scales. As an illustration, we verify the obtained results through an example.

Key words: time scale, boundary value problem, periodic solution, critical-point theorem, variational method