Doubly symmetric periodic orbits around one oblate primary in the restricted three-body problem

It is shown that there exists a class of doubly symmetric periodic solutions of Lunar type around one oblate primary in the restricted three-body problem. A small parameter is introduced as the closeness of the infinitesimal body to the oblate primary. The radius of the oblate primary is even smaller compared to the distance from the infinitesimal body to this primary, such that the order of magnitudes of the oblate perturbation and that of the third-body perturbation are comparable. The proof is based on the perturbation techniques and a corollary of Arenstorf's fixed-point theorem, where the error estimates are settled by averaging the first-order system and using the Gronwall's inequality.