Periodic Solutions for Circular Restricted 4-body Problems with Newtonian Potentials

We study the existence of non-collision periodic solutions with Newtonian potentials for the following planar restricted 4-body problems: Assume that the given positive masses $m_{1},m_{2},m_{3}$ in a Lagrange configuration move in circular obits around their center of masses, the sufficiently small mass moves around some body. Using variational minimizing methods, we prove the existence of minimizers for the Lagrangian action on anti-T/2 symmetric loop spaces. Moreover, we prove the minimizers are non-collision periodic solutions with some fixed wingding numbers.